Mathematical Documentationof the objects realized in the visualization program

3D-XplorMath
Select the Table Of Contents (TOC) of the desired
kind of objects:
Table Of Contents of Planar Curves
Table Of Contents of Space Curves
Surface Organisation
Go To Platonics
Table of Contents of Conformal Maps
Table Of Contents of Fractals
ODEs
Table Of Contents of Lattice Models
Table Of Contents of Soliton Traveling Waves
Shepard Tones

Homepage of 3D-XPlorMath (3DXM):
http://3d-xplormath.org/
Tutorial movies for using 3DXM:
http://3d-xplormath.org/Movies/index.html
Version November 29, 2020

The Surfaces Are Organized
According To their Construction
Surfaces may appear under several headings:
The Catenoid is an explicitly parametrized, minimal surface of revolution.
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Curvature Properties of Surfaces
Surfaces of Revolution
The Unduloid, a Surface of Constant Mean Curvature
Sphere, with Stereographic and Archimedes’ Projections
TOC of Explicitly Parametrized and Implicit Surfaces
Menu of Nonorientable Surfaces in previous collection
Menu of Implicit Surfaces in previous collection
TOC of Spherical Surfaces (K = 1)
TOC of Pseudospherical Surfaces (K = −1)
TOC of Minimal Surfaces (H = 0)
Ward Solitons
Anand-Ward Solitons
Voxel Clouds of Electron Densities of Hydrogen

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Planar Curves
Go To Page 1(Click the Names)
CircleEllipseParabola
Hyperbola
Conic Sections
Kepler Orbits, explaining 1/r-Potential
Nephroid of Freeth
Sine Curve
Pendulum ODE Function
Lissajous Plane Curve
Catenary
Convex Curves from Support Function
Tractrix
Cissoid and Strophoid
Conchoid
Lemniscate
Clothoid
Archimedean Spiral
Logarithmic Spiral
Cycloid
Epi- and Hypocycloids
Cardioid and Limacon
Astroid
Continue TOC next page

Deltoid
Nephroid
Ancient Method of Construction:
Mechanically Generated Curves
Seven Cubic Curves, five with Addition:
Cubic Polynomial, Cuspidal Cubic,
Connected Rational Cubic, Rational Cubic with Poles,
Elliptic Cubic; Folium, Nodal Cubic
Elliptic Functions, parametrizing Elliptic Curves
Geometric Addition on Cubic Curves
Folium
Implicit Planar Curves, highly singular examples:
Tacnodal Quartic, Teissier Sextic
Cassinian Ovals, an implicit family
Userdefined Curves, explicitly parametrized:
User Cartesian, User Polar, User Graphimplicit: User Implicit Curves
User Curves by Curvature
Graphs of Planar Curves
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The Circle *x = aa cos(t), y = aa sin(t), 0 ≤ t ≤ 2π
3DXM - suggestion: Select from the Action Menu Show

Generalized Cycloid and vary in the Settings Menu, entry:
Set Parameters, the (integer) ratio between the radius aa
and the rolling radius hh.
The length of the drawing stick is ii∗rolling radius.
The circle is the simplest and best known closed curve inthe plane. The default image shows the circle together
with the theorem of Thales about right angled triangles.
Other properties of the circle are also known since over2000 years. In fact, many of the plane curves that haveindividual names were already considered (and named) by
the ancient Greeks, and a large class of these can be obtained by rolling one circle on the inside or the outside ofsome other circle. The Greeks were interested in rolling
constructions because it was their main tool for describing the motions of the planets (Ptolemy). The following
curves from the Plane Curve menu can be obtained byrolling constructions:
Cycloid, Ellipse, Astroid, Deltoid, Cardioid, Limaçon,
Nephroid, Epi- and Hypocycloids.
Not all geometric properties of these curves follow easily* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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from their definition as rolling curve, but in some cases theconnection with complex functions (Conformal Category)does.
Cycloids arise by rolling a circle on a straight line. The
parametric equations code for such a cycloid is
P.x := aa · t − bb sin(t)P.y := aa − bb sin(t), aa = bb.
Cycloids have other cycloids of the same size as evolute(Action Menu: “Show Osculating Circles with Normals”).
This fact is responsible for Huyghen’s cycloid pendulum tohave a period independent of the amplitude of the oscillation.
Ellipses are obtained if inside a circle of radius aa anothercircle of radius r = hh = 0.5aa rolls and then traces a curvewith a radial stick of length R = ii · r. The parametricequations for such an ellipse is
P.x := (R + r) cos(t)P.y := (R − r) sin(t).
In the visualization of the complex map z → z + 1/z in
Polar Coordinates the image of the circle of Radius R issuch an ellipse with r = 1/R.
Astroids are obtained if inside a circle of radius aa another circle of radius r = hh = 0.25aa rolls and then traces
a curve with a radial stick of length R = ii · r = r. Parametric equations for such Astroids areP.x := (aa − r) cos(t) + R cos(4t)P.y := (aa − r) sin(t) − R sin(4t).
Astroids can also be obtained by rolling the larger circle ofradius r = hh = 0.75aa (put gg = 0 in this case). Anothergeometric construction of the Astroids uses the fact thatthe length of the segment of each tangent between the xaxis and the y-axis has constant length. — Try hh :=aa/3 to obtain a Deltoid.
Cardioids and Limaçons are obtained if outside a circleof radius aa another circle of radius r = hh = −aa rolls
and then traces a curve with a radial stick of length R =ii · r, ii = 1 for the Cardioids, ii > 1 for the Limaçons.
Parametric equations for Cardioids and Limaçons are
P.x := (aa + r) cos(t) + R cos(2t)P.y := (aa + r) sin(t) + R sin(2t).
The Cardioids and Limaçons can also be obtained by rolling
the larger circle of radius r = hh = +2aa; now ii < 1 forthe Limaçons. Note that the fixed circle is inside the larger
rolling circle.
The evolute of the Cardioid (Action Menu: Show Osculating Circles with Normals) is a smaller Cardioid. The imageof the unit circle unter the complex map z → w = (z 2 +2z)
is a Cardioid; images of larger circles are Limaçons. Inverses z → 1/w(z) of Limaçons are figure-eight shaped,
one of them is a Lemniscate.
Nephroids are generated by rolling a circle of one ra-dius outside of a second circle of twice the radius, as the
program demonstrates. With R = 3r we thus have theparametrization
P.x := R cos(t) + r cos(3t)P.y := R sin(t) + r sin(3t).
As with Cardioids and Limaçons one can also make theradius for the drawing stick shorter or longer: After selecting Circle set the parameters aa = 1, hh = −0.5, ii = 1 for
the Nephroid and ii > 1 for its looping relatives. – Pick in
the Action Menu: Show Osculating Circles with Normals.
The Normals envelope a smaller Nephroid.
The complex map z → z 3 + 3z maps the unit circle to sucha Nephroid. To see this, in the Conformal Map Category,select z → z ee + ee · z from the Conformal Map Menu,
then choose Set Parameters from the Settings Menu andput ee = 3.
Archimedes’ Angle Trisection. A demo of this construction can be selected from the Action Menu.
Circle Involute Gear. Another demo from the Action
Menu. Involute Gear is used for heavy machinery because of the following two advantages: If one wheel rotateswith constant angular velocity then so does the other, thus
avoiding vibrations. If the teeth become thiner by usage,
the axes can be moved closer to each other.
H.K.

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The Ellipse *x(t) = aa cos(t), y(t) = bb sin(t), 0 ≤ t ≤ 2π
3DXM-suggestion:

Select in the Action Menu: Show Osculating Circles with
Normals. In the Animate Menu try the default Morph.
For related curves see: Parabola, Hyperbola, Conic Sections and their ATOs.
The Ellipse is shown together with the so called Leitkreis
construction of the curve and its tangent, see below. This
construction assumes that the constants aa and bb are positive. The larger of the two is called the semi-major axis
length, the smaller one is the semi-minor axis length.
The Ellipse is also the set of points satisfying the following
implicit equation: (x/aa)2 + (y/bb)2 = 1.
A geometric definition of the Ellipse, that can be used toshape flower beds is:
An Ellipse is the set of points for which the sumof the distances from two focal points is a constant L equal to twice the semi-major axis length.
A gardener connects the two focal points by a cord oflength L, pulls the cord tight with a stick which then draws
the boundary of the flower bed with the stick. Anotherversion of this definition is:* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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An Ellipse is the set of points which have equal
distance from a circle of radius L and a (focal)
point inside the circle.
Both these definitions are illustrated in the program.
The normal to an ellipse at any point bisects the angle
made by the two lines joining that point to the foci. Thissays that rays coming out of one focal point are reflected
off the ellipse towards the other focal point. Therefore one
can build elliptically shaped “whispering galleries”, wherea word spoken softly at one focal point can be heard only
close to the other focal point.
To add a simple proof we show that the tangent leaves theellipse on one side; more precisely, we show that for every
other point on the tangent the sum of the distances to thetwo focal points F1 , F2 is more than the length L of the
major axis. (In the display: F = F2 .) Pick any point Q
on the tangent, join it to the two focal points and reflect
the segment QF in the tangent, giving another segment
QS. Now F1 QS is a radial straight segment only if Q is
the point of tangency—otherwise F1 QS is by the triangle
inequality longer than the radius F1 S (of length L) of thecircle around F1 .
The evolute of an ellipse, i.e., the curve enveloped by thenormals of the ellipse—see Action Menu: Draw osculating
circles with normals, is a generalized Astroid, it is lesssymmetric than the true Astroid.
An Ellipse can also be obtained by a rolling construction:

Inside a circle of radius aa another circle of radius r :=hh = 0.5aa rolls and traces the Ellipse with a stick ofradius R := ii · r, see Plane Curves Menu: Circle andselect from the Action Menu: Show Generalized Cycloids.
The parametric equation resulting from this constructionis:x(t) = (R + r) cos(t)y(t) = (R − r) sin(t)This is related to the visualization of the complex map
z → z + 1/z in Polar Coordinates, the image of the circleof radius R is such an ellipse with r = 1/R.
Such rolling constructions are reached with the Plane Curves
Menu entry: Circle and then the Action Menu Draw Generalized Cycloids or with Epi- and Hypocycloids. Recall
that negative values of the rolling radius hh gives curves
on the outside, positive radii (hh < aa) on the inside of
the fixed circle.
Other rolling curves are:
Cycloid, Astroid, Deltoid, Cardioid, Limacon,
Nephroid, Epi- and Hypocycloids.
H.K.

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Parabola *See also: Ellipse, Hyperbola, Conic Section and their ATOs,
and in the Category Surfaces see: Conic Sections and Dandelin Spheres
The usual parametric equations for the Parabola arex(t) := t2 /4p
y(t) := t,where p = aa/4,
so the Parabolavisualizes the graphs of the two functions√
y(x) := 4p · x and x(y) := y 2 /4p.
The vertical line x = −p is called the directrix and thepoint (x, y) = (p, 0) is called focal√point of the Parabola.
The distance from a point (x, y = 4p · x) on the Parabola
to the directrix √is (x + p), and this is the same distanceas from (x, y = 4p · x) to the focal point (p, 0), because(x − p)2 + y 2 = (x + p)2 .
The point (p, 0) is called ”focal point”, because light rays
which come in parallel to the x-axis are reflected off the
Parabola so that they continue to the focal point. This
fact is illustrated in the program. It gives the following
ruler construction of the Parabola:
Prepare the construction by drawing x-axis, y-axis, directrix and focal point F. Then draw any line parallel to thex-axis and intersect it with the directrix in a point S. Theline orthogonal to the connection SF and through its mid* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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point is the tangent of the Parabola and intersects therefore the incoming ray in the point of the Parabola which
we wanted to find.
The same construction works for Ellipse and Hyperbola, if
the directrix is replaced by a circle of radius 2*a around
one focal point. The curve is the set of points which havethe same distance from this circle and the other focal point.
The Action Menu of the Parabola has an entry “Show
Normals Through Mouse Point”. This illustrates an unexpected property of the Parabola. One may already be
surprised that at the intersection points of normals always
three normals meet. We know no other curve which is accompanied by such a net of normals. The surprise should
increase if one looks at the y-coordinates of the parabolapoints from where three such intersecting normals originate: these y-coordinates add up to 0! In other words, theintersection behaviour of the normals reflects the additionon the y-axis.
The explanation of where this intersection property comesfrom is quite interesting. The normals of the Parabola are
the tangents to its evolute, the semi-cubical parabola, asingular cubic curve (see Cuspidal Cubic). So the intersection property of the parabola normals can be thought ofas defining an addition law for the evolute, and as such it
is a simpler limiting case of the addition law that exists on
any cubic curve.
R.S.P.
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Hyperbola *See also Parabola, Ellipse, Conic Sections and their ATOs.
The most common parametric equations for a Hyperbola
with semi-axes aa and bb are:x(t) = ±aa cosh(t), y(t) = bb sinh(t),t ∈ R;

and another version is:x(t) = aa/ cos(t), y(t) = bb sin(t)/ cos(t), t ∈ [0, 2π].
The corresponding implicit equation is:(x/aa)2 − (y/bb)2 = 1.
The function graphs: {(x, y); y = 1/x + m · x} are also
Hyperbolae.
A geometric definition of the Hyperbola is:
A Hyperbola is the set of points for which the difference of the distance from two focal points
is constant.
Or:
A Hyperbola is the set of points which have thesame distance from a circle and a (focal) point
outside that circle.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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If one applies an inversion (x, y) → (x, y)/(x2 + y 2 ) to aright Hyperbola (i.e. aa = bb) then one obtains a Lemniscate.
In the visualization of the complex map z → z + 1/z in
Polar Coordinates, the image of the radial lines are theHyperbolae:x(R) = (R + 1/R) cos φy(R) = (R − 1/R) sin φ,

R ∈ R.

And the image of √the standard Cartesian Grid under thecomplex map z → z is a grid of two families of orthogonalHyperbolae.
H.K.

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Conic Sections*See also Parabola, Ellipse, Hyperbola and their ATOs.
A cone of revolution (e.g., {(x, y, z); x2 +y 2 = m·z 2 }) is oneof the simplest surfaces. Its intersections with planes arecalled conic sections. Apart from pairs of lines these conic
sections are Parabolae, Ellipses or Hyperbolae. These curves
have also other geometric definitions (e.g., The locus ofpoints having the same distance from a focal point and acircle). See their Menu entries.
On the other hand, they are also more robust than thesedefinitions show: Photographic images of conic sections
are again conic sections; or in a completely different formulation: The intersection of a plane and any “quadraticcone”, i.e.,{(x, y, z) | a · x2 + b · y 2 + c · z 2 + d · xy + e · yz = 0},
is not more complicated than planar sections of circularcones but are the same old Parabolae, Ellipses or Hyperbolae as above. A special case of this robustness is the factthat orthogonal projections of conic sections in 3-space areagain conic sections. This is illustrated in the program as
follows:
Interpret the illustration as if it showed level lines on a hiking map. The equidist parallel lines are the level lines of a
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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sloping plane; the smaller the distance between these levellines the steeper the plane. The equidistant concentric circles are the level lines of a circular cone, as for example anant lion would dig in sandy ground; without height numbers written next to the level lines we can of course notdecide whether the circular level lines represent a conicalmountain or a conical hole in the ground. We suggest thatthe blue level line and the vertex of the cone are at heightzero and the other levels are higher up so that the cone isa hole.
The intersection curve between plane and cone has then aneasy pointwise construction: Simply intersect level lines ofthe same height on the two surfaces. (These are lines withthe same color in the program illustration.) This construction reveals a new geometric property of the intersection
curve on the map, of this conic section:
Take the ratio of the distances from a point on the
curve, (i) to the level line at height 0 of the plane(called directrix) and (ii) to the vertex at heightzero of the cone (called focus). This ratio is thesame as the ratio of adjacent level lines of planeand cone and therefore the same for all points ofthis conic section.
H.K.

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Conic Sections, Kepler orbits *See also Parabola, Ellipse, Hyperbola and their ATOs.
For many properties of the conic sections a parametrization is not relevant. However, when Kepler discovered thatplanets and comets travel on conic sections around the sunthen this discovery came with a companion: the speed on
the orbit is such that angular momentum is preserved. In
more elementary terms: the radial connection from the sun
to the planet sweeps out equal areas in equal times. Withthe 3dfs demo we explain geometrically how this celestial
parametrization is connected with the focal properties ofconic sections. Here we give the algebraic explanation first.
An ellipse, parametrized as affine image of a circle andtranslated to the left is
P (ϕ) := (a cos ϕ − e, b sin ϕ).
√

If we choose e := a2 − b2 then we have |P (ϕ)| = (a −e cos ϕ). This gives the connection with the oldest definition of an ellipse: The sum of the distances from P (ϕ) to
the two points (±e, 0) is 2a.
Next we compute the quantity A, equal to twice the area
swept out by the position vector P , and also the derivatives* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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of P and A:
Z ϕA(ϕ) =0det(P (ϕ), P 0 (ϕ))dϕ,0

P (ϕ) = (−a sin ϕ, b cos ϕ),
A0 (ϕ) = b(a − e cos ϕ),
and we denote the function inverse to A(ϕ) by Φ(A), sothat,
Φ(A(ϕ)) = ϕ,1.
Φ (A) =b(a − e cos Φ)0Let us write Q to denote the position when expressed as afunction of A, i.e., Q(A) := P (Φ(A)). Now Kepler’s Second Law says that A proportional to time, or equivalently
that A is the time in approriate units, so the velocity is
Q0 (A) = P 0 (Φ(A)) · Φ0 (A), and the kinetic energy is:
2
2
2
2
1 0
a
sin
ϕ
+
b
sin
ϕ
2
K.E. = Q (A) =22b2 (a − e cos ϕ)2a2 − e2 cos2 ϕ= 2
2b (a − e cos ϕ)2(a + e cos ϕ)= 2
2b (a − e cos ϕ)
a
1
1
= 2·
− 2.
b a − e cos ϕ 2b
a
1
1
= 2·
−
.
b |P (ϕ)| 2b2

Thus, in units where we take twice the swept out area as
the time, the potential energy can be read off by using thelaw of energy consevation, i.e., the fact that the kinetic
energy plus the potential energy is constant. In fact, it
follows from this that the potential energy at orbit point
Q(A(ϕ)) = P (ϕ) is equal to:
a
1
− 2·
,
b |P (ϕ)|
which is the famous 1/r law for the potential energy.
Next, we present a geometric proof. The starting point isthe determination of the correct orbital speed by the property that the product of the speed |v| with the distance pof the tangent line from the center is the constant angular momentum, Kepler’s second law. Of course we canillustrate such a fact only if we also represent the size of
velocities by the length of segments and we have to keep inmind that segments which illustrate a length and segments
which illustrate a velocity are interpretated with different
units.
Recall the following theorem about circles: if two secantsof a circle intersect then the product of the subsegments of
one secant ist the same as the product of the subsegmentsof the other secant.
This will be applied to the circle the radius of which is thelength 2a of the major axis. (The midpoint is the otherfocus, not the sun.) The two secants intersect in the focus representing the sun: one secant is an extension of themajor axis the other is perpendicular to the tangent line.
The subsegments of the first secant have the lengths 2a−2eand 2a+2e, where 2e is the distance between the foci. The
subsegments of the second secant have one length 2p andone labeled |v|.

p r
r

2a
|v|

Kepler Ellipse with constructionof proper speed and potential.
The circle theorem says: (2a − 2e) · (2a + 2e) = 2p · |v|.
Since the left side is constant we can interprete the segment
labeled |v| as representing the correct orbital speed.
Now that we know at each point of the orbit the correct
speed we can deduce Newton’s 1/r-law for the gravitational potential, if we use kinetic energy plus potential energy equals constant total energ. In the illustration wehave two similar right triangles, the small one has hypothenuse = r and one other side = p, the big one has
as hypothenuse a circle diameter of length 4a and the corresponding other side has length 2p + |v|. Now we use theabove const := (2a − 2e) · (2a + 2e) = 2p · |v| to eliminate
p from the proportion:p : r = (2p + |v|) : 4aThis gives
2a/r = 1 + |v|/2p = 1 + v 2 /const.
Up to physical constants (units), v 2 is the kinetic energy,
so that (again up to units) −1/r is the potential energy –since such a potential makes kinetic plus potential energy
constant.
Another simple property of Kepler ellipses and hyperbolas is: Their velocity diagram, the so called hodograph, isa circle. Usually one simply translates the velocity vector from the orbit point to the sun. In our picture we
see the velocity vector rotated by 90 degrees; indeed, it
ends on the circle. This leads to a geometric representation of the Runge-Lenz vector: In our picture we really see
the cross product of the (tangential) velocity vector withangular momentum (a constant vector orthogonal to theorbit plane). If we add to it a vector of constant length
2a and parallel to the position vector then we reach the
midpoint of our circle, the other focal point of the orbit

ellipse. This sum vector is, up to the constant negative
factor −(a − e)/2e the classical Runge-Lenz vector.
Mathematically, the parabolic and hyperbolic Kepler orbits
allow similar derivations of the −1/r-potential, which wewill give next. Historically this played no role since thenon-repeating orbits could not be determined with enoughprecision at the time.
Derivation of the −1/r-potential from a parabolic Kepler
orbit. Let in the picture (below) |p| be the distance fromthe sun at (1/4, 0) to a tangent of a parabolic Kepler orbit
and let |v| be the orbital speed at that moment. Conservation of angular momentum says p · |v| = const. Let ϕ be
the angle between the segment marked p and the verticalaxis; since the sun is at the focal point of the parabola
we have p · sin ϕ = 1/4. This and the previous angularmomentum equation say that, up to a choice of unit forvelocity, we have:
Kepler speed: |v| = sin ϕ,
Angular momentum: p · |v| = 1/4.
If we call r the distance to the planet, than we also have
p/r = sin ϕ = |v|. Multiplication with the angular momentum gives
1
1 2|v| = ,
2
8r
−1
Potential energy:.
8r

Kinetic energy:

2Planet = (a ,a)rp
Sun = (1/4,0)Derivation of the −1/r-potential from a hyperbolic Kepler
orbit. As before we call p the distance from the sun toa tangent of the hyperbolic orbit and v the speed at thatorbit point.

Tv - 2pPlanet = (a*cosh x, b*sinh x)pp(-e-2a,0)(-e,0)r
Sun = (e,0)

Conservation of angular momentum says p · |v| = const.
We use the property of the circle (radius 2a) about prod-ucts of segments on secants (which intersect at the sun S):
2p · |T − S| = (2e − 2a)(2e + 2a) = 4b2 .
Therefore, again up to the unit for velocity, we have identified the correct
Kepler velocity: v = |S − T |.
Finally, similar triangles give:p/r = (v − 2p)/4a or 4a/r = v/p − 2,and elimination of p with the angular momentum, i.e. with
1/p = v/2b2 , shows that kinetic energy plus a radial function are constant – thus identifying the 1/r-potential:
4ab2 /r = v 2 /2 − 2b2 .
Additional properties: As in the case of elliptical orbits
we see that the hodograph is a circle because the velocity
vector, rotated by 90 degrees, ends on the circle which weused for the construction of the hyperbola. And if we add
to the endpoint of this rotated velocity a vector parallelto the position vector and of constant length 2a then wereach the midpoint of the circle, the other focal point of theorbit. The constant(!) difference vector between the two
focal points, the geometric Runge-Lenz vector, differs fromthe common definition by the constant factor (e − a)/2e.
H.K.

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Nephroid of Freeth *This curve, first described 1879, is the member aa = 0 in
the following family of curves:
x(t) = (1 − aa · sin(t/2)) cos(t)y(t) = (1 − aa · sin(t/2)) sin(t)The default morph starts at aa = 0 with a circle, traversed
twice. For small aa > 0 one double point develops. At
aa = 1 the curve reaches the origin with a cusp.√ Thiscusp deforms into a second double point. At aa = 2 thetwo tangents of the double point coincide and are vertical.
This point of double tangency deforms into three double
points. The Nephroid of Freeth is reached at aa = 2, when
two of the mentioned three double points coincide with theearliest one to form a triple intersection.
Apart from being in a simple family, which shows all thesesingularities of curves, we learnt fromhttp://www.2dcurves.com/derived/strophoid.html
that the Nephroid of Freeth has the curious property that
one can construct a regular sevengon with it: The verticaltangent at the triple intersection meets the curve again in
two points whose radius vectors enclose the angle 3π/7.
H.K.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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Sine *The demo in 3D-XplorMath illustrates: If a unit circle
in the plane is traversed with constant velocity then it isparametrized with the so-called trigonometric or circularfunctions,c(t) = (cos t, sin t).

History
The earliest known computations with angles did not yet
use degrees. In Babylonian astronomy angles were measured by their chord, i.e. by the base of an isocele triangle
with the vertex in the center of a unit circle and the othervertices on the circumference. Hipparchus (180 - 125 BC)
is credited with the first table giving chords in terms of degrees. The table proceeded in steps of 60◦ /16. Ptolemy (ca90 - ca 168 AD) computed more accurate tables in stepsof 1◦ . With these tables trigonometric functions had come* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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into existence, in modern notation the function
2π ϕchord: ϕ 7→ 2 sin( 360
2 ).
Astronomy needed computations with spherical triangles.
Initially these were done in geometric form, based on theorems on quadrilaterals inscribed in circles. Around 800 AD
Arab astronomers developed the more streamlined computations based on properties of the functions sin and cos,
expressed in formulas similar to the ones in use today.
Since Newton’s time sin and cos are even more conveniently defined by their linear differential equations
sin00 = − sin, sin(0) = 0, sin0 (0) = 1;
cos00 = − cos, cos(0) = 1, cos0 (0) = 0.
The first derivatives sin0 = cos, cos0 = − sin and addition
theorems likesin(a + x) = sin(a) cos(x) + cos(a) sin(x),cos(a + x) = cos(a) cos(x) − sin(a) sin(x),follow from the uniqueness theorems for such differential
equations (both sides satisfy the same ODE and the same
initial conditions). Also the identity
sin2 (x) + cos2 (x) = sin2 (0) + cos2 (0) = 1
which is needed to show “that sin and cos, defined by theirdifferential equations, indeed parametrize the unit circle
with unit velocity”, follows directly by differentiation.

With Euler’s discovery of the close connection between the
exponential and the trigonometric functions computations
became even more convenient, since one only needs
exp0 = exp, exp(0) = 1, exp(z + w) = exp(z) exp(w)exp(x + i y) = exp(x) · (cos(y) + i sin(y)).
In the 19th century approximationof functions by Fourier
PNpolynomials PN (x) := a0 + n=1 an cos(n x) + bn sin(n x)lead to unexpected difficulties and corresponding deep insights. The modern notion of a function and also the concept of continuity have their roots in these studies.
Today, when high school kids first meet sine and cosine,
they do not meet them as real valued functions, but asratios of edge lengths in rectangular triangles. There is noreference to a numerical algorithm that computes the values of sine and cosine from “angles”. The above addition
theorems emphasize this: if (c1 , s1 ), (c2 , s2 ) ∈ S1 , then also
(c1 c2 − s1 s2 , s1 c2 + c1 s2 ) ∈ S1 . Note that this addition ofrational points gives again a rational point, and rational
points on S1 are the same as Pythagorean triples, namely(a/b)2 + (c/d)2 = 1 is equivalent to (ad)2 + (bc)2 = (bd)2 .
On this level sine and cosine belong to the geometry ofsimilar triangles, and indeed, the addition formulas can beproved by using similarity of triangles. – So, what is therelation between sine, cosine and angles? One answer is todefine sine, cosine in terms of their ODEs quoted above,
another is, to discuss first the arc length of the circle, whichmeans, define the inverse functions arcsine, arccosine first.

We come back to computations of sine and cosine below,
but first we return to the beginning. The following formulas show (in modern notation) how the trigonometricfunctions enter planar and spherical geometry.
Laws of Sines and Cosines
For any planar triangle with side lengths a, b and c, whose
opposite angles are α, β and γ respectively, we have:
Projection theorem:

c = a · cos β + b · cos αSine theorem:b · sin α = hc = a · sin β

Cosine theorem:

c2 = a2 + b2 − 2ab cos γ

In astronomy the corresponding formulas for spherical triangles on the unit sphere played a much more importantrole. They are:Projection theorem:

cos a sin c =sin a cos c cos β + sin b cos αSine theorem:

sin b · sin α = sin a · sin β

Cosine thm:cos c = cos a cos b + sin a sin b cos γ.

A consequence of the first two theorems is theAngle Cosine thm: cos γ = − cos α cos β+sin α sin β cos c.
For triangles with small edge lengths a, b, c the sphericalformulas become the planar ones if all terms that are atleast cubic in the edge lengths are ignored, i.e.
sin a ≈ a, cos a ≈ 1 − a2 /2, sin a cos c ≈ a.

Trivia
Orthogonal Projections of the Helix

The sinusoid is an orthogonal projection of the helix spacecurve. In 3DXM, a helix can be seen in the Space Curves
category and, independently, via the Action Menu entry
Show Planar Curve As Graph, after selecting Circle.

The sinusoid is the development of an obliquely cut rightcircular cylinder—i.e., the edge of the cylinder rolls out toa sinusoid.
Numerical Computations
From the ODEs one immediately gets the Taylor series(limN →∞ below) which are convenient to compute sine andcosine for small arguments (so that small N are sufficient):
N
N
X
X(−1)k 2k+1(−1)k 2k
sin x ≈x
, cos x ≈x .
(2k + 1)!
(2k)!
k=0

k=0

Numerical efforts can be greatly reduced with the simple

angle tripling formulas
P3 (y) : = 3y − 4y 3 ,sin(3x) = 3 sin(x) − 4 sin3 (x) = P3 (sin x),cos(3x) = −3 cos(x) + 4 cos3 (x) = −P3 (cos x).
Accuracy, already with N = 2, is surprising:
0 = sin π ≈ P3 (P3 (P3 (T aylor5 (π/27)))) ≈ −1.6 · 10−9 .
Since sin x ≈ x is a very good approximation for small x we
getlimn→∞ P3◦n (x/3n ) = sin(x).
Or, much faster: limn→∞ P3◦n (T aylor5 (x/3n )) = sin(x).
Since sin0 (π) = −1, one can compute π with the Newton
iteration
xn+1 := xn + sin(xn ), x0 = 3.
This can also be seen as a consequence of arcsin(x) ≈ x
for small x and π/2 < x < π ⇒ π = x + arcsin(sin(x)).
Then, using the better approximation:
arcsin(x) ≈ x + 61 x3 ,one gets the really fast iteration
xn+1 := F (xn ) := xn + sin(xn ) + 16 sin(xn )3 .
Example: F (2.5) = 3.1342, F (F (2.5)) = 3.14159265358.
When Archimedes estimated π, he used the inverse of theangle doubling formulas to compute the length of a regularinscribed 2n-gon from the length of a regular n-gon, namelypxxxcos( ) = 0.5(1 + cos(x)), sin( ) = 0.5 sin(x)/ cos( ).
222
XL, HK.

Go To Planar TOC

Pendulum ODE function *Although one may feel that the sine function is an explicit
function, it is really defined via the ODE
sin00 (x) = − sin(x), with sin(0) = 0, sin0 (0) = 1.
The solutions of the ODE of the mathematical pendulum
f 00 (x) = − sin(f (x))
was also studied a lot, but did not reach the popularity ofthe sine function. In 3D-XplorMath we use parameters as
follows:−cc
f 00 (x) =· sin(aa f (x)), f (0) = hh, f 0 (0) = bb.
aa
For better comparison with the sine function, aa = 0 isallowed. The size of cc changes the frequency, as with thesine function. In addition observe that, for aa =/ 0. theperiod becomes longer when the amplitude grows. As withthe sine function one has an “energy integral”:
f 0 (x)2cc
f 0 (0)2cc− 2 cos(aa · f (x)) =− 2 cos(aa · f (0)).
2
aa
2
aa
With aa = 1, cc = −1 we obtain solutions of the SineGordon equation qxt = sin(q), namely q(x, t) := f (x + t).
For f (0) = π, f 0 (0) = 2 we get f (u) = 4 arctan(exp(u)).
H.K.
Go To Planar TOC* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Go To Planar TOC

Lissajous Plane Curve *The Lissajous curves show the orbits of two orthogonalharmonic undamped oscillators.

Lissajous(t) :=aa · sin(ee · t + cc)
bb · sin(dd · t),

Default values: dd = 3, ee = 5, cc = 0.
If the parameters dd, ee are integers then the curves areclosed. Actually, a rational ratio is sufficient.
The default morph varies the phase cc, which changes thecurves a lot.
These planar curves have obvious analogues in R3 .
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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Catenary *The catenary is also known as the chainette, alysoid, andhyperbolic cosine. It is defined as the graph of the function
y = a cosh(x/a). (Recall cosh(x) := (ex + e−x )/2, wheree = 2.71828 . . . is the base of the natural logarithms.)

The Catenary
The catenary is the shape an ideal string takes when hanging between two points. By “ideal” is meant that the stringis perferctly flexible and inextensible, has no thickness, isof uniform density. In other words the catenary is a mathematical abstraction of the shape of a hanging string, and it* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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closely approximates the shapes of most hanging string-likeobjects we see, such as ropes, outdoor telecommunication
wires, necklaces, chains, etc. For any particular hanging
string, we will need to choose the parameter a correctly tomodel that string.
Notice that except for scaling there is really only a single catenary. That is, the scaling transformation (x, y) 7→(ax, ay) maps the graph of y = a cosh(x/a) onto the graphof y = cosh(x). The scaling transformation just amounts
to a change in the choice of units used to measure distances.
History
Galileo was the first to investigate the catenary, but hemistook it for a parabola. James Bernoulli in 1691 obtained its true form and gave some of its properties. [cf.,Robert C. Yates, 1952]
Galileo’s suggestion that a heavy rope would hang in the
shape of a parabola was disproved by Jungius in 1669, but
the true shape of the catenary, was not found until 1690–
91, when Huygens, Leibniz and John Bernoulli replied to achallenge by James Bernoulli. David Gregory, the Oxford
professor, wrote a comprehensive treatise on the ‘catenarian’ in 1697. The name was first used by Huygens in a
letter to Leibniz in 1690. [cf., E.H.Lockwood, 1961].
[By the way, it is true that if you carefully weight a hanging
string so that their is equal weight of string per unit ofhorizontal distance (rather than per unit of length) thenits shape will be a parabola, so Gallileo wasn’t so far fromthe truth.]
The Catenary has numerous interesting properties.
Properties of the Catenary
Caustics

Parallel rays above the exponential curve
The Catacaustic of the exponential curve (x, ex ) with lightrays from above and parallel to the y axes is the catenary.
The exponential function ex has interesting properties it-self. It is the only function who agrees with its derivative.
Involute

The involute of catenary starting at the vertex is the curve
Tractrix . (In 3DXM, the involutes of a curve can beshown in the menu Action → Show Involutes.) Note that
all involutes are parallel curves of each other. This is atheorem.

Evolute

The evolute of the catenary is also the tractrix. (In 3DXM,
this can be seen from the menu Action → Show Osculating
Circles.)Radial and Kampyle of Eudoxus

The radial of the catenary is the Kampyle of Eudoxus. In
the figure above, the blue curve is half the catenary. The
green curve is the Kampyle of Eudoxus. The rainbow lines
are radii of osculating circles and their parallels through 0.
The Kampyle of Eudoxus is defined as the parametric curve
x = − cosh(t) sinh(t), y = cosh(t).

Catenoid

If you rotate the graph of x = cosh(y) about the y-axis,
the resulting surface of revolution is a minimal surface,called the Catenoid. It is one endpoint of an interesting
morph you can see in 3DXM, by switching to the Surface category, choosing Helicoid-Catenoid from the Surface
menu, and then choosing Morph from the Animate menu.
If you look closely you will see that during this morph distances and angles on the surface are preserved. See About
This Object... in the Documentation menu when HelicoidCatenoid is selected for a discussion of this.
XL.

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On Curves Given By Their Support Function *This note is about smooth, closed, convex curves in theplane and how to define them in terms of their so-called
Minkowski support function h. For quick reference we firstshow how, in 3D-XplorMath, h can be modified by specifying parameters. Then we begin with a more general classof geometric objects, namely convex bodies.
1. Parameter Dependent Formulas
In 3D-XplorMath, the support function h is given in termsof Fourier summands:h(ϕ) := aa + bb cos(ϕ) + cc cos(2ϕ)+
dd cos(3ϕ) + ee cos(4ϕ) + ff cos(5ϕ).
In terms of this function we define the following curve:cos(ϕ)− sin(ϕ)c(ϕ) := h(ϕ) ·+ h0 (ϕ) ·.
sin(ϕ)+ cos(ϕ)
Differentiation shows that c is given in terms of its unit
normal and tangent vectors and the function h:− sin(ϕ)
c0 (ϕ) = (h + h00 )(ϕ) ·.
+ cos(ϕ)* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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One obtains curves with nonsingular parametrization (|c0 | >0) if aa is choosen large enough. And since h(ϕ) equals thescalar product between c(ϕ) and the unit normal n(ϕ) =(cos(ϕ), sin(ϕ)) one has a simple geometric interpretation:h(ϕ) is the distance of the tangent at c(ϕ) from the origin.
2. Background And Explanations
A convex body in Rn is a compact subset B having nonempty interior and such that it includes the line segmentjoining any two of its points. A hyperplane H in Rn iscalled a supporting hyperplane of B if it contains a pointof B and if B is included in one of the two halfspaces definedby H. It is not difficult to show that every boundary pointof B lies on at least one supporting hyperplane, and that
B is the intersection of all such halfspaces.
A smooth, closed, planar curves c is called convex if its
tangent at each point intersects c only at that one point.
The complement in R2 of such a curve has a single bounded
component, the interior of the curve, and one unbounded
component, its exterior . The curve is the boundary of its
interior, and we denote by B the curve together with its
interior. It is easy to see that B is a convex body in R2 , asdefined above, and in fact the tangent line at any point ofc is the unique supporting hyperplane (= line!) containing that point. (There are of course more general planar
convex bodies. For example if P is a closed polygon in
R2 together with its interior, then P is a convex body, butthere are infinitely many supporting lines through each ver-tex, while the supporting line containing an edge contains
infinitely many points.)
Now let O be some interior point of c and take O as theorigin of a cartesian coordinates by fixing a ray from O
as the positive x-axis. With respect to these coordinates,
at each point p on c the outward directed unit normal
at p will have the form n(ϕ) = (cos(ϕ), sin(ϕ)) whereϕ = ϕ(p) satisfies 0 ≤ φ ≤ 2π. If we as usual think of
S1 as the interval [0, 2π] with endpoints identified, then it
can be shown that the map p 7→ ϕ(p) is a smooth oneto-one map of c with S1 , so that the inverse map gives aparametrization c(ϕ) of the curve by S1 . (This just saysthat given any direction in the plane, there is a uniquepoint p on c where the outward normal has that direction,and the point p varies smoothly with the direction.)
The Minkowski support function for the curve c is the function h defined on S1 by letting h(ϕ) be the distance fromthe origin of the line of support (or tangent) through c(ϕ),
that is h(ϕ) := n(ϕ) · c(ϕ), the scalar product of c(ϕ) andn(ϕ). From this definition it is easy to reconstruct the
curve in terms of its support function as in part 1.
3. Things To Observe
Recall one has in any parametrization the curvature formula
n0 (t) = κ(t)c0 (t),which in the present case reduces to:
1/κ(ϕ) = h(ϕ) + h00 (ϕ) = |c0 (ϕ)|.

Clearly aa has to be large enough to make κ positive andthe parametrization nonsingular. Adding a linear combination of cos(ϕ) and sin(ϕ) to the support function corresponds to a change of only the origin, the shape of thecurve stays the same. The bb cos(ϕ)-term in the support
function is therefore not really necessary, but one can use
it to see how the parametrization of the curve changes.
The cos-terms in even multiples of ϕ make up the evenpart (h(ϕ) + h(ϕ + π))/2 of h. The origin is the midpointof curves with even support function. If h is odd except
for the constant term, i.e.,h(ϕ) = aa + (h(ϕ) − h(ϕ + π))/2,
then one obtains curves of constant width w where:
w = h(ϕ) + h(ϕ + π) = 2 aa.
The default curve in 3D-XplorMath is such a curve of constant width and the default morph shows a family of such
curves. We emphasize the width of our curves by drawing
them together with their pairs of parallel tangents. Since
the (non-)constancy of the distance between these parallel tangents is difficult to see we have added a circle ofthe same width (= diameter). One cannot easily recognize how many extrema the curvature κ(ϕ) has. To see it
clearly we recommend selecting the entry Show Osculating
Circles from the Action Menu, since the evolute has a cusp
at every extremal value of κ.
H.K.

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Tractrix *The Tractrix is a curve with the following nice interpretation: Suppose a dog-owner takes his pet along as hegoes for a walk “down” the y-axis. He starts from theorigin, with his dog initially standing on the x-axis at a
distance aa away from the owner. Then the Tractrix is thepath followed by the dog if he “follows his owner unwillingly”, i.e., if he constantly pulls against the leash, keeping
it tight. This means mathematically that the leash is always tangent to the path of the dog, so that the length of
the tangent segment from the Tractrix to the y-axis hasconstant length aa. Parametric equations for the Tractrix(take bb = 0) are:x(t) = aa · sin(t)(1 + bb)
y(t) = aa · (cos(t)(1 + bb) + ln(tan(t/2))).
The curves obtained for bb =/ 0 are generated by the samekinematic motion, except that a different point of the moving plane is taken as the drawing pen. See the default
Morph.
The Tractrix has a well-known surface of revolution, called
the Pseudosphere, Namely, rotating it around the y-axis
gives a surface with Gaussian curvature -1. This meansthat the Pseudosphere can be considered as a portion of* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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the Hyperbolic Plane. The latter is a geometry that was
discovered in the 19th century by Bolyai and Lobachevsky.
It satisfies all the axioms of Euclidean Geometry except the
Axiom of Parallels. In fact, through a point outside a given
line (= geodesic) there are infinitely many lines that areparallel to (i.e., do not meet) the given line.
There are many connections, sometimes unexpected, between planar curves. For the Tractrix select: Show Osculating Circles And Normals. One observes a Catenary(see another entry in the curve menu) as the envelope ofthe normals.
H.K.

Cissoid and Strophoid *
3DXM Family: c(t) := 2aat(t2 − bb) bb
t2 − bb,+(1 + t2 ) 2aa (1 + t2 )

The additive constant bb/2aa in the y-coordinate has theeffect that the drawing mechanism is the same for thewhole family, try the default Morph in the Animate Menu.
History
Diocles ( 250 – ∼100 BC) invented the Cissoid to solve thedoubling of the cube problem (also know as the the Delian
problem). The name Cissoid (ivy-shaped) derives from theshape of the curve. Later the method used to generate thiscurve was generalized, and we call all curves generated ina similar way Cissoids. Newton (see below) found a way
to generate the Cissoid mechanically. The same kinematic
motion with a different choice of the drawing pen generates
the (right) Strophoid, formulas below.
From Thomas L. Heath’s Euclid’s Elements translation(1925) (comments on definition 2, book one):
This curve is assumed to be the same as that bymeans of which, according to Eutocius, Diocles in* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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his book On Burning-Glasses solved the problemof doubling the cube.
From Robert C. Yates’ Curves and their properties (1952):
As early as 1689, J. C. Sturm, in his Mathesis
Enucleata, gave a mechanical device for the constructions of the Cissoid of Diocles.
From E.H.Lockwood A book of Curves (1961):
The name cissoid (“Ivy-shaped”) is mentioned byGeminus in the first century B.C., that is, about acentury after the death of the inventor Diocles. In
the commentaries on the work by Archimedes On
the Sphere and the Cylinder, the curve is referred
to as Diocles’ contribution to the classic problem of doubling the cube. ... Fermat and Roberval constructed the tangent (1634); Huygens andWallis found the area (1658); while Newton gives
it as an example, in his Arithmetica Universalis,of the ancients’ attempts at solving cubic problems and again as a specimen in his Enumeratio
Linearum Tertii Ordinis.

1 Description

The Cissoid of Diocles is a special case of the general cissoid. It is a cissoid of a circle and a line tangent to thecircle with respect to a point on the circle opposite to thetangent point. Here is a step-by-step description of theconstruction:
1. Let there be given a circle C and a line L tangent tothis circle.
2. Let O be the point on the circle opposite to the tangent
point.
3. Let P1 be a point on the circle C.
4. Let P2 be the intersection of line [O,P1] and L.
5. Choose Q on line [O,P1] with dist[O, Q] = dist[P 1, P 2].
6. The locus of Q (as P1 moves on C) is the cissoid of
Diocles.
An important property to note is that Q and P1 are symmetric with respect to the midpoint of the segment [O,P2].
Call this midpoint M. We can reflect every element in theconstruction around M, which will help us visually seeother properties.

2 Formula derivation
Let the given circle C be centered at (1/2, 0) with radius
1/2. Let the given line L be x = 1, and let the given point
O be the origin. Let P 1 be a variable point on the circle,
and Q the tracing point on line [O, P 1]. Let the point (1, 0)
be A. We want to describe distance r = dist[O, Q] in termsof the angle θ = [A, O, P 1]. This will give us an equation
for the Cissoid in polar coordinates (r, θ). From elementarygeometry, the triangle [A, O, P 1] is a right triangle, so bytrignometry, the length of [O, P 1] is cos(θ). Similarly, triangle [O, A, P 2] is a right triangle and the length of [O, P 2]
1
is cos(θ). Since dist[O, Q] = dist[O, P 2] − dist[O, P 1], we1− cos(θ). Thus the polar equation
have dist[O, Q] = cos(θ)
1is r = cos(θ)− cos(θ). If we combine the fractions anduse the identity sin2 + cos2 = 1, we arive at an equivalent
form: r = sin(θ) tan(θ).

3 Formulas for the Cissoid and the Strophoid
In the following, the cusp is at the origin, and the asymptote is x = 1. (So the diameter of the circle is 1 (= aa in
3DXM).)
Parametric: (sin2 (t), sin2 (t) tan(t)) − π/2 < t < π/2.


3
2
t
t
Parametric:,−∞<t<∞
2
2(1 + t ) (1 + t ) 2

2
t − 1 t(t − 1)
Strophoid:,−∞<t<∞
2
2(1 + t ) (1 + t )1− cos(θ)
Polar: r = cos(θ)− π/2 < t < π/2.

Cissoid: y 2 (1−x) = x3 , Strophoid: y 2 (1−x) = x2 (1 +x).

The Cissoid has numerous interesting properties.
4 Properties
4.1 Doubling the CubeGiven a segment [C, B], with the help of the Cissoid of
Diocles we can construct a segment [C, M ] such that
dist[C, M ]3 = 2 ∗ dist[C, B]3 . This solves the famous doubling the cube problem.

Step-by-step description:
1. Given two points C and B.
2. Construct a circle c1, centered
on C and passing through B.
3. Construct points O and A on thecircle such that line [O, A] is perpendicular to line [C, B]
4. Construct a cissoid of Diocles using circle c1, tangent at A, andpole at O.
5. Construct point D such that B
is the midpoint of segment[C,D].
6. Construct line[A,D]. Let the intersection of cissoid and line[A,D]
be Q. (The intersection cannot
be found with Greek Ruler andCompass. We assume it is a given.)
7. Let the intersection of line [C, D]
and line [O, Q] be M .
8. dist[C, M ]3 = 2 · dist[C, D]3 .
This can be proved trivially withanalytic geometry.

4.2 Diocles’ Construction
By some modern common accounts(Morris Kline, Thomas L. Heath),
here’s how Diocles constructed thecurve in his book
On Burning-Glasses:
Let AB and CD be perpendicular
diameters of a circle. Let E be apoint on arc[B,C], and Z be a point
on arc[B,D], such that BE, BZ areequal. Draw ZH perpendicular to
CD. Draw ED. Let P be intersection[ZH,ED]. The cissoid is the locus of all points P determined by all
positions of E on arc[B,C] and Z on
arc[B,D] with arc[B,E]=arc[B,Z].
(The portion of the curve that lies
outside of the circle is a later generalization).
In the curve, we have CH/HZ=HZ/HD=HD/HP. Thus
HZ and HD are two mean proportionals between CH and
HP. Proof: taking CH/HZ=HZ/HD, we have CH ∗ HD =HZ 2 . triangle[D,C,Z] is a right triangle since it’s a triangle on a circle with one side being the diameter (elementary geometry). We know an angle[D,C,Z] and one side
distance[D,C], thus by trignometry of right angles, we canderive all lengths DZ, CZ, and HZ. Substituting the results

of computation in CH ∗ HD = HZ 2 results an identity.
Similarly, we know length HP and find HZ/HD=HD/HP
to be an identity.
4.3 Newton’s Carpenter’s Square and Tangent

Newton showed that Cissoid of Diocles and the right Strophoid can be generated by sliding a right triangle. The
midpoint J of the edge CF draws the Cissoid, the vertex F
the Strophopid. This method also easily proves the tangent
construction.
Step-by-step description:
1. Let there be two distinct fixed points B and O, both ona given line j. (distance[B,O] will be the radius of thecissoid of Diocle we are about to construct.)
2. Let there be a line k passing O and perpendicular to j.
3. Let there be a circle centered on an arbitrary point C
on k, with radius OB.

4. There are two tangents of this circle passing B, let thetangent points be E and F.
5. Let I be the midpoint between E and the center of thecircle. Similarly, let J be the midpoint between F andthe center of the circle.
6. The locus of I and J (as C moves on k) is the cissoidof Diocles and a line. Also, the locus of E and F is theright strophoid.
Tangent construction for Cissoid and Strophoid: Think oftriangle[C,F,B] as a rigid moving body. The point C moves
in the direction of vector[O,C], and point B moves in the
direction of vector[B,F]. The intersection H (not shown)of normals of line[O,C] and line[B,F] is its center of rotation. J is the point tracing the Cissoid and is also a pointon the triangle, thus HJ is normal to the Cissoid. For
the Strophoid change the last sentence: Since the tracing
point F is a point on the triangle, thus HF is normal to
the Strophoid.
In 3D-XploreMath, this construction is shown automatically when Cissoid is chosen from the Plane Curve menu,
just after the curve is drawn (or when it is redrawn bychoosing Create from the Action menu or typing CommandK). In the Action Menu switch between Cissoid and Strophoid. Hold down the option key to slow the animation, hold down Control to reverse direction, and press
the spacebar to pause.
In the animation, the tangent and normal are shown asblue. The line from the critical point, from the so called
momentary fixed point of the motion, is normal to thecurve. This point is the intersection of the green lines; oneof them is a vertical drop, the other perpendicular to the
red line.
4.4 Pedal and Cardioid

The pedal of a cissoid of Diocles with respect to a point P
is the cardioid. If the cissoid’s asymptote is the line y = 1
and its cusp is at the origin, then P is at {0, 4}. It follows
by definition, the negative pedal of a cardioid with respect
to a point opposite its cusp is the cissoid of Diocles.

4.5 Negative Pedal and Parabola

The pedal of a parabola with respect to its vertex is thecissoid of Diocles. (and then by definition, the negative
pedal of a cissoid of Diocles with respect to its cusp is aparabola.)
4.6 Inversion and Parabola

The inversion of a cissoid of Diocles at cusp is a parabola.

4.7 Roulette of a Parabola
Let there be a fixed parabola. Let there be an equalparabola that rolls on the given parabola in such way thatthe two parabolas are symmetric to the line of tangency.
The vertex of the rolling parabola traces a cissoid of Diocles.
XL.

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Conchoid *3D-XplorMath parametrization:r=bb+ aa, x = r · cos t, y = r · sin t.
cos tHistory
According to common modern accounts, the conchoid of
Nicomedes was first conceived around 200 B.C by Nicomedes,
to solve the angle trisection problem. The name conchoid isderived from Greek meaning “shell”, as in the word conch.
The curve is also known as cochloid.
From E. H. Lockwood (1961):
The invention of the ‘mussel-shell shaped’ conchoidis ascribed to Nicomedes (second century B.C.) by
Pappus and other classical authors; it was a favorite
with the mathematicians of the seventeenth century
as a specimen for the new method of analytical geometry and calculus. It could be used (as was the
purpose of its invention) to solve the two problemsof doubling the cube and of trisecting an angle; andhence for every cubic or quartic problem. For thisreason, Newton suggested that it should be treatedas a ‘standard’ curve.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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Description
The Conchoid of Nicomedes is a one parameter family of
curves. They are special cases of a more general conchoid
construction, being the conchoids of a line.
Step-by-step explanation:
1. Given a line `, a point O not on `, and a distance k.
2. Draw a line m through O and any point P on `.
3. Mark points Q1 and Q2 on m such that
distance[Q1, P ] = distance[Q2, p] = k.
4. The locus of Q1 and Q2 as P varies on ` is the conchoid of Nicomedes.
The point O is called the pole of the conchoid, and the line
` is called its directrix . It is an asymptote of the curve.

The following figures shows the curve family. The pole istaken to be at the origin, and directrix is y = 1. The figure

on top has constants k from −2 to 2. The one below has
constants k from −100 to 100.

FormulasLet the distance between pole and line be b, and the given
constant be k. The curve has only the one parameter k,because for a given b, all families of the curve can be generated by varying k (they differ only in scale). (Similarly,

we could use b as the parameter.) In a mathematical context, we should just use b = 1, however, it is convenient tohave formulas that have both b and k. Also, for a given k,
the curve has two branches. In a mathematical context, it
would be better to define the curve with a signed constantk corresponding to a curve of only one branch. We will
be using this intepretation of k. In this respect, the conchoid of Nichomedes is then two conchoids of a line withconstants k and −k.
The curve with negative offset can be classified into three
types: if b < k there is a loop; if b = k, a cusp; and if b > k,
it is smoothly imbedded. Curves with positive offsets arealways smooth.
The following are the formulas for a conchoid of a liney = b, with pole O at the origin, and offset k.
Polar: r = b/ sin(θ) + k, −π/2 < θ < π/2.
This equation is easily derived: the line x = b in polar
equation is r = b/ cos θ, therefore the polar equation isr = b/ cos(θ) + k with −π/2 < θ < π/2 for a signed k (i.e.,
describing one branch.). Properties of cosine show that asθ goes from 0 to 2π, two conchoids with offset ±|k| resultsfrom a single equation r = b/ cos(θ) + k. To rotate thegraph by π/2, we replace cosine by sine.
Parametric:√√22(t + (kt)/ b + t , b + (bk)/ b2 + t2 ), −∞ < t < ∞.
If we replace t in the above parametric equation by b tan(t),we get the form:
b
−π
3π
π
(k +
) · (sin(t), cos(t)),<t<, t 6= .
cos(t)
2
2
2
For conchoids of a line with positive and negative offsets kand pole at the origin, we have the quartic
Implicit Cartesian equation: (x2 + y 2 )(y − b)2 = k 2 y 2 .
If k < b, the point at the origin is an isolated point.
If √k < 0 and b < |k|, the √conchoid has a loop with area
(b k 2 − b2 − 2bk ln((k + k 2 − b2 )/b) + k 2 arccos(b/k)).
The area between any conchoid of a line and its asymptote
is infinite.
Tangent Construction
Look at the conchoid tracing
as a mechanical device, wherea bar line [O, P ] slides on aline at P and a fixed joint
O. The point P on the bar
moves along the directrix, andthe point at O moves in the
direction of the vector [O, P ].
We know the direction of motion of the bar at the points
O and P at arbitrary time.
The intersection of normals to these directions form the instantaneous center of rotation N . Since the tracing points

Q1 and Q2 are parts of the apparatus, N is also their center of rotation and therefore line [N, Q1] and line [N, Q2]
are the curve’s normals.
Angle Trisection
The curve can be used to solve the Greek Angle Trisection
problem. Given an acute angle AOB, we want to constructan angle that is 1/3 of AOB, with the help of the conchoidof Nicomedes.
Steps: Draw a line m intersecting segment [A, O] and perpendicular to it. Let D be intersection of m and the line
[A, O], L the intersection of m and the line [B, O]. Suppose we are given a conchoid of Nicomedes, with pole atO, directrix m, and offset 2 · distance[O, L]. Draw a line
` through L and perpendicular to m. Let C be the intersection of the curve and ` on the opposite side of the pole.
Theorem. angle[A, O, B] = 3 · angle[A, O, C].
Proof:<)[A, O, C] =<)[O, C, L]
because the line [O, C]
cuts parallel lines. Let:q be the line [O, C],
N := m ∩ q,
M midpoint of [N, C],k := distance[O, L].
By our construction,
distance[N, M ] = distance[M, C] = k. Since N LC is aright triangle, we see that M N , M L, M C, and OL allhave the same length, thus triangle [M, L, C] and triangle[M, L, N ] are isosceles, and it follows that <)[N, M, L] =2· <)[M, C, L]. Since distance[O, L] = distance[M, L], triangle [M, L, O] is also isosceles, and thus its two base angles
are equal. This shows that an angle equal to <)[A, O, C] is
1/3 an angle equal to <)[A, O, B].
The essential point where the conchoid makes the trisection possible is in the construction of the point C on ` suchthat distance[N, C] = 2 distance[O, L], where N is the intersection of m and the line [O, C]. Note that for eachnew angle to trisect, a new conchoid is needed. This is incontrast to some other trisectrixes such as the quadratrix,
where all angles can be trisected once the curve is given.
The conchoid can also be used to solve the classic problemof doubling the cube.
XL.

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Lemniscate *The Lemniscate is a figure-eight curve with a simple mechanical construction attributed to Bernoulli: Choose
two ’focal’ points F1 , F2 at distance L := 2 ∗ dd, then take√three rods, one of length L, two of length R = L/ 2.
The short ones can rotate around the focal points andthe long one connects their free ends with rotating joints(red lines in the figure). This machine has one degree of
freedom and the midpoint of the long rod traces out theLemniscate while the short rods rotate (not uniformly).
– This drawing mechanism will also work for arbitrary
lengths 0 < R < L, R := cc. The default morph in 3DXM
varies cc. Another interesting morph is obtained by varying the position of the drawing pen on the long rod withff ∈ (0, 1). Click the Init To Current Parameters button in Set Morphing, then put f 0 := 0, f 1 := 1.
Since the Bernoulli Lemniscate is much better known than* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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curves drawn by√ mechanisms with parameters different
from L : R = 2 : 1, we will give, below, parametrizations and equations only for the Bernoulli curve.
The curves in 3DXM are obtained as follows:
The endpoint of the right rod rotates with constant speed,i.e. P (t) = (dd + cc · cos(t), cc · sin(t)). The endpoint Q(t)of the left rod is obtained by intersecting two circles (of radius R around F1 and radius L around P (t)). One of theintersection points is P 0 (t) = (−dd + cc · cos(t), cc · sin(t)),
since |F2 − F1 | = |P (t) − P 0 (t)| = L = 2 · dd. Therefore
Q(t) is obtained by reflecting P 0 (t) in the Diagonal F1 P (t)of the parallelogram F2 , F1 , P 0 (t), P (t).
The drawing pen is at ff · P (t) + (1 − ff ) · Q(t).
Mechanical constructions of curves give rise to simple tangent constructions. We imagine that a plane is attachedto the long rod. Then every point of this plane traces out
a curve when the rods move. The velocity vectors of thesetraced curves give, at each moment, a vectorfield, that hasconcentric circles as integral curves (or, exceptionally, parallel lines). The centers of these concentric circles are themomentary centers of rotation for the moving plane. If wejoin a point of a traced curve to the corresponding momentary center of rotation, then this radius (drawn blue)
is orthogonal to the tangent (also blue).
How can one find the momentary center of rotation forthe current drawing machine? The endpoints of the two
short rods are points of the moving plane. We know thateach can only move orthogonally to its rod (namely rotate
around the other endpoint, a focal point). This says thatboth short rods point to the momentary center of rotation,
which therefore is obtained as the intersection of two lines(drawn green in the figure).
Compare the other mechanically constructed curves.
Parametrizations are not unique, here is a well known one:x(t) := cos(t)/(1 + sin(t)2 )y(t) := sin(t) · cos(t)/(1 + sin(t)2 ).
The Bernoulli Lemniscate has this implicit equation:(x2 + y 2 )2 = x2 − y 2 .
Divide this by r2 := x2 + y 2 to get the polar form:
r2 = cos(φ)2 − sin(φ)2 .
√The points F1 , F2 := ±1/ 2 are called Focal points of theLemniscate because of the special property:|P − F1 | · |P − F2 | = |F1 − F2 |2 /4.
If one takes the complex square root of a circle which
touches the y-axis from the right at 0 then one also obtains(half √
of) a Lemniscate. In the Conformal Category, choosez → z, and then in the Action Menu, select Choose Circle by Mouse, and create a circle that is tangent to they-axis at 0.
The inversion map: (x, y) 7→ (x, y)/(x2 + y 2 ) often transforms some interesting curve into another interesting curve.

And indeed, the Lemniscate, with the above parametrization, is transformed by inversion into the curve
x = 1/ cos(t), y = sin(t)/ cos(t).
Observe the implicit equation x2 − y 2 = 1. It showsthat the new curve is a hyperbola with orthogonal asymptotes. So we could have obtained the Bernoulli Lemniscate from the orthogonal hyperbola by inversion in a circle around its midpoint. – More generally, inversions ofhyperbolae x2 /a2 − y 2 /b2 = const give figure 8 curves
with non-orthogonal double tangents. The angle 2α between the double tangents is the same as the angle between the asymptotes and satisfies tan α = b/a. The angle 2β between the double tangents of the figure 8 curvesof our drawingpmechanisms satisfies sin β = R/L. Set
a := R, b := a/ L2 /R2 − 1 to obtain α = β. Invert, with~x 7→ ~x · a2 /|~x|2 , the lemniscate and put the result into theterm x2 /a2 − y 2 /b2 to find that it is 1. This gives implicit
lemniscate equations: x2 /a2 − y 2 /b2 = (x2 + y 2 )2 /a4 .
And, invert hyperbola parametrizations, e.g. x = a/ cos(t),y = b sin(t)/ cos(t), to parametrize lemniscates.
We note that not every figure 8 curve (with orthogonal
double tangents) is a Bernoulli Lemniscate. Another figureeight is obtained by the simpler parametrization:x(t) := cos(t), y(t) := sin(t) · cos(t),which has the implicit equation y 2 = x2 (1 − x2 ).
H.K.

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Clothoid *The Clothoid, also called Spiral of Cornu, is a curve whosecurvature is equal to its arclength. It has the parametricformula:Z t0cos(x2 /2) dx,

Z t

sin(x2 /2) dx) .

0Discussion
If a plane curve is given by a parametric formula (f (t), g(t)),
then the length of the part corresponding to a parameter* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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Rtp

interval [a, t] is s(t) = a f 0 (τ )2 + g 0 (τ )2 dτ . If we apply this formula to the Clothoid we see that Rthe arclengthtcorresponding to the interval [0, t] is s(t) = 0 1 dt = t, sothat the parameter t is precisely the (signed!) arclength
measured along the curve from its midpoint, (0, 0).
Next, recall that the curvature κ of a plane curve is defined as the rate of change (with respect to arclength) of
the angle θ that its tangent makes with some fixed line
(which we can take to be the x-axis). And since the slopedydydx of the curve is tan(θ), and by the chain rule dx =g0
(dy/dt)/(dx/dt) = f 0 , we see that θ(t) = arctan(g 0 (t)/f 0 (t)).
So if we assume that parameter t is arclength, then using
the formulas for the derivative of the arctangent and of aquotient, we see that:
κ(t) = θ0 (t) = −g 0 (t)f 00 (t) + f 0 (t)g 00 (t),(where we have ignored the denominator, since parameterization by arclength implies that it equals unity). Applying this to the Clothoid, we obtain κ(t) = t. Since thearclength function is also t, this shows that the Clothoid
is indeed a curve whose curvature function is equal to its
arclength function.
The Fundamental Theorem of Plane Curves
Next let’s look at this question from the other direction,
and also more generally. Suppose we are given a function
κ(t). Can we find a plane curve parameterized by arclength(f (t), g(t)) such that κ is its curvature function? Recall

dx
0from above that dθ
dt = κ, and of course dt = f (t) and
dy
dy 2
dy dx
dx 2
0
=
g
(t).
Now,
since
(
)
+
(
)
=
1,
while
dt
dt
dt
dt / dt =dy/dx = tan(θ), it follows from elementary trigonometrydythat dx=cos(θ)
while
dt
dt = sin(θ). Thus we have the following system of three differential equations for the threefunctions θ(t), f (t), and g(t):θ0 (t) = κ(t), f 0 (t) = cos(θ(t)), g 0 (t) = sin(θ(t)).
Rτ

The first equation is solved by θ(τ ) = θ0 + 0 κ(σ) dσ, andsubstituting this in the other two equations, we find thatthe general solutions for f and g are given by:
Z t
f (t) = x0 +Z τcos(θ0 +κ(σ) dσ) dτ

0

0

Z t

Z τ

g(t) = y0 +sin(θ0 +0κ(σ) dσ) dτ.
0This is an elegant explicit solution to our question! It
shows that not only is there a solution to our question (saythe one obtained by setting x0 , y0 and θ0 all equal to zero),but also that the solution is unique up to a translation (by(x0 , y0 )) and a rotation (by θ0 ), that is unique up to a
general rigid motion.
This fact has a name—it is called The Fundamental Theorem of Plane Curves. It tells us us that most geometricand most economical descriptions of plane curves is not via
parametric equations, which have a lot of redundancy, butrather by the single function κ that gives the curvature asa function of arclength.
Exercise Take κ(t) = t and check that the above formulas
give the parametric equations for the Clothoid in this case.
Back to the Clothoid
We close with a few more details about the Clothoid. First,
here is a plot of the integrand sin(x2 /2):and next a plot of its indefinite integral,
the so-called Fresnel integral:

Rt0

sin(x2 /2) dx,

From this plot we see that the √y-coordinate oscillates. Its
limit as t goes to infinity is π/2, from which we see
thatof the two spirals of the Clothoid are at√ the centers√±( π/2, π/2).
XL & RSP.

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Archimedean Spirals *An Archimedean Spiral is a curve defined by a polar equation of the form r = θa . Special names are being given forcertain values of a. For example if a = 1, so r = θ, then it
is called Archimedes’ Spiral.
Formulas in 3DXM:
r(t) := cc · taa , θ(t) := t,Default Morph:−1 ≤ aa ≤ 1.25.
cc for scaling.

Archimede’s Spiral

For a = −1, so r = 1/θ,
we get the reciprocal (orhyperbolic) spiral:

Reciprocal Spiral* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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The case a = 1/2, so r =hyperbolic) spiral.

√

θ, is called the Fermat (orFermat’s Spiral√
While a = −1/2, or r = 1/ θ, it is called the Lituus:

LituusIn 3D-XplorMath, you can change the parameter a by going to the menu Settings → Set Parameters, and change
the value of aa. You can see an animation of Archimedeanspirals where the exponent a = aa varies gradually, between −1 and 1.25. See the Animate Menu, entry Morph.
The reason that the parabolic spiral and the hyperbolic
spiral are so named is that their equations in polar coordinates, rθ = 1 and r2 = θ, respectively resembles theequations for a hyperbola (xy = 1) and parabola (x2 = y)in rectangular coordinates.
The hyperbolic spiral is also called reciprocal spiral because it is the inverse curve of Archimedes’ spiral, withinversion center at the origin.
The inversion curve of any Archimedean spirals with respect to a circle as center is another Archimedean spiral,
scaled by the square of the radius of the circle. This iseasily seen as follows. If a point P in the plane has polar
coordinates (r, θ), then under inversion in the circle of radius b centered at the origin, it gets mapped to the point
P 0 with polar coordinates (b2 /r, θ), so that points having
polar coordinates (ta , θ) are mapped to points having polar coordinates (b2 t−a , θ).
From the above, we can see that the Archimedes’ spiralinverts to the reciprocal spiral, and Fermat’s spiral inverts
to the Lituus.

The following two images illustrates Archimedes’s spiraland Reciprocal spiral as mutual inverses. The red curve isthe reciprocal spiral, the purple is the Archimedes’ spiral.
The yellow is the inversion circle.

The following image illustrates a Lituus and Fermat’s spiral as mutual inverses. The red curve is the Fermat’s spiral.
The blue curve is its inversion, which is a lituus scaled by
52 . The yellow circle is the inversion circle with radius 5.
Note that points inside the circle gets mapped to outside of
the circle. The closer the point is to the origin, the farther
is its corresponding point outside the circle.

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The Logarithmic Spiral *The parametric equations for the Logarithmic Spiral are:x(t) =aa · exp(bb · t) · cos(t)y(t) =aa · exp(bb · t) · sin(t).
This spiral is connected with the complex Exponential
Map as follows:
x(t) + i y(t) = aa exp((bb + i)t).
The animation that is automatically displayed when you
select Logarithmic Spiral from the Plane Curves menu
shows the osculating circles of the spiral. Their midpoints
draw another curve, the evolute of this spiral. These osculating circles illustrate an interesting theorem, namely if
the curvature is a monotone function along a segment of aplane curve, then the osculating circles are nested - because
the distance of the midpoints of two osculating circles is(by definition) the length of a secant of the evolute while
the difference of their radii is the arc length of the evolute
between the two midpoints. (See page 31 of J.J. Stoker’s“Differential Geometry”, Wiley-Interscience, 1969).
For the logarithmic spiral this implies that through every
point of the plane minus the origin passes exactly one osculating circle. Étienne Ghys pointed out that this leads* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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to a surprise: The unit tangent vectors of the osculating
circles define a vector field X on R2 \ {0} – but this vector field has more integral curves, i.e. solution curves of
the ODE c0 (t) = X(c(t)), than just the osculating circles,
namely also the logarithmic spiral. How is this compatible
with the uniqueness results of ODE solutions? Read words
backwards for explanation:
eht dleifrotcev si ton ztihcspiL gnola eht evruc.
Remarks about Spirals
All spirals with names share the following properties:
If they are left rotating their curvature is positive, andnegative for right rotating ones. If they are outgoing, theabsolute value of the curvature is monotone decreasing,and increasing for those which spiral inwards. Moreover,for every such curvature function κ(s) one can find a spiral,with κ(s) its curvature function, as follows:First find an antiderivative T (s) of κ(s), i.e. T 0 (s) = κ(s).
0Next define a unit field c (s) := cos(T (s)), sin(T (s)) .
Finally obtain the spiral c(s) by integrating c0 (s).
Therefore one may take the quoted curvature properties
as definition of a spiral. This has the consequence that aspiral with curvature κ1 (s) can be deformed into a spiralwith curvature κ2 (s) through spirals with curvatureκa (s) := (1 − a) · κ1 (s) + a · κ2 (s).
R.S.P., H.K.

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Cycloid *Cycloids are generated by rolling a circle on a straight line
and tracing out the path of some point along the radius.
The parametric equation for such a cycloid is:x(t) = aa · t − bb · sin ty(t) = aa − bb · cos t,
where aa is the radius of the rolling circle and bb is the
distance of the drawing point from the center of the circle.
The choice bb = aa gives the standard cycloid.
Cycloids have other cycloids of the same size as evolutes,
see the Action Menu Entry Show Osculating Circles with
Normals. This fact is responsible for Huyghen’s cycloid
pendulum having its period independent of the amplitudeof the oscillation.
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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About Epicycloids and Hypocycloids *See also the ATOs for Spherical Cycloids
Definition and tangent construction

Epicycloids resp. Hypocycloids are obtained if one circle ofradius r rolls on the outside resp. inside of another circleof radius R.
In 3D-XplorMath: r = hh, R = aa.
The angular velocity of the rolling circle is f r times theangular velocity of the fixed circle (negative for hypocycloids). f r has to be an integer for the hypocycloid to beclosed. The formulas do not actually roll one circle aroundanother, they represent the curve as superposition of tworotations:f r := (R − r)/(−r);
c.x := (R − r) cos(t) + r cos(f r · t);c.y := (R − r) sin(t) + r sin(f r · t);
Double generation: If one changes the radius of the rolling
circle from r to R−r then these formulas are preserved, except for the parametrization speed. To view this in 3DXM
replace hh by aa − hh.
* This file is from the 3D-XplorMath project. Please see:
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Epicycloids are obtained if one circle of radiusr = −hh rolls on the outside of another circle of radius
R = aa. The angular velocity of the rolling circle is f r > 0
times the angular velocity of the fixed circle (again an integer for closed epicycloids).
f r := (R + r)/r;
c.x := (R + r) cos(t) − r cos(f r ∗ t);c.y := (R + r) sin(t) − r sin(f r ∗ t);
These formulas agree with those of the hypocycloids except
for the sign of r. We view them in 3DXM by using negativehh.
We can also use a drawing stick of length ii∗r. The default
morph shows this: 0.5 < ii < 1.5.
These more general (ii <> 1) rolling curves were important for Greek astronomy because the planets orbit the sun(almost) on circles. Therefore, when one looks at otherplanets from earth, their orbits are (almost) such rolling
curves. It is no surprise that many of these curves haveindividual names: Astroid, Cardioid, Limacon, Nephroid
are examples in 3DXM.
Tangent construction.
Rolling curves have a very simple tangent construction.
The point of the rolling circle which is in contact with thebase curve has velocity zero – just watch cars going by.
This means that the connecting segment from this point

of contact of the wheel to the endpoint of the drawing stickis the radius of the momentary rotation. The tangent of
the curve which is drawn by the drawing stick is therefore
orthogonal to this momentary radius.
The 3DXM-demo draws the rolling curve and shows its
tangents.
H.K., R.S.P.
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Cardioids and Limaçons *Cardioids and Limaçons are obtained if on the outside of
one fixed circle of radius r = aa another circle of the sameradius rolls. These curves are traced by a radial stick oflength R = ii ∗ r, ii = 1 for Cardioids and ii > 1 for
Limaçons.
One choice of parametric equations for these curves is:x(t) = 2r cos(t) + R cos(2t)y(t) = 2r sin(t) + R sin(2t).
The evolute of the Cardioid is a smaller Cardioid, see in the
Action Menu the entry Show Osculating Circles with
Normals. In the entry Add Caustics one can rotate all
normals by a fixed amount and these rotated lines always
envelope a Cardioid.
To see the Cardioid generated by rolling a larger circle
around a smaller one choose in the exhibit Epi- and Hypocycloids parameters hh = 2 ∗ aa, ii = 1.
The image of the unit circle under the complex map
z 7→ w(z) = z 2 + 2z
is a Cardioid; images of larger circles (around 0) are Limacons. Inverses z 7→ 1/w(z) of Limaçons are figure-eight
shaped, including a Lemniscate.
H.K.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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Astroid *Parametrization in 3DXM: c(t) := aa · (cos3 (t), sin3 (t)).
Implicit equation:

x2/3 + y 2/3 = aa2/3 .

Description

An Astroid is a curve traced out by a point on the circumference of one circle (of radius r) as that circle rolls withoutslipping on the inside of a second circle having four times
or four-thirds times the radius of the first. The latter isknown as double generation. The Astroid is thus a special
kind of a hypocycloid—the family of analogous curves one
gets if one allows the ratios of the radii to be arbitrary.
* This file is from the 3D-XplorMath project. Please see:
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In 3D-XplorMath, the radius r is represented by the parameter aa. A nice geometric property of the Astroid isthat its tangents, when extended until they cut the x-axis
and the y-axis, all have the same length. This means, if
one leans a ladder (say of length L) against a wall at all
possible angles, then the envelope of the ladder’s positions
is part of an Astroid. Since (by symmetry) the tangent to
the Astroid at a point p closest to the origin has a slopeof plus or minus one, it follows that the distance of p fromthe origin is L/2, and so L is the “waist-diameter” of the
Astroid, i.e., the distance from p to −p. Since the diagonalof the Astroid clearly has length 2L, it is twice as long as
the waist-diameter.
It can be shown that the normals of an Astroid envelope
an Astroid of twice the size. (To see a visual demonstration of this fact, in 3D-XplorMath, select Show Osculating
Circles and Normals from the Action Menu.) If you think
about what this means, you should see that it gives a ruler
construction for the Astroid: Intersect each ladder (between the x-axis and the y-axis) for the smaller Astroidwith the orthogonal and twice as long ladder (between the
45-degree lines) for the larger Astroid.

More Formulas
The initial formulas give an astroid centered at the origin
with its four cusps lying on the axes at distance aa fromthe origin. To derive a polynomial equation first cube bothsides of the above implicit equation (with aa = 1), factorize
and simplify:
1 = x2 + y 2 + 3x4/3 y 2/3 + 3x2/3 y 4/3
1−x2 − y 2 = 3x2/3 y 2/3 (x2/3 + y 2/3 ) = 3x2/3 y 2/3 .
Then cube again:(1 − x2 − y 2 )3 = 27x2 y 2 .
History
Quote from Robert C. Yates, 1952:
The cycloidal curves, including the astroid, were discovered by Roemer (1674) in his search for the bestform for gear teeth. Double generation was first noticed by Daniel Bernoulli in 1725.
Quote from E. H. Lockwood, 1961:
The astroid seems to have acquired its present nameonly in 1838, in a book published in Vienna; it went,even after that time, under various other names,such as cubocycloid, paracycle, four-cusp-curve, andso on. The equation x2/3 + y 2/3 = a2/3 can, however, be found in Leibniz’s correspondence as earlyas 1715.

Properties
Trammel of Archimedes and Envelope of Ellipses
Define the axes of the astroid to be the two perpendicularlines passing through the pairs of alternate cusps. A fundamental property of the Astroid is that the length of thesegment of a tangent between these two axes is a constant.
The Trammel of Archimedes is a mechanical device that isbased on this property: it has a fixed bar whose ends slideon two perpendicular tracks. The envelope of the moving
bar is then the Astroid, while any particular point on thebar will trace out an ellipse.
The Astroid is also the envelope of co-axial ellipses whose
sum of major and minor axes is constant.

The Evolute of the Astroid
The evolute of an astroid is another astroid. (In fact, theevolute of any epi- or hypo- cycloid is a scaled version of
itself.) In the first figure below, each point on the curveis connected to the center of its osculating circle, while inthe second, the evolute is seen as the envelope of normals.

Curve Construction
The Astroid is rich in properties that can be used to devise
other mechanical means to generate the curve and to construct its tangents, and the centers of its osculating circles.
Suppose we have a circle C centered at B and passing
through some point K. We will construct an Astroid thatis also centered at B and that has one of its cusps at K.
Choose the origin of a cartesian coordinate system at B,

and take the point (1, 0) at K. Given a point L on the circle
C, drop a perpendicular from L to the x-axis, and let M
be their intersection. Similarly drop a perpendicular fromL to the y-axis and call the intersection N . Let P be thepoint on M N such that LP and M N are perpendicular.
Then P is a point of the Astroid, M N is the tangent to
the Astroid at P , and LP the normal at P . If D is theintersection of LP and the circle C, and D0 is the reflectionof D thru M N , then D0 is the center of osculating circle
at P .

Pedal, Radial, and Rose
The pedal of an Astroid with respect to its center is a 4petaled rose, called a quadrifolium. The Astroid’s radialis also a quadrifolium. (For any epi- or hypo- cycloid, thepedal and radial are equal, and is a rose.)

Catacaustic and Deltoid
The catacaustic of a Deltoid with respect to parallel raysin any direction is an Astroid.

Orthoptic
We recall that the orthoptic of a curve C is the locus ofpoints P where two tangents to C meet at right angles.
The orthoptic of the Astroid is the quadriffolium r2 =(1/2) cos(2θ)2 . [Robert C. Yates.]

XL.

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Deltoid *The Deltoid curve was conceived by Euler in 1745 in connection with his study of caustics.
Formulas in 3D-XplorMath:x = 2 cos(t) + cos(2t), y = 2 sin(t) − sin(2t), 0 < t ≤ 2π,
and its implicit equation is:(x2 + y 2 )2 − 8x(x2 − 3y 2 ) + 18(x2 + y 2 ) − 27 = 0.

The Deltoid or Tricuspid
The Deltoid is also known as the Tricuspid, and can be
defined as the trace of a point on one circle that rolls inside* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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another circle of 3 or 3/2 times as large a radius. The latteris called double generation. The figure below shows both ofthese methods. O is the center of the fixed circle of radiusa, C the center of the rolling circle of radius a/3, and P
the tracing point. OHCJ, JPT and TAOGE are colinear,where G and A are distant a/3 from O, and A is the centerof the rolling circle with radius 2a/3. PHG is colinear andgives the tangent at P. Triangles TEJ, TGP, and JHP are
all similar and T P/JP = 2 . Angle JCP = 3∗Angle BOJ.
Let the point Q (not shown) be the intersection of JE andthe circle centered on C. Points Q, P are symmetric withrespect to point C. The intersection of OQ, PJ forms thecenter of osculating circle at P.

The Deltoid has numerous interesting properties.
Properties
Tangent

Let A be the center of the curve, B be one of the cusp
points,and P be any point on the curve. Let E, H be theintersections of the curve and the tangent at P. The segment EH has constant length distance[E,H]== 4/3*distance[A,B]. The locus of midpoint D of the tangent segment EH is the inscribed circle. The normals at E,P,H
are concurrent, and the locus of these intersections is thecircumscribed circle. If J is the intersection of another tangent, cutting EH at right angle, then the locus traced by
J (the Deltoid’s orthoptic) is the inscribed circle.

The Deltoid and the Astroid

The caustic of the Deltoid with respect to parallel rays inany direction is an Astroid.

Evolute

The evolute of Deltoid is another Deltoid. (In fact, the evolutes of all epicycloids and hypocycloids are scaled versionof themselves.) In the above figure, the evolute is shownas the envelope of its normals.

Simson Lines

The Deltoid is the envelope of the Simson lines of any
triangle. (Robert Simson, 1687–1768)

Step by step description:
1. Let a triangle be inscribed in a circle. 2. Pick any
point P on the circle. 3. Mark a point Q1 on any sideof the triangle such that line[P,Q1] is perpendicular to it,
extending the side if necessary. 4. Similarly, find points Q2and Q3 with respect to P for the other two sides. 5. The
points Q1, Q2, and Q3 are colinear, and the line passing
through them is called the Simson line of the triangle withrespect to P. 6. Find Simson lines for the other points P
on the circle. Their envelope is the deltoid. Amazingly,
this is true for any triangle.
Pedal, Radial, and Rose

The pedal curve of a Deltoid with respect to a cusp, vertex,
or center is a folium curve with one, two, or three loops
respectively. The last one is called the trifolium, a threepetalled rose. The Deltoid’s radial is a trifolium too.
XL.

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The Nephroid *The Nephroid is generated by rolling a circle of one radius on the outside of a second circle of twice the radius.
In 3D-XplorMath, either choose Nephroid from the Plane
Curve menu, or choose Circle, then select Set Parameters... from the Settings menu and set hh = −0.5 · aa,
ii = 1. With R = 3r we thus have the parametrization for
Nephroids:x(t) =R · sin(t) + r · sin(3t)y(t) =R · cos(t) + r · cos(3t)As with Cardioids and Limaçons, one can also make theradius for the drawing stick shorter or longer: Set ii > 1
for the looping relatives of the Nephroid or see the default
Morph in the Animation menu.
The complex map z 7→ z 3 + 3z maps the unit circle to sucha Nephroid. To see this, in the Conformal Map Category,* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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select z 7→ z e e + eez from the Conformal Map menu, andthen choose Set Parameters from the Settings menu andset ee to 3.
The normals of one Nephroid have as envelope another,smaller Nephroid—the same phenomenon as for the Cardioid and the Cycloid. (To see this select Show Osculating Circles With Normals from the Action menu). In
technical jargon, the caustics for each of these curves is asimilar curve.
H.K.

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About Mechanically Generated Curves *ExamplesPresently we have the following mechanically generated
plane curves programmed together with a decoration whichshows this generation and the corresponding constructionof the tangents of the curve:
Epi- and Hypocycloids,
all other rolling curves.
Also: Tractrix, Cissoid,
Conchoid, Lemniscate.

This image is obtained with Color Morph in the Animation menu, it shows the family obtained from the current
drawing mechanism (here Lemniscate).
Moving Planes
It is often convenient to discuss such mechanical generations in terms of two planes, a fixed plane on which thedrawing is done (paper plane) and a second plane which is
attached to that piece of the mechanical contraption that
holds the drawing pen (drawing plane). In the case ofrolling curves we have the drawing plane attached to the* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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rolling wheel, in the case of the Lemniscate the drawing
plane is attached to the middle one of the three connectedmoving segments.
We think of the orbits of the points of the drawing
plane as curves that are mechanically generated
by the apparatus under consideration.
The velocity vectors of these orbits clearly give a time dependent vector field. Since this vector field is obtained bydifferentiating the orbits of a family of isometries we obtain at each time t the vector field of a Euclidean group of
motions, in other words: for most t the vector field consists of the velocity vectors of a rotation, a rotation aroundthe so called momentary center of rotation. This way oflooking at the generation gives immediately tangent constructions for all orbits: join the momentary center of rotation to the moving point, the perpendicular line throughthe point is tangent to its orbit.
It is therefore useful to visualize the movement of the drawing plane together with the time dependent velocity fieldof its points. We have done this by decorating the drawing plane with not too many but enough random points sothat the movement of the drawing plane becomes visible,
but the curve under consideration is not obscured. Moreover, to make the vector field visible at each moment t, wehave drawn the random points not once, but at two subsequent positions. This picture is interpreted by the brain
correctly.
Finally, one has to determine the momentary center of ro-tation. This is different for each construction. For rolling
curves the definition of “rolling” is such that that point,where the rolling wheel touches the fixed curve (“street”),
is the momentary center of rotation. In general one has tolook for points of the mechanical apparatus for which thedirection of the momentary movement (“orbit tangent”)
can be decided. The momentary center is then on the line(“radius”) perpendicular to the tangent, so that two suchlines are needed. The 3DXM demos use green lines to determine the momentary center.
H.K.

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Cubic Curves *Real Cubic Curves in R2 were studied extensively by Newton. Later these curves were considered as the real pointsof complex curves in C2 . If they do not have double points
they can be parametrized by additive groups. This meansthat the points on these curves can be added. Surprisingly
this addition is a geometric addition, i.e. the sum P + Q
can be geometrically constructed from P, Q and the curve.
In the case of cubic curves we have:
P +Q+R=0 ⇔A straight line intersects the curve in P, Q, R .
In 3D-XplorMath are the following examples:
Cubic Polynomial Graph, x(t) = t, y(t) = x(t)3 + aa · x(t),Cuspidal Cubic, x(t) = 3t2 /(4aa), y(t) = t(t2 + bb)/(4aa2 ),Cubic Rational Graph I, x(t) = tan(t/2)/aa, y(t) = sin(t),Cubic Rat’l Graph II, x(t) = tanh(t/2)/aa, y(t) = sinh(t),Elliptic Cubic, x + 1/x − aa · (y − 1/y) = ff (implicit),
Folium, [x, y] = aa[(t2 − t3 ), (t − 2t2 + t3 )]/(1 − 3t − 3t2 ),Nodal Cubic, x = 1 − t2 , y = ((1 − t2 ) + bb) · t.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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The last two of these are cubics with one double point.
Their points do not form a group, therefore their Action
Menu has only the Standard Actions for plane curves.
All the others are parametrized by a 1-dim. Abelian groupand the curves are shown with a default demo explaining
the geometric addition.
If we intersect a Cubic Polynomial Graph without quadraticterm with a line, then the x-coordinates of the intersection
points are always roots of a polynomial without quadratic
term. In other words: these three x-coordinates add up to0., the geometric addition is the standard addition on thex-axis.
The Cuspidal Cubic is also parametrized by R (or C) anda simple computation shows: if 1/t1 + 1/t2 + 1/t3 = 0 then
the three points [x(ti ), y(ti )] lie on a straight line. And dually, if t1 + t2 + t3 = 0 then the tangents at the three points[x(ti ), y(ti )] pass through one point. Again, the addition
has a simple geometric interpretation that allows to construct, if two points and the curve are given, their sum.
The first Rational Cubic Graph is parametrized by a circle
S1 (we have to add the infinite point (∞, 0) ). The demothat comes with the curve shows how the sum point can beconstructed by intersecting lines. The Action Menu offersa second demo that shows how addition on the parametrizing circle and on the curve are the same.
The second Rational Cubic Graph would not be here if we
could visualize these curves over the complex Numbers.
Over C one can think of this curve as the group S1 ⊕ R,a cylinder. The first rational graph visualizes the equator
circle, the second one visualizes the generator through theneutral element plus the opposite generator: two copies ofR (and a double point at ∞).
The Elliptic Cubic is parametrized by a pair of so called
Elliptic Functions. Such functions can be viewed either as
doubly periodic functions from C to S1 or as functions defined on some torus. For more details see the text Elliptic
Functions.
The addition on elliptic curves can be compared with addition on the circle. The formulas for trigonometric functionscos(α + β) = cos(α) cos(β) − sin(α) sin(β)
sin(α + β) = sin(α) cos(β) + cos(α) sin(β)
show that(x1 , y1 ) ⊕ (x2 , y2 ) := (x1 x2 − y1 y2 , x1 y2 + y1 x2 )gives the addition of points (x1 , y1 ), (x2 , y2 ) ∈ S1 . Noticethat the rational points (i.e. the pythagorean triples) area subgroup. The elliptic functions have analogous functional equations which are similarly the basis for addition
formulas for points on elliptic cubics. This is explained in
the text Addition on Cubic Curves.
H.K.

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Symmetries Of Elliptic Functions *[The approach below to elliptic functions follows
that given in section 3 of ”The Genus One Helicoid
and the Minimal Surfaces that led to its Discovery”,
by David Hoffman, Hermann Karcher, and FushengWei, published in Global Analysis and Modern Mathematics, Publish or Perish Press, 1993. For convenience, the full text of section 3 (without diagrams)
has been made an appendix to the chapter on the
Conformal Map Category in the documentation of
3D-XplorMath.]
An elliptic function is a doubly periodic meromorphic function, F (z), on the complex plane C. The subgroup L of C
consisting of the periods of F (the period lattice) is isomorphic to the direct sum of two copies of Z, so that the quotient, T = C/L, is a torus with a conformal structure, i.e.,a Riemann surface of genus one. Since F is well-definedon C/L, we may equally well consider it as a meromorphic
function on the Riemann surface T .
It is well-known that the conformal equivalence class ofsuch a complex torus can be described by a single complex
number. If we choose two generators for L then, withoutchanging the conformal class of C/L, we can rotate andscale the lattice so that one generator is the complex num* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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ber 1, and the other, τ , then determines the conformal classof T. Moreover, τ1 and τ2 determine the same conformalclass if and only if they are conjugate under SL(2, Z).
The simplest elliptic functions are those defining a degree
two map of T to the Riemann sphere. We will be concerned with four such functions, that we call JD, JE, JF,
and WP. The first three are closely related to the classical
Jacobi elliptic functions, but have normalizations that are
better adapted to certain geometric purposes, and similarly WP is a version of the Weierstrass ℘-function, witha geometric normalization. Any of these four functions
can be considered as the projection of a branched covering over the Riemann sphere with total space T , and assuch it has four branch values, i.e., points of the Riemannsphere where the ramification index is two. For JD there
is a complex number D such that these four branch values
are {D, −D, 1/D, −1/D}. Similarly for JE and JF there
are complex numbers E and F so that the branch values
are {E, −E, 1/E, −1/E} and {F, −F, 1/F, −1/F } respectively, while for WP there is a complex number P such thatthe branch values are {P, −1/P, 0, ∞}. The cross-ratio, λ,of these branch values (in proper order) determines τ andlikewise is determined by τ .
The branch values E, F, and P of JE, JF, and WP can beeasily computed from the branch value D of JD (and hencefrom dd) using the following formulas:

E = (D − 1)/(D + 1), F = −i(D − i)/(D + i),
P = i(D2 + 1)/(D2 − 1),and we will use D as our preferred parameter for describing
the conformal class of T . In 3D-XplorMath, D is related
to the parameter dd (of the Set Parameter... dialog) by
D = exp(dd), i.e., if dd = a + ib, the D = exp(a) exp(ib).
This is convenient, since if D lies on the unit circle (i.e., if
dd is imaginary) then the torus is rectilinear, while if D has
equal real and imaginary parts (i.e., if b = π/4) then thetorus is rhombic. (The square torus being both rectilinear
and rhombic, corresponds to dd = i · π/4).
To completely specify an elliptic function in 3D-XplorMath,choose one of JD, JE, JF, or WP from the Conformal
Map menu, and specify dd in the Set Parameter... dialog.
(Choosing Elliptic Function from the Conformal map menu
will give the default choices of JD and a square torus.)
When elliptic functions where first constructed by Jacobi
and by Weierstrass these authors assumed that the latticeof the torus was given. On the other hand, in Algebraic
Geometry, tori appeared as elliptic curves. In this representation the branch values of functions on the torus are
given with the equation, while an integration of a holomorphic form (unique up to a multiplicative constant) is
required to find the lattice. Therefore the relation between
the period quotient τ (or rather its SL(2, Z)-orbit) and thecross ratio lambda of the four branch values has been wellstudied. More recently, in Minimal Surface Theory, it wasalso more convenient to assume that the branch values ofa degree two elliptic functions were given and that the periods had to be computed. Moreover, symmetries became
more important than in the earlier studies.
Note that the four branch points of a degree two elliptic function (also called ”two-division points”, or Zweiteilungspunkte) form a half-period lattice. There are three
involutions of the torus which permute these branch points;
each of these involutions has again four fixed-points andthese are all midpoints between the four branch points.
Since each of the involutions permutes the branch points, ittransforms the elliptic function by a Moebius transformation. In Minimal Surface Theory, period conditions could
be solved without computations if those Moebius transformations were not arbitrary, but rather were isometricrotations of the Riemann sphere—see in the Surface Category the minimal surfaces by Riemann and those named
Jd and Je. This suggested the following construction: As
degree two MAPS from a torus (T = C/L) to a sphere, wehave the natural quotient maps T /−id; these maps havefour branch points, since the 180 degree rotations havefour fixed points. To get well defined FUNCTIONS wehave to choose three points and send them to {0, 1, ∞}.
We choose these points from the midpoints between the
branch points, and the different choices lead to different
functions. The symmetries also determine the points thatare sent to {−1, +i, −i}. In this way we get the most symmetric elliptic functions, and they are denoted JD, JE, JF.

The program allows one to compare them with Jacobi’s
elliptic functions. The function W P = JE ∗ JF has a double zero, a double pole and the values {+i, −i} on certain
midpoints (diagonal ones in the case of rectangular tori).
Up to an additive and a multiplicative constant it agrees
with the Weierstrass ℘-function, but in our normalization
it is the Gauss map of Riemann’s minimal surface on eachrectangular torus.
We compute the J-functions as follows. If one branchvalue is called +B, then the others are {−B, +1/B, −1/B}.
Therefore the function satisfies the differential equations(J 0 )2 = (J 0 (0))2 (J 4 + 1 − (B 2 + 1/B 2 )J 2 ) = F (J),J 00 = (J 0 (0))2 (2J 3 − (B 2 + 1/B 2 )J = F 0 (J)/2).
Numerically we solve this with a fourth order scheme√thathas the analytic continuation of the square root J 0 = J 0 2
built into it:
Let J(0), J 0 (0) be given. Compute J 00 (0) := F 0 (J(0))/2
and, for small z,00
Jm := J(0) + J 0 (0) · z/2 + J 00 (0) · z 2 /8, Jm:= F 0 (Jm )/2,00
J(z) := J(0) + J 0 (0) · z + (J 00 (0) + 2 · Jm
) · z 2 /6.

Finally let J 0 (z) be that square root of F (J(z)) that is00
closer to J 0 (0) + Jm
· z (analytic continuation!). Repeat.
H.K.

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Addition on Cubic Curves *See also the Action Menu of the Parabola “Show Normals
through Mouse Point” and the comments in the ATO.
As an introductory example view the unit circle as a group.
Then the addition of angles φ ∈ (R mod 2π) gets translated
via the parametrizationx = cos(φ), y = sin(φ)
into(x1 , y1 ) ⊕ (x2 , y2 ) := (x1 x2 − y1 y2 , x1 y2 + x2 y1 ).
Once this addition law is known one does not need thetranscendental functions sin and cos to “add” points on
the circle. Even to do this addition with ruler and compass
is easy. And it is amusing to note that the Pythagorean (orrational) points of the circle are a subgroup, e.g. (3/5, 4/5)⊕(12/13, 5/13) = ((36 − 20)/65, (15 + 48)/65).
In a similar way there exists a geometric addition on cubic
curves, and if the cubic is parametrized with appropriate
functions (defined either on C, or on C/2πZ, or on C/Γ, Γa lattice in C) then the well known addition in the domain
is, under the special parametrization, the same as the geometric addition on the cubic. The simplest instance iswhen the cubic is the graph of a cubic polynomial without* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
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quadratic term: y = x3 + mz + c. Then, if we have twopoints (x1 , y1 ), (x2 , y2 ) on this cubic and join them by aline, this line intersects the graph in a third point (x3 , y3 )such that x1 +x2 +x3 = 0. This gives a geometric definitionof addition on this cubic graph.

Addition on a polynomial cubic graph without
quadratic term. Every line intersects so that x1 +x2 + x3 = 0. Note discrete subgroup.
Similarly, let us map C bijectively onto the Cuspidal Cubic by z 7→ (z 2 , z 3 ). In this case, if we have z1 + z2 +z3 = 0, then the tangents at the three points (zj2 , zj3 )are concurrent—we have seen this as a property of the
Parabola, because the Cuspidal Cubic is the evolute of
the Parabola. One can also see the previous colinearity as
reflecting addition, because the three points (zj2 , zj3 ), j =1, 2, 3, of this cubic lie on a line if 1/z1 + 1/z2 + 1/z3 = 0.

Addition on the cuspidal cubic z 7→ (z 2 , z 3 ). Notethe discrete subgroup. If z1 + z2 + z3 = 0, then
the tangents at these three points are concurrent.
If 1/z1 + 1/z2 + 1/z3 = 0, then these three points
lie on a straight line.
The next case is the group C/2πZ. The trigonometricfunctions identify points in C mod 2π. We map this group
to a cubic curve by x := tan(z/2), y := sin(z), so thaty = 2x/(x2 + 1) and this cubic is again a graph. The
addition theorems tan(z + w) = (tan(z) + tan(w))/(1 −tan(z) tan(w)) and sin(z+w) = sin(z) cos(w)+cos(z) sin(w)with cos(z) = 1 − 2 sin(z/2)2 = 1 − sin(z) · tan(z/2) again
give an addition on this cubic graph: it is a geometric addition because the three points (xj , yj ) lie on one line iff z1 +z2 + z3 = 0. The name “geometric addition” is even morejustified because the third point (x3 , y3 ) can be constructedwith ruler and compass from the other two. In fact, for repeated additions a ruler suffices: As a preparation we haveto add to all points in sight the 2-division point (∞, 0) =(tan(π/2), sin(π)) as follows: (x, y) ⊕ (∞, 0) = (−1/x, −y).
One needs ruler and unit circle for this. Then the lines
through (x1 , y1 ), (x2 , y2 ) and (x1 , y1 ) ⊕ (∞, 0), (x2 , y2 ) ⊕(∞, 0) intersect in the point (x3 , y3 ) = −(x1 , y1 ) ⊕ (x2 , y2 ).

P3 := - (P1 + P2)

!,0)

P1+(

P2

!,0)

P2+(

P1

!,0)

P3+(

Addition group S1 on a cubic that is the graphof x 7→ y = 2x/(x2 + 1), parametrized by x :=tan(z/2), y := sin(z). Note the finite discrete
subgroup. (∞, 0) = (tan(π/2), sin(π)), the point
at infinity, is the only point of order 2.
So far we have seen the circle part of the cylinder group
C/2πZ. To see a generator of the cylinder we replace t, x, y
by it, ix, iy, then we obtain x := tanh(z/2), y := sinh(z),so that y = 2x/(1 − x2 ). The component of the graphthrough 0 is a subgroup isomorphic to R. It represents
one generator of the cylinder. The other two components
represent the opposite generator with one point missing:the 2-division point opposite 0 is the point (∞, 0) on thiscubic. This allows the same ruler construction of addition
as before, except for a sign change in (x, y) ⊕ (∞, 0) =(+1/x, −y) (because 1/i = −i).

P3 := - (P1 + P2)

!,0)

P1+(

!,0)

P2+(

P2

P1

!,0)

P3+(

(x,y)+(!,0) = (1/x,-y)

Addition group R∪R on a cubic that is the graphof x 7→ y = 2x/(1 − x2 ) and is parametrized by
x := tanh(z/2), y := sinh(z). (∞, 0) is the only
point of finite order. Note the infinite discrete
subgroup with one finite subgroup of order 2.
Finally we come to the group C/Γ. The parametrizing
functions of the previous example, tan(z/2), sin(z), must

be replaced by Γ-invariant, “doubly periodic” functions,also called elliptic functions. The simplest of these arethose of degree two, as maps from the torus T 2 := C/Γ
to the Riemann sphere S2 = C ∪ {∞}. Two facts areimportant:(i) Pairs of such functions satisfy cubic equations such as(w2 + 1)v = const · (v 2 − 1)w. The solution set of any
cubic equation is called a cubic curve.
(ii) There are addition formulas, analogous to those forsin and cos.
They determine the pair (v(z1 + z2 ), w(z1 + z2 )) fromthe pairs (v(z1 ), w(z1 )) and (v(z2 ), w(z2 )).
It turns out that these addition formulas are again “geometric” as in the previous cases, namely, the three pairs(v(z1 ), w(z1 )), (v(z2 ), w(z2 )), (−v(z1 + z2 ), −w(z1 + z2 ))lie on a line. Therefore we can again define addition on
the cubic geometrically:
Join the points to be added by a line and take the third
point of intersection with the cubic as the negative of
the sum.
The addition formulas are simple enough so that the geometric addition is again a “ruler and compass construction”. The compass is only needed to add 2-division points
as in the previous case, all further additions can be doneby intersecting lines only.

v

P3+T
P2+T
P1 = (w,v)

w
P1+T = (-1/w,-v)
P2P3 = - (P1+P2)
2-division point T = (!,0), T+T = 0(Notice the discrete subgroup)
Addition on a general cubic:
w1 + w2 v1 − v2 1 + w1 w2 v1 − v2
(w1 , v1 ) (w2 , v2 ) = (·,·)
1 − w1 w2 v1 + v2 1 − w1 w2 1 − v1 v2The elliptic functions v, w, parametrizing the above cubic
curve have numerous properties that can be used to define
them. For example, they are numerically accessible, since
they are solutions of the following system of differentialequations (compare tan0 / tan = 1/ cot + cot):
1v0= w0 (0)
−w ,
v
w


w0
1
= v 0 (0)+v ,wv
with v 0 (0)/w0 (0) = −2 for the above cubic. These imply
functional equations for v, w so that more similarities withthe trigonometric case, like (sin0 )2 = 1 − sin2 , become apparent: 0 22v
1= w0 (0)2
−w
v
w
!
2

1
0
2
+w −4
= w (0)
w

2 !
1
0
2
= v (0)−v− 4w0 (0)2 ,
v


0
2
w (0)
and hence:(v 0 )2 = v 0 (0)2 (1 − v 2 )2 − 4 0 2 · v 2 .
v (0)Every differential equation(f 0 )2 = F (f ) implies 2f 00 = F 0 (f ).
The first order equation determines f 0 only up to sign while
the second order equation determines f 00 uniquely, in particular for trigonometric and elliptic functions.
H.K.
Go To Planar TOC

Folium of Descartes *This is a famous curve with a long history (see e.g.
http:\\www-history.mcs.st-andrews.ac.uk/Curves/Curves.html). The curve is the solution set ofthe equation
x3 + y 3 = 3axy.
One can see that the solutions for different a differ only
by scaling, namely divide the equation by a3 and replacex/a, y/a by x, y.
The two most frequently given parametrizations are:3t2
3t
, y(t) =,x(t) =
1 + t3
1 + t3sin 2ϕr(ϕ) =, −π/4 < ϕ < 3π/4.
33
sin ϕ + cos ϕ
The first parametrization has the disadvantage that at
t = −1 the denominator vanishes, the curve jumps “fromminus infinity to plus infinity”, while the important double point at 0 ∈ R2 is left out (or given by t = ∞). Thiscan be remedied by the transformation u = 1/(1 + t), t =* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Go To Planar TOC

−1 + 1/u, which changes the parametrization tou2 − u3
u − 2u2 + u3x(u) =, y(u) =,1 − 3u + 3u2
1 − 3u + 3u2− ∞ < u < ∞.
H.K.

Go To Planar TOC

About Implicit Curves in the Plane *Compare Implicit Surfaces in Space
There are three principal methods for describing curves in
the plane:a) As parametrized curves c(t) = (x(t), y(t)) with x, y :(t0 , t1 ) 7→ R. For example the unit circle can be given asx(t) = R · cos(t), y(t) = R · sin(t), t ∈ [0, 2π].
b) As the graph y = F (x) of a function F : [x0 , x1 ] 7→ R.
For example the upper unit semi-circle
can be given as the√graph of the function F (x) = 1 − x2 for x ∈ (−1, +1).
c) Implicitly as a level set {f = c} of a function f : R2 7→ R.
For example the unit circle is the level {f = 1} of thefunction f (x, y) = x2 + y 2 .
Implicit Curves in 3DXM:
Cassini Ovals f (x, y) = ((x−aa)2 +y 2 )((x+aa)2 +y 2 )−bb4
Tacnodal Quartic f (x, y) = y 3 + y 2 − x4
Teissier singular Sextic f (x, y) = (y 2 − x3 )2 − x5 · y
Userdefined Implicit Curves: available
Parametrized Curves with Level Functions:Cuspidal Cubic f (x, y) = 27aa · y 2 − 4(x + bb)x2
Nodal Cubic f (x, y) = y 2 − (1 − x)x2
Clearly, method b) can easily be written as a special caseof methods a) or c) by using for a) the “graph parametriza* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Go To Planar TOC

tion” x(t) = t, y(t) = F (t), and by using for c) the trivial
level function f (x, y) = y − F (x) and {f = 0}.
However, implicit curves {f (x, y) = c} really give a different and somewhat richer class of objects than are given
by explicit parametrization. For example, level sets may
have several components; also, one is more interested in
the singularities of level sets. In differential geometry one
usually assumes that parametrized curves are without singularities, while in algebraic geometry the singularities ofthe level sets of polynomials are a major subfield of interest. The Tacnodal Quartic and the Teissier Sextic are
examples in 3DXM.
Up to release 10.6 there are only a small number of implicit curves preprogrammed into 3DXM. What actuallygets drawn are the solutions of the equation f (x, y) = f f
with xmin ≤ x ≤ xmax , ymin ≤ y ≤ ymax (where theselimits can be set in the Settings Menu, dialog entry Set
t,u,v,ranges... and, as always, ff can be set in the dialog
entry Set Parameters, Modify Object.
Note that user-defined implicit curves can be entered.
The default morphs of the implicit curves vary the parameter ff , so that, what one sees is a family of level curvesof f . It may be helpful to think of the function f (x, y) as
giving the “height above sea-level” at the point (x, y), in
which case the levels {f (x, y) = ff } are just the level lines
one is used to from topographic maps. If one chooses fromthe Animation Menu the entry Color Morph, the program
will draw such a topographic map with each level line adifferent color.
Some parametrized curves are provided with level functions. For these the Animation Menu has the entry Morph
Level Lines. In the Cassini case this morph looks betterwith morphing parameter bb = ff 1/4 .
Note that, while a parametrized curve depends on parameters only if the author chooses to embed it in some family, implicit curves always come naturally as 1-parameterfamily of curves. These families have been used to study
singularities of curves via limits of nonsingular curves.
Tangents, Normals and Curvature
The gradient of the (height) function f is a vectorfieldalong and normal to the level lines. Therefore we have,even without parametrizing the curve, normals and tangents:
grad fn=, t = (−ny , nx ).
|grad f |Assuming we had a parametrized curve with unit normal
and tangent fields n, t then the formula ṅ(s) = κ(s) · ċ(s)holds whether or not s is arc length parameter. This implies for our vector fields
κ = h∇t n, ti = hessef (t, t)/|grad f |.
There is an Action Menu entry Osculating Circles At
Mouse Point. It shows several level curves and the osculating circle to the level curve at the mouse point.
R.S.P.

Go To Planar TOC

Cassinian Ovals *Level function in 3DXM:
f (x, y) := ((x − aa)2 + y 2 ) · ((x + aa)2 + y 2 ) − bb4
The default Color Morph varies bb = ff 1/4 instead of ff .
The Cassinian Ovals (or Ovals of Cassini) were first studied
in 1680 by Giovanni Domenico Cassini (1625–1712, aka
Jean-Dominique Cassini) as a model for the orbit of the
Sun around the Earth.
A Cassinian Oval is a plane curve that is the locus of
all points P such that the product of the distances of P
from two fixed points F1 , F2 has some constant value c, or
P F1 P F2 = c. Note the analogy with the definition of anellipse (where product is replaced by sum). As with the ellipse, the two points F1 and F2 are called foci of the oval. If
the origin of our coordinates is the midpoint of the two fociand the x-axis the line joining them, then the foci will havethe coordinates(a, 0) and (−a, 0). Following convention,√
b := c. Then the condition for a point P = (x, y) to lie on
the oval becomes: ((x − a)2 + y 2 )1/2 ((x + a)2 + y 2 )1/2 = b2 .
Squaring both sides gives the following quartic polynomial
equation for the Cassinian Oval:((x − a)2 + y 2 )((x + a)2 + y 2 ) = b4 .
When b is less that half the distance 2a between the foci,i.e., b/a < 1, there are two branches of the curve. When
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Go To Planar TOC

a = b, the curve has the shape of a figure eight and is
known as the Lemniscate of Bernoulli.
The following image shows a family of Cassinian Ovals
with a = 1 and several different values of b.

In 3D-XplorMath, you can change the value of parameterb = bb in the Settings Menu → SetParameters. An animation of varying values of b can be seen from the Animate
Menu → Color Morph.
Bipolar equation: r1 r2 = b2
Polar equation: r4 + a4 − 2r2 a2 cos(2θ) = b4
A parametrization for Cassini’s oval is r(t)·(cos(t), sin(t)),
p22r (t) := a cos(2t) + (−a4 + b4 ) + a4 (cos(2t))2 ,t ∈ (0, 2π], and a < b. This parametrization only generates
parts of the curve when a > b.

By default 3D-XplorMath shows how the product definition of the Cassinian ovals leads to a ruler and circle
construction based on the following circle theorem about
products of segments:

Cassinian Ovals as sections of a Torus
Let c be the radius of the generating circle and d the distance from the center of the tube to the directrix of thetorus. The intersection of a plane c distant from the √torus’
directrix is a Cassinian oval, with a = d and b2 = 4cd,
where a is half of the distance between foci, and b2 is theconstant product of distances.
Cassinian ovals with a large value of b2 approch a circle,
and the corresponding torus is one such that the tube radius is larger than the center to directrix, that is, a selfintersecting torus without the hole. This surface also approaches a sphere.
Note that the two tori in the figure below are not identical.

Arbitrary vertical slices of a torus are called Spiric Sections. In general they are not Cassinian ovals.

Proof: Start with the equation of a torusp( x2 + y 2 − d)2 + z 2 = c2 .

Insert y = c, rearrange and square again:p
x + z + d = 2d x2 + c2 , (x2 + z 2 + d2 )2 = 4d2 (x2 + c2 ).
2

2

2

Now multiply the factors of the implicit equation of an
Cassinian oval and rearrange((x − a)2 + y 2 ) · ((x + a)2 + y 2 ) = b4 ,(x2 − a2 )2 + y 4 + 2y 2 (x2 + a2 ) = b4 ,(x2 + y 2 )2 + 2a2 (y 2 − x2 ) = b4 − a4 .
These two equations match because of a = d, b2 = 2dc,
after rotation of the y-axis into the z-axis.
Curves that are the locus of points the product of whose
distances from n points is constant are discussed on pages
60–63 of Visual Complex Analysis by Tristan Needham.
XL.

Go To Planar TOC

User Defined Plane Curves in 3DXM *Selection of one of these entries will open a dialog to enter
the data the user wishes. Default examples are provided.
User Cartesian: enter x(t) := . . . , y(t) := . . ..
User Polar: enter r(t) := . . . , ϕ(t) := . . ..
The curve is (r(t) cos(ϕ(t)), r(t) sin(ϕ(t))).
User Graph: enter y(t) := . . ., implied is x(t) := t. The
curve (t, y(t)) is the Graph of the function y. Three approximations are shown:Taylor, Interpolation, Fourier.
These are the explicitly parametrized user curves. The
standard decorations are available: Parallel Curves, Generalized Cycoids, Osculating Circles, Family of Normals
and their Envelope, Caustics from Rotated Normals.
User Implicit: enter level function F (x, y) := . . ..
See the separate text: Implicit Planar Curves above, available also from the Documentation Menu (after selection ofuser defined implicit curve).
User Curvature: enter the curvature function κ(s) := . . ..
The program assumes that the parameter s is arc length.
See also the text below: User Curves By Curvature, again
available from the Documentation Menu of 3DXM.
H.K.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Go To Planar TOC

User Defined by Curvature *A planar curve (parametrized by arc length) can be reconstructed from its curvature function t 7→ κ(t) as follows:Rt(1) take the antiderivative of κ, α(t) := κ(σ)dσ,(2) choose an initial point p, an initial tangent vector ċ(0)and an orthonormal basis e1 = ċ(0), e2 ,so the definition of curvature (namely κ := |c̈|, plus a sign
convention) implies that,(3) ċ(t) = e1 · cos α(t) + e2 · sin α(t).
Then one more integration,Rt(4) c(t) = p + 0 ċ(σ)dσ,
determines the curve. This description explains why thecurvature is also called the “rotation speed” of the tangent
vector field ċ(t).
In 3D-XplorMath one can select User Curvature. A dialog
box opens and one can enter the desired curvature function. The initial point p is taken as the origin and theinitial tangent is taken as the unit vector in the positive
x-direction.
The parameter gg in this case defines a “precision divisor”, that can be between 1 and 30. The size of the* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Go To Planar TOC

subintervals used in approximating the above integrals isδ := (tM ax − tM in)/(tResolution − 1) if gg = 1, and ingeneral it is δ/gg. If the curvature function κ becomesvery large somewhere, and in particular if it is infinite atan endpoint of the interval [tM in, tM ax], it is a good idea
to use a fairly large value of gg to counteract the resulting
numerical inaccuracies that will occur in the evaluation of
the integrals.
Note that 3D-XplorMath offers the same Action Menu Entries as for explicitly parametrized curves. For example try
the caustics.
R.S.P.

Go To Planar TOC

Graphs of Functions, Graphs of Planar Curves *1-dimensional functions are always visualized by drawing
their graphs {(x, f (x)}, while planar curves, although usually described as maps: t ∈ [t0 , t1 ] 7→ c(t) ∈ R2 , are nevertheless visualized by their image, the set of their values: {c(t) ∈ R2 ; t ∈ [t0 , t1 ]}. The graph allows to see,for each argument x, the value f (x); the image shows noconnection between the points of the curve and the arguments t ∈ [t0 , t1 ]. This leads to difficulties if one wants tovisualize the connection between a curve and “its” component functions, because the image of a curve does notshow traces of the map that was used to draw it. Therefore, strictly speaking, the image of a curve does not have“its” component functions. We need to have some specificparametrization t 7→ c(t) of the curve in mind before wecan speak of “the” component functions of the curve, or
better of its representing map t 7→ c(t).
In 3D-XplorMath many planar curves have an Action Menu
entry: Project Curve to x-y-Axes. The resulting demo
shows the curve and the graphs of the component functionsof the curve that the program uses for drawing the imageof the curve. A moving point on the curve and corresponding moving points on the graphs are shown. The moving
point on the curve demonstrates the parametrization thatis employed by the program.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Go To Planar TOC

Epicycloid with graphs of component functions.
We have enough dimensions to also show the graph of aparametrized planar curve, {(t, c(t)); t ∈ [t0 , t1 ]} ⊂ R3 .
This is not used much, except for world lines in Relativity Theory. However, the graphs of the component functions of the curve are just orthogonal projections of thegraph of the curve! So, after getting used to the unusualsight of the graph of a curve, see in 3D-XplorMath the
Action Menu entry Show Planar Curve as Graph, sucha demo may help to bridge the gap between the graphrepresentation of 1-dim functions and the image representation of higher dimensional maps. Actually, in R3

there are only two more cases for which the dimensions fit,
the frequently used graphs of real valued “height” functions h from R2 : {(x, y, h(x, y)); (x, y) ∈ R2 , h(x, y) ∈ R}
(think of “mountains”) and the mentioned graphs of planar curves: {(t, c(t)) ∈ R3 ; t ∈ [t0 , t1 ]}, which are curves
in R3 that frequently look like deformed helices.

Epicycloid in y-z-plane and its graph in R3 .
This image is easier to interprete when the point of thecurve moves and when the visualization is in stereo.
H.K.

Go To Planar TOC

Space Curves
Go To Page 1
Explicitly Parametrized Curves (Click the Names)
The Helix
Torus Knots
Genus Two Knots
Cinquefoil Knot
Trefoil Knot
Figure 8 -, Square - and Granny Knots
Morph Through Prime Knots 31 , 41 , 52 , 61 72
Lissajous Curves, e.g. Prime Knot 74
V. Jones’ Braid List of all Knots up to 10 Crossings
Implicitly Defined Curves
Intersection of Two Cylinders
Userdefined Implicit Space Curves
Spherical Curves
About Spherical Curves
Loxodromes (Constant Angle with Meridians)
Viviani Curve (Intersect Sphere and Cylinder)
Spherical Cycloids
Spherical Lemniscates
Spherical Ellipses
Curves Defined By ODEs
Space Curves of Constant Curvature On Cylinders
Space Curves of Constant Curvature on Tori
Continue TOC next page

Closed Space Curves of Constant Curvature
Autoevolutes
Closed Space Curves of Constant Torsion
From Angular Velocity to Rigid Motion
Solid Body Motion: Euler’s Polhode
Userdefined by Curvature and Torsion
Other Userdefined Parametrized Space Curves
Space Curve TOC

The Helix *The helix is the simplest nonplanar space curve. It canbe translated along itsself by a group of isometries (calledscrew motions) and therefore has its geometric invariants– the curvature and the torsion – constant.
Our (circular) helix as a parametrized curve c is given (withdefaults aa = bb = 1.5, cc = 0.25) asc(t) = (aa cos(t), bb sin(t), cc(t − tmin) − 3).
In the default Morph we extend the helix like pulling a bed
spring and therefore want to keep its length constant. To
do this we compute f := (aa2 + cc2 )−1/2 and show the
reparametrized curve c(f · t).
Before we do the morph we select from the Action Menu
Show As Tube. These tubes are either made with the’Frenet Frame’ or with a ’Parallel Frame’. The tube behaves like an elastic rod if we choose in the Action Menu
Parallel Frame. The default morph now shows (at theright end, the left is kept fixed) that the tube also twists
around itsself while it is extended. When this occurs withelectrical wires or water hoses that are pulled sideways
from their coil, it is a well known and annoying phenomenon.
H.K.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

Torus Knots *Torus knots are quite popular space curves because they
represent the simplest way to write down knotted curves
in R3 . Our knots are parametrized as

aa + bb · cos(dd · t) · cos(ee · t)c(t) :=  aa + bb · cos(dd · t) · sin(ee · t) 
cc · sin(dd · t)with defaults aa = 3, bb = 1.5, cc = 1.5, dd = 5, ee = 2.
The default Morph changes the torus size. If, before themorph, one chooses in the Action Menu Show As Tube andParallel Frame then one notices that the twisting of the
tube (see the ATO of the helix for more details) is clearlyvisible already for rather small changes of the shape of thetorus.
The Action Menu has also the entry Show Dotted Torus.
Selecting it adds the torus to the picture. This is more
spectacular when viewing in Anaglyph Stereo Vision,
through red/green filter glasses. Observe that our braingets these several thousand dots sorted out into corresponding pairs of red and green dots that then form thetorus surface in R3 - and this seems to happen instantly.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

The best method to get a feeling for the curvature of aspace curve is to select in the Action Menu Show Osculating Circles & Evolute. The Radius r of the circle
is the radius of curvature of the curve at the current pointand κ := 1/r is called the curvature (at that point). The
direction from c(t) to the midpoint of the osculating circle
determines always the direction of the second basis vectorof the Frenet frame.
If one uses the Parallel Frame, then one has to represent
the curvature by a vector of length κ in the plane spannedby the two normal vectors of the Parallel frame. If one has
selected, in the Action Menu, Parallel Frame and clicks
Show Repére Mobile then this curvature vector is drawn,
together with its past history, in each normal plane. – The
last entry in the Action Menu, Show Frenet Integration
does the opposite: if the curvature vector function is givenin the initial normal plane then the demo reconstructs thecurve by integrating the Frenet equation.
H.K.

Space Curve TOC

Genus Two Knots *Torus Knots , see the previous entry, are the most easily
described knots and, in particular when viewed on a torus,
they are also very easy to visualize.
If one wants to visualize other knots on some surface, one
needs more complicated surfaces than tori. From this pointof view the next simplest knots can be put on a genus 2
surface. The surface we chose looks like two tori which arejoined by a small handle. (The size of these tori is controlled by the parameters aa and bb as for torus knots.)
The surface is implicitly described by an equation (see implicit surfaces in the surface category) and can be madefatter by increasing ff. As examples of genus 2 knots wechose the connected sums of two (dd, ee) - torus knots. The
sign of hh controls whether the two torus knots are connected with reflectional symmetry or with 180◦ rotational
symmetry. The two simplest examples are the Square Knotand the Granny Knot where two (3, 2) - torus knots (=Trefoil Knots) are connected with the two types of symmetry,see Figure 8 -, Square - and Granny Knots .
The sum of the two torus knots is first constructed outside
the surface, then projected onto the surface and finallysmoothed with a length minimizing algorithm. The result
is good enough for tubes made with a Parallel frame, but
the tube from the Frenet frame is not smooth.
– H.K.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

Cinquefoil Knot *Parametric Formulas for the Cinquefoil Knot:
P.x := (2 − cos(2 t/(2 aa + 1))) · cos(t);P.y := (2 − cos(2 t/(2 aa + 1))) · sin(t);P.z := − sin(2 t/(2 aa + 1));
The choice aa = 1 gives a Trefoil knot, aa = 2 the Cinquefoil, and in general aa = k gives the (2k+1)-foil knot (theprogram rounds aa before using it). The parameter range
for t should be 0 to (4 k + 2) π. If you change aa in the Set
Parameters... dialog, then these values of tMin and tMax
are set also, but you can change them later in the Set t,u,v
Ranges... dialog.
A nice animation of the Cinquefoil knot can be obtainedby first choosing Show As Tube from the Action menu,
Anaglyph Stereo Vision from the View menu, and thenRotate from the Animation menu.
R.S.P.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

Trefoil Knot *Parametric formulas for the Trefoil Knot:x = 0.01 (41 cos(t) − 18 sin(t) − 83 cos(2 t)−
83 sin(2 t) − 11 cos(3 t) + 27 sin(3 t)) · hh
y = 0.01 (36 cos(t) + 27 sin(t) − 113 cos(2 t)+
30 sin(2 t) + 11 cos(3 t) − 27 sin(3 t)) · hhz = 0.01 (45 sin(t) − 30 cos(2 t) + 113 sin(2 t)−
11 cos(3 t) + 27 sin(3 t)) · hh
The Trefoil knot, Figure 8 -, Square - and Granny Knots
displayed by 3D-XplorMath are all harmonic or Fourier
knots. That is they are parametrized using finite Fourier
series for all three coordinates. The particular parametrizations are taken from the 1995 PhD thesis of Aaron Trautwein at The University of Iowa.
Compare the rotation of the Frenet frame along this trefoil
knot (defined with harmonic polynomials) and along thetrefoil that results when you select Torus Knot with theparameters (dd=3, ee=2): Near the points on the torus
knot where the curvature is very small, the rotation speedof the Frenet frame is large. - The Trefoil Knot can beshown with a Satellite Knot, default dd = 55, ee = 2.
In stereo mode a Möbius band bounded by the Trefoil Knotis added. Their handedness depends on the sign of hh.
R.S.P.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

Figure 8 Knot, Granny Knot, Square Knot *The Trefoil knot, Figure 8 Knot, Granny Knot, Square
Knot, displayed by 3D-XplorMath are all harmonic or Fourier knots. That is they are parametrized using finite
Fourier series for all three coordinates. The particularparametrizations are taken from the 1995 PhD thesis of
Aaron Trautwein at The University of Iowa.
Satellite Knots can be added in the Action Menu to thesefour knots.
The Figure 8 Knot is an alternating prime knot with minimal crossing number 4. It is the next simplest knot after the Trefoil Knot (see Morph Through Prime Knots
31 , 41 , 52 , 61 72 ).
Parametric formulas for the Figure 8 Knot:x =(32 cos(t) − 51 sin(t) − 104 cos(2t) − 34 sin(2t)+
104 cos(3t) − 91 sin(3t))/100
y =(94 cos(t) + 41 sin(t) + 113 cos(2t) − 68 cos(3t)−
124 sin(3t))/140z =(16 sin(t) − 138 cos(2t) − 39 sin(2t) − 99 cos(3t)−
21 sin(3t))/70
The Granny Knot and the Square Knot are not prime,
both are sums of two Trefoil Knots. The Square Knot has* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

a mirror symmetry so that one Trefoil is left handed theother right handed. The Granny Knot is the sum of twosame-handed Trefoil Knots.
Parametric formulas for the Granny Knot:x =(−22 cos(t) − 128 sin(t) − 44 cos(3t) − 78 sin(3t))/80
y =(−10 cos(2t) − 27 sin(2t) + 38 cos(4t) + 46 sin(4t))/80
z =(70 cos(3t) − 40 sin(3t))/100
Parametric formulas for the Square Knot:x =(−22 cos(t) − 128 sin(t) − 44 cos(3t) − 78 sin(3t))/100
y =(11 cos(t) − 43 sin(3t) + 34 cos(5t) − 39 sin(5t))/100z =(70 cos(3t) − 40 sin(3t) + 18 cos(5t) − 9 sin(5t))/100
R.S.P.

Space Curve TOC

Morph Through Five Prime Knots *A prime knot is a knot that cannot be written as the knot
sum of smaller knots. For example, the Square Knot and
the Granny Knot are not prime since each is a sum of two
Trefoil Knots. There are 14 prime knots with at most
7 minimal number of crossings. They have been handdrawn so often that they have assumed an esthetically defined standard shape. Of these first 14 prime knots thefollowing ones are in a morphing family, the prime knots
31 , 41 , 52 , 61 , 72 . Choose dd = 3 and 0 ≤ f f ≤ 4.3 in Set
Morphing and the program will deform the Trefoil Knotthrough the following images:

If one chooses dd = 5 and 0 ≤ f f ≤ 2.3 in Set Morphing then the program will deform the (5,2)-Torus Knotthrough the following images of the prime knots 51 , 62 , 75

The prime knot 74 is the default Lissajous space curve.
There are 249 prime knots with at most 10 minimal num* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

ber of crossings. One can visualize those via the Space
Curves Menu entry: V.Jones Braid List.
The notion of prime knot is important because Horst Schubert proved that the decomposition of a knot as knot sum(= connected sum) of prime knots is unique. The knot
invariants are a good way to check whether a given knot
is a prime knot.
There is an easy sufficient criterion that guarantees thatthe knot under consideration cannot be drawn with fewer
crossings. First we define alternating and reduced alternating knots: if the thread of the knot passes alternatingly
through overcrossings and undercrossings then the knot iscalled alternating. For example, if we twist a circle into afigure 8 we obtain an alternating trivial knot. In this case
we observe an easily recognizable property of the crossingin the knot diagram: if the crossing is removed the knot diagram decomposes into two components. A crossing withthis property is called an isthmus. Clearly, one can rotate
one component of the knot diagram through 180 degrees,i.e. untwist and thereby remove the isthmus to obtain arepresentation with fewer crossings. An alternating knot
without an isthmus is called a reduced alternating knot.
Theorem : Reduced alternating knots cannot be represented with fewer crossings, they are always non-trivial.
All prime knots with at most 7 crossings are reduced alternating knots.
H.K.
Space Curve TOC

Lissajous Curves, e.g. the Prime Knot 74 *Lissajous curves are a popular family of planar curves,
resp. space curves. They are complicated enough to beinteresting, but regular enough to be esthetically pleasing.
They are described by simple formulas:x(t) := aa · sin(2π · dd · t)y(t) := bb · sin(2π · ee · t + gg)
z(t) := aa · sin(2π · f f · t + cc)
In 3DXM the parameters dd, ee, f f are rounded to integers so that the curves are closed on the interval [0, 1].
The default morph varies the phase gg from 0 to π/2. –The Lissajous curves are also physically interesting, theydescribe the joint motion of orthogonal uncoupled oscillators (x(t), y(t), z(t)) with different frequencies.
A prime knot is not a knot sum of smaller knots. E.g.
Square Knot and Granny Knot are not prime: each is asum of two Trefoil Knots. There are 14 prime knots withthe minimal number of crossings ≤ 7, see the documentation About This Object for V.Jones Braid List. The
4th 7-crossings-knot, the prime knot 74 , is our default Lissajous space curve, (dd, ee, f f, gg) = (2, 3, 7, π/2). – Other
alternating examples are: (dd, ee, f f ) = (2, 5, 13), (4, 3, 23):

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

There are 249 prime knots with at most 10 minimal number of crossings. One can visualize those via the Space
Curves Menu entry: V.Jones Braid List.
The notion of prime knot is important because Horst Schubert proved that the decomposition of a knot as knot sum(= connected sum) of prime knots is unique. The knot
invariants are a good way to check whether a given knot
is a prime knot. There is no more elementary criterion torecognize a knot as prime.
There is an easy sufficient criterion that guarantees thatthe knot under consideration cannot be drawn with fewer
crossings. First we define alternating and reduced alternating knots: if the thread of the knot passes alternatingly
through overcrossings and undercrossings then the knots iscalled alternating. For example, if we twist a circle into afigure 8 we obtain an alternating trivial knot. In this case
we observe an easily recognizable property of the crossingin the knot diagram: if the crossing is removed the knot diagram decomposes into two components. A crossing withthis property is called an isthmus. Clearly, one can always
rotate one component of the knot diagram through 180 degrees, untwist and thereby remove the isthmus to obtain arepresentation with one less crossings. An alternating knot
without an isthmus is called a reduced alternating knot.
Theorem : Reduced alternating knots cannot be represented with fewer crossings, they are always non-trivial.
H.K.

Space Curve TOC

Braid List Of Prime Knots *A prime knot is a knot that cannot be written as the knot
sum of smaller knots. For example, the Square Knot and
the Granny Knot are not prime since each is a sum of two
Trefoil Knots. There are 249 prime knots with at most 10
minimal number of crossings. In 3DXM we use the braid
representation of knots. Vaughn Jones gave this list to
one of us in the 80s. The usual hand drawn versions are
prettier than the braids:

Copied from the article ’Prime Knot’ in Wikipedia.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

The above first 14 prime knots are all so called alternatingknots: if one follows the thread of the knot then one passes– alternatingly(!) – overcrossings and undercrossings. A
knot that is represented as a “reduced” alternating knot
cannot be drawn with fewer crossings, in particular: a reduced alternating knot is always non-trivial. If one twists
a circle to a figure 8 then one obtains a non-reduced alternating knot that is clearly trivial. Similarly, one cantake the alternating Granny Knot in 3DXM and turn oneof the Trefoil parts 180 degrees around the horizontal axis.
One obtains an alternating knot with an additional crossing in the middle. Again, this knot is not reduced because
by cutting out the new crossing the knot diagram decomposes into two components. Such an easily recognizablecrossing is called an isthmus. One can always untwist an
isthmus and obtain a knot with one less crossing. A knot
diagram without an isthmus is called reduced.
The notion of prime knot is important because Horst Schubert proved that the decomposition of a knot as knot sum(= connected sum) of prime knots is unique.
All torus knots are prime knots. The genus 2 knots in
3DXM are sums of two torus knots.
The space curve ”Morph Prime Knots 5 4 3” has a default
morph that runs through the prime knots 3.1, 4.1, 5.2, 6.1,
7.2. If one changes dd from 3 to 5 then the ff-morph runs
through 5.1, 6.2, 7.5. The prime knot 7.4 is shown as a
Lissajous knot.
H.K.

Space Curve TOC

The Intersection of Two Cylinders *The image shows the space curve defined implicitly as theintersection of the two cylinders:y 2 + z 2 = ff
and(cos(aa)x + sin(aa)y)2 + (z − cc)2 = gg.
These two cylinders are made visible by displaying a random set of dots on each of them. In the Action Menu one
can choose to put more random dots on the boundary of
the intersection of the two solid cylinders.
In the default settings the two cylinders touch and thedefault morph rotates one of them by changing aa.
We find it interesting to change the radius of the smaller
cylinder while the cylinders keep touching: morph gg up
to ff while keeping dd = 0, since we compute (behind the
user)
√
√
cc = ff − gg + dd.
At gg = ff the intersection curve degenerates into two
ellipses (for each aa).
The distance between the tangent planes of the two cylinders (at their common normal) is dd.
H.K.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

Userdefined Implicit Space Curves *The exhibit shows the intersection curve of two surfaces,
given by equations F 1(x, y, z) = ff, F 2(x, y, z) = gg.
To see also the surfaces (as dotted point clouds) choose thecorresponding entry in the Action Menu.
The initial dialogue offers three different defaults for thesurfaces given by F 1, F 2:
1.) A conic and a plane with the default morph tilting theplane.
2.) The graph of a function R3 7→ R and a cylinder. This
exhibit can be used to explain extrema under side conditions.
3.) A torus and a tangent plane. This is an example
where the intersection has double point singularities at
those points where the intersection of the surfaces is nottransversal.
By varying these defaults one can create a rich collectionof space curves. (The number of points in the point clouds
cannot be changed in this exhibit.)
Note that the surfaces are only computed inside a sphereof radius ORB around the origin. This parameter can be
set in the first entry of the Settings Menu.
H.K.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

About Spherical Curves *In many ways there is a close analogy between planar Euclidean geometry and two-dimensional spherical geometry.
In the ATO for Spherical Ellipses we translate the sum-ofdistances definition from the plane to the sphere and use
the same arguments as in the plane to construct points andtangents of the curve. Similarly, in the ATOs about Spherical Cycloids, we roll spherical circles along spherical circles.
See also Spherical Lemniscates. Such analogies of course
require basic notions which correspond to each other.
Lines and Triangles
Straight lines in the plane are the shortest connections between their points. On the sphere the shortest connections are great circle arcs that are not longer than half
way around. A line cuts the plane into two congruent
half-planes that are interchanged by the reflection in theline. Similarly, the sphere is cut by the plane of a great
circle into two congruent half-spheres, and the reflection in
the plane interchanges these two half-spheres. Therefore
we speak of the reflection (of the sphere) in a great circle.
These analogies are enough to translate the planar notion
straight line to the spherical notion great circle. Threepoints A, B, C and three shortest connections of lengths* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

a, b, c make a triangle — in the plane or on the sphere.
The angles at the points (or vertices) are denoted α, β, γ.
For the plane, the basic triangle formulas are close to thedefinition of sine and cosine:
Projection theorem:

c = a · cos β + b · cos αSine theorem:b · sin α = hc = a · sin β

Cosine theorem:

c2 = a2 + b2 − 2ab cos γ

Note that the more complicated third formula follows fromthe first two: Use the Sine theorem in the form 0 = b ·sin α − a · sin β and add the square of this to the square of
the Projection theorem. Simplify with cos2 + sin2 = 1 anduse the trigonometric identity cos α cos β − sin α sin β =cos(α + β) = cos(π − γ) = − cos(γ) to obtain the Cosine
theorem.
To derive similar formulas for spherical triangles, use geographic coordinates on the standard unit sphere, withthe polar center at the north pole C := (0, 0, 1). A point
A at spherical distance b from C satisfies hA, Ci = cos b.
Thus, after rotation into the x-z-plane, it has coordinates
A := (sin b, 0, cos b). A third point B at distance a from thepole C and such that the angle ∠ACB equals γ has spherical polar coordinates B := (sin a cos γ, sin a sin γ, cos a).
The spherical cosine formula follows by taking a scalar
product:
hA, Bi = cos c = cos a cos b + sin a sin b cos γ.

The name is justified since a Taylor approximation up tosecond order gives the corresponding formula for the plane.
For more details: note that the graph of the function
x → cos x lies above the graph of the quadratic function x → 1 − x2 /2 and not above any wider parabolax → 1 − x2 /(2 + ). Therefore 1 − x2 /2 is called thequadratic Taylor approximation of cos near x = 0. We
substitute this approximation for x = a, x = b, x = c,and similarly sin x ≈ x, in the spherical cosine formulaand obtain:
cos c ≈ 1 − c2 /2 ≈ (1 − a2 /2)(1 − b2 /2) + ab cos γ, orc2 ≈ a2 + b2 − 2ab cos γ, which is the planar formula.
For a more systematic derivation we use the reflection R
which interchanges C, A and observe that R(B) has the coordinates R(B) := (sin c cos α, sin c sin α, cos c). But R(B)can also be computed from the reflection matrix and thecoordinates of B. Equating the two expressions gives three
formulas between a, b, γ on one side and c, α on the otherside. Of course these formulas hold for any permutation ofA, B, C: − cos b 0 sin b
sin a cos γ 0
10  ·  sin a sin γ  =sin b
0 cos b
cos a

 

− sin a cos b cos γ + cos a sin b
sin c cos α =  sin c sin α 
sin a sin γ
cos a cos b + sin a sin b cos γ
cos c.
We use for these formulas the same names as in the planar

case since an even simpler Taylor approximation simplifies
also the first two equations to their planar counterparts:
Projection theorem:sin c cos α =− sin a cos b cos γ + cos a sin b
Sine theorem:sin c · sin α = sin a · sin γ

Cosine thm:cos c = cos a cos b + sin a sin b cos γ.

A consequence of the first two theorems is theAngle cosine: cos γ = − cos α cos β + sin α sin β cos c.
Application: Go To Platonics
Two-dimensional spherical geometry captures certain aspects of three-dimensional Euclidean geometry. For example, if we project an icosahedron from its center to its
circumsphere then the 20 triangular faces of the icosahedron are mapped to a tessellation of S2 by 20 equilateraltriangles whose angles are 72◦ because five triangles meetat every vertex. From the angle cosine theorem we read
off the edge-length σ of these triangles, with α = 2π/5 wehave for the
Icosahedron: (cos α + cos2 α)/ sin2 α = cos σ.
Given the above spherical tools this is a conceptually very
simple construction.
Osculating Circles
At every point of a twice differentiable curve c on S2 one
can determine its osculating circle: the parametrized circle that agrees with c up to the second derivative at thatpoint. While it is easy to place a ruler next to a curveso that the ruler approximates a tangent line, one cannot
so easily guess these best approximating circles. For allplanar curves and space curves in 3DXM one can choose
Osculating Circles from the Action Menu and one can believe that the resulting images show best approximating
circles. In the case of spherical curves one observes thatthese osculating circles actually lie on the sphere. To understand this, consider the usual osculating circle in R3
and intersect its plane, the osculating plane of the curve c,with S2 ; this intersection circle is clearly a better approximation of the curve than any other circle in this plane
and therefore it is the osculating circle. Although we cannot yet describe the curvature of a curve by a real valued
function, we can already agree that, at each point, a spacecurve is curved as strongly as its osculating circle. We callthe spherical radii of these circles the spherical curvatureradii and we are ready to translate geometric constructions(with curves) from the plane to the sphere.
Parallel curves of a spherical curve c on S2 . We defineη(t) := ċ(t) × c(t)/|ċ(t)| as the oriented spherical unit normal of c. The parallel curve at spherical distance  is thenin complete analogy with the plane given as
Parallel Curves on S2 : c (t) := cos  · c(t) + sin  · η(t).
It is easy to check that the curvature radii of c are obtained by adding  to the curvature radii of c — which iswhat our intuition expects of parallel curves.

Spherical Evolvents (also called involutes). For a physicalrealization of an evolvent attach a string segment to thecurve and move the end point so that the string is always
tangent to the curve, in the forward or in the backward
direction. The Euclidean formula for the backwards evolvent is (assuming |ċ(t)| = 1)e(t) := c(t) − (t − t0 ) · ċ(t), t ≥ t0 .
A remarkable property of the evolvent is that t − t0 is its
curvature radius at e(t).
We translate this construction to the sphere. The formulafor the spherical evolvent is (assume again |ċ(t)| = 1)e(t) := cos(t − t0 ) · c(t) − sin(t − t0 ) · ċ(t).
A short computation shows that the spherical curvatureradius at e(t) is t − t0 , as in the plane. Also, it is true forthe plane and for the sphere that the segment from c(t) toe(t) is orthogonal to ė(t), i.e., this segment is the curvatureradius of the evolvent at e(t).
Spherical Evolutes. For any given (planar or) sphericalcurve c we call the curve of the (planar or) spherical midpoints of the osculating circles of c the (planar or) sphericalevolute of c. In 3DXM this can best be seen in the demo forSpherical Ellipses. In the previous paragraph we have seen
that, in the plane and on the sphere, the evolvent of theevolute of c is this given curve c. Thus, the natural translations of notions from the plane to the sphere continue tohave natural properties.
What is Curvature?
More precisely, what real number should measure the size

of the curvature at one point of the curve c, and which realvalued function should describe the curvatures of c? For
the plane, differential geometers have agreed to take the rotation speed of a unit normal of c as the quantitative size of
its curvature. For example, the rotation speed of the unit
normal n of a circle of radius r (use arc length parametrization) is 1/r, since c(t) = r · (cos(t/r), sin(t/r)), |ċ(t)| = 1
and n(t) = (cos(t/r), sin(t/r)), hence ṅ(t) = (1/r) · ċ(t).
Although this is a good reason for taking 1/r as the curvature of a circle of radius r in the plane, the argument
does not carry over to S2 , since: What is the spherical rotation speed of the spherical normal? Of course we could
also call on the sphere 1/curvature radius the curvatureof the curve. This is not a good idea on S2 since circles ofradius π/2 are great circles, i.e., shortest connections, andwe would expect them to have curvature 0. Fortunately,
there is for the plane another good reason for taking 1/r
as “the” curvature, and this time the corresponding computation can be repeated on S2 . If we imagine a family ofparallel curves then it looks as if the length grows faster if
the curvature is larger.
We can make this intuition more precise with a computation. First, in the plane:c (t) := c(t) +  · n(t), {ċ(t), n(t)} orthonormalċ (t) = ċ(t) +  · ṅ(t), ṅ(t) = κ(t) · ċ(t)d|ċ (t)|=0 /|ċ(t)| = κ(t).
dHere, the second line defines the curvature as the rotation speed of the normal and the third line says that thiscurvature function can also be computed as the change oflength of tangent vectors in a parallel family of curves. Of
course we can do the same computation as in line three forspherical curves:c (t) := cos  · c(t) + sin  · η(t)ċ (t) = cos  · ċ(t) + sin  · η̇(t)η̇(t) = c̈(t) × c(t)|ċ (t)|/|ċ(t)| = hċ (t), ċ(t)i/hċ(t), ċ(t)i =cos  + sin hη̇(t), ċ(t)i/hċ(t), ċ(t)id|ċ (t)|=0 /|ċ(t)| = −hη(t), c̈(t)i/hċ(t), ċ(t)i.
d
Before we take this as the definition of spherical curvaturefor spherical curves we check which function of the radius
we get for circles of spherical radius r:
cr (t) = (sin r cos t, sin r sin t, cos r)dcr (t) = (cos r cos t, cos r sin t, − sin r)
η(t) =drc̈r (t) = −(sin r cos t, sin r sin t, 0),
finally:sin r cos r− hη(t), c̈(t)i/hċ(t), ċ(t)i == cot r.
2
sin r
This is a satisfying answer, since cot r behaves like 1/r forsmall r and cot(r = π/2) = 0 as we expect for great circles.

Now we are ready for the definition and we remark that thehistorical name is geodesic curvature, not the more naive
spherical curvature which we used above.
Definition. The geodesic curvature κg (t) of a sphericalcurve c(t) with spherical unit normal η(t) isκg (t) := −hη(t), c̈(t)i/hċ(t), ċ(t)i.
The Spherical Frenet Equation
Finally we observe that for a unit speed spherical curve c
we have the following natural orthonormal frame along the
curve:(e1 (t), e2 (t), e3 (t)) := (ċ(t), c(t), η(t)),
and the geodesic curvature controls the derivative of thisframe via the following spherical Frenet equation:dċ(t) =
−1 · c(t) − κg (t) · η(t)dtdc(t) = +1 · ċ(t)dtd
η(t) = +κg (t) · ċ(t)dtObserve that the coefficient matrix
0 −1 −κg
 1
0
0 
κg 0
0

is skew symmetric. This fact implies that any solution(e(t), f (t), g(t)) with orthonormal initial conditions stays
orthonormal. This says that t → f (t) is a spherical curveparametrized by arclength (namely: |f˙(t)| = |e(t)| = 1).
Moreover g(t) is orthogonal to f (t), f˙(t) and therefore thespherical unit normal of f . The third Frenet equation saysthat the given function κg (t) (because of ġ(t) = κg (t)·e(t))
is indeed the geodesic curvature of the curve t → f (t): toany given κg (t) we have found a curve with that geodesic
curvature.
We repeat: from elementary distance and triangle geometry to the differential geometry of curves we have explained a very close analogy between the Euclidean plane
and the sphere. The 3DXM demos try to emphasize this, inparticular Spherical Ellipses, Spherical Cycloids, Spherical Lemniscates.
HK.

Space Curve TOC

Loxodrome *A Loxodrome (or “rhumb line”) is a route that a boat
would take if it kept a constant compass heading (so thaton a Mercator projection it is simply a straight line). To
be more formal, a loxodrome is a path on the unit sphereS2 ⊂ R3 that makes a constant angle with the great circles
of longitude (“meridians”). Recall that a Logarithmic
Spiral in the (complex) plane C makes a constant angle
with the rays through the origin. Stereographic Projection
St : C 7→ S2 unites these two facts: It maps the radial lines
in C to the meridians on S2 and it is also angle preserving (“conformal”). The logarithmic spirals - which meetthe radial lines in C under constant angles - are therefore
mapped by stereographic projection to the loxodromes on
S2 - which meet the meridians under constant angles.
On the other hand, the complex Exponential Map exp :C 7→ C is also conformal and it maps the lines parallel tothe real axis to the radial lines in C and all other straightlines to curves which meet the radial lines under constantangles, i.e. the logarithmic spirals. These are therefore
parametrized as c(t) = exp((1 + i · aa) · t + const) and the
loxodromes on the sphere are given parametrically as
t 7→ St(exp((1 + i · aa) · t + const)).
1: Their osculating circles all lie on the sphere. And 2: theMercator map from the sphere is: exp−1 ◦ St−1 : S2 7→ C.
R.S.P.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

The Viviani Curve *The Viviani curve is the intersection of a sphere of radius
2 · aa and a cylinder of radius aa that touch at a single
point, the double point of the curve. Parametric formulas
for it are:z = aa (1 + cos(t)) = aa 2 cos(t/2)2 ,y = aa sin(t) = aa 2 sin(t/2) cos(t/2), andx = aa 2 sin(t/2)
Implicit equations for the two intersecting surfaces are:x2 + y 2 + z 2 = 4 aa2 ,a sphere of radius 2 aa,(z − aa − bb)2 + y 2 = aa2 ,a cylinder of radius aa.

The planar projections of this curve are therefore in general curves of degree 4, but because of its symmetries theViviani curve has two orthogonal two-to-one projections
that are simpler; namely curves of degree 2. Indeed projecting it to the y-z-plane we get a twice covered circle (useSettings Menu: Set Viewpoint and Up Direction 200,0,0),
projecting to the x-z-plane gives a twice covered parabolicpiece, (1 − z/(2aa)) = (x/(2aa))2 , while the projection to
the x-y-plane is the degree 4 figure 8 with the equation (foraa = 1/2): x2 − y 2 = x4 .
Note that the osculating circles lie on the sphere.
R.S.P.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

About Spherical Cycloids *See also the ATOs for Spherical Ellipses and for
Planar Rolling Curves, e.g. Astroid, Cardioid
Spherical Definition( in analogy to planar case)
The spherical ellipses demonstrated already how definitions from planar Euclidean geometry can be repeated on
the sphere; the demo illustrates that also spherical evolutes
are analogous to the planar ones. Rolling curves, spherical cycloids, provide more such examples: simply let one
spherical circle roll (on the inside or the outside) along another spherical circle. Here roll means that the arclengths(= angle at the center times sine of the spherical radius)of corresponding arcs of the two circles agree. The true
rolling curves are obtained by looking at the curve tracedout by one point of the rolling circle, but, just as in theplane, one may also look at the traces of other points on a
fixed radius, inside or outside the rolling circle — choose
bb different from 1 in the Settings Menu, Set Parameters
Dialog.
The rolling construction is illustrated by choosing Show
Rolling Circle in the Action Menu.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

Rolling curves have a very simple tangent construction.
The point of the rolling circle which is in contact withthe base curve has velocity zero – just watch cars going
by. This means that the connecting segment (which is apiece of a great circle of the sphere) from this point ofcontact of the wheel to the endpoint of the (great circle)
drawing stick is the (great circle) radius of the momentaryrotation. The tangent of the curve drawn by the drawing stick is therefore orthogonal to this momentary radius.
The 3DXM-demo draws the rolling curve and shows its
tangents.
One can observe, for all spherical curves (in 3DXM: Viviani Curve (Intersect Sphere and Cylinder), Loxodromes(Constant Angle with Meridians), Spherical Ellipses, Spherical Cycloids, Spherical Lemniscates), that the osculating
circles lie on the sphere of the spherical curve by choosing Show Osculating Circle in the Action Menu. To understand this, note, that the osculating circle lies in theosculating plane (Action Menu!) and, of course, no circle
in a given osculating plane can be a better approximationof the curve than the intersection of this plane with thesphere on which the curve lies.
H.K.

Space Curve TOC

About Spherical Lemniscates*
See ATOs for Spherical Ellipses, Planar Lemniscate

The spherical definition is completely analogous to the planar case. The curves are traced by a mechanical drawing mechanism. It consists of three great circle rods, two
shorter ones of (spherical) length ρ = cc and a longer oneof length λ = 2 · dd (drawn red in the figure). The shorter
ones have one endpoint each at ‘focal’ points F1 , F2 , aroundwhich they can rotate. The longer great circle rod connects
the two shorter ones, thus creating a mechanism with one
degree of freedom: If one of the short rods rotates with constant angular velocity, then the connecting long rod forces
the other short rod to rotate with non-constant angular* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

velocity. – The parameter ff ∈ [0, 1] chooses the drawing
pen on the middle rod and this pen draws the curve.
This demo should be seen as another example of how constructions from Euclidean geometry can be repeated inspherical geometry. See also About Mechanically Generated Curves from the Documentation Menu of Planar
Curves.
Mechanically generated curves come together with a construction of their tangents! Our drawing mechanism isanchored with its focal points on a fixed sphere. On this
fixed sphere the drawing takes place. A second moving
sphere (same radius and midpoint as the fixed sphere) is
attached to the drawing part of the mechanical apparatus, in the present case attached to the middle rod. One
can imagine that any point of the moving sphere traces a
curve on the fixed sphere! The velocity vectors of thesetraced curves give a time-dependent vector field on the
fixed sphere. It is a marvellous theorem that for each fixed
time t the vectors of the time-dependent field are the velocity vectors of a standard rotation of the sphere. In otherwords: For each fixed time t are the integral curves of thismomentary velocity field concentric circles, the orbits of astandard rotation. The antipodal centers of these circles
are therefore called momentary centers of rotation. Any
point c(t) on one of the traced curves can be connected(by a great circle) to the antipodal momentary centers ofrotation (drawn blue) and the tangent of the traced curveat c(t) (also blue) is orthogonal to this momentary radius.

How can one find, for a specific drawing mechanism, these
momentary centers of rotation? Consider the endpoints ofthe middle rod of the present mechanism. They are pointsof the moving sphere. Since they are also the endpointsof the two rotating rods, they can only move orthogonallyto their rods. The two rotating rods are therefore always
pointing to the momentary center of rotation. We can findthese centers by intersecting the two great circles on which
the short rods lie (drawn dotted green).
H.K.

Space Curve TOC

About Spherical Ellipses *See ATO for Planar Ellipses
In 3DXplorMath the Default Morph shows a family of
ellipses with fixed focal points F1 , F2 as the larger axis aa
varies from its allowed minimum e = bb/2 to its allowed
maximum π − e = π − bb/2. Another interesting morph is0.11 ≤ aa ≤ 1.43, 0.2 ≤ bb ≤ π − 0.2: the distance of the
focal points increases until they are almost antipodal andthe major axis is only slightly longer than the distance ofthe focal points.

Elementary Definition. Many elementary constructions from planar Euclidean geometry have natural analogues on the twodimensional sphere S2 . For example, we
can take the definition of planar ellipses and use it on thesphere as follows: Pick two points F1 , F2 ∈ S2 of spherical distance 2e := dist(F1 , F2 ) < π and define the set ofpoints P ∈ S2 for which the sum of the distances to thetwo points F1 , F2 equals a constant =: 2a, i.e. the set:
P ∈ S2 ; dist(P, F1 ) + dist(P, F2 ) = 2a,
to be a Spherical Ellipse.
In the Euclidean plane there is only one restriction between
the parameters of an ellipse: 2e < 2a. Since distances on* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

S2 cannot be larger than π we have two restrictions in
spherical geometry: 2e < 2a < 2π − 2e.
For fixed focal points, i.e. for fixed e, these curves cover
the sphere (we allow that the smallest and the largest ellipse degenerate to great circle segments). One observes
that the ellipse with 2a = π is a great circle and that ellipses with 2a > π are congruent to ellipses with 2a < π
and focal points −F1 , −F2 .
This is because dist(P, F ) = π − dist(P, −F ) implies
π < 2a = dist(P, F1 ) + dist(P, F2 ) ⇒dist(P, −F1 ) + dist(P, −F2 ) = 2π − 2a < π.
Similarly, on the sphere one does not need to distinguishbetween ellipses and hyperbolas:{P ∈ S2 ; dist(P, F1 ) + dist(P, F2 ) = 2a} ={P ∈ S2 ; dist(P, F1 ) − dist(P, −F2 ) = 2a − π}.

Practical Application. These curves are used sincemore than 50 years in the LORAN System to determine
the position of a ship on the ocean as follows. Consider
a pair of radio stations which broadcast synchronized signals. If one measures at any point P on the earth the timedifference with which a pair of signals from the two stations
arrives, then one knows the difference of the two distancesfrom P to the radio stations. Therefore sea charts wereprepared which show the curves of constant difference of

the distances to the two radio stations. This has to bedone for several pairs of radio stations. In araes of theocean where the families of curves (for at least two pairsof radio stations) intersect reasonably transversal it is sufficient to measure two time differences, then a look on thesea chart will show the ship’s position as the intersection
point of two curves, two sperical hyperbolas. On the site
http://webhome.idirect.com/...
jproc/hyperbolic/index.html
or˜
˜ jproc/hyperbolic/lorc− hyperbola.jpg

this is explained by the following map:

Elementary Construction, 3DXM-demo
Begin by drawing a circle of radius 2a around F1 (called
Leitkreis in German). Next, for every point C on this circle we find a point X on the spherical ellipse as follows:
Let M be the midpoint of the great circle segment from C
to F2 and let T be the great circle through M and perpendicular to that segment. In other words, T is the symmetry line between C and F2 . Finally we intersect T with the
Leitkreis radius from F1 to C in X. — Because we used
the symmetry line T we have dist(X, C) = dist(X, F2 ) andtherefore:dist(X, F1 ) + dist(X, F2 ) = dist(X, F1 ) + dist(X, C)= dist(C, F1 ) = 2a.
It is easy to prove that the great circle T is tangent to theellipse at the pointX.

Connection with Elliptic Functions
We met a family of ellipses all having the same focal points(’confocal’) and also the orthogonal family of confocal hyperbolas in the visualization of the complex function z →z + 1/z. In the same way two orthogonal families of confocal spherical ellipses show up in the visualization of Elliptic Functions from rectangular tori to the Riemann sphere(choose in the Action Menu: Show Image on Riemann
Sphere and in the View Menu: Anaglyph Stereo Vision).
— Note that in the plane all such families of confocal ellipses and hyperbolas are essentially the same, they differ

only in scale. On the sphere we get different families fordifferent rectangular tori, i.e. for different quadrupels of
focal points {F1 , F2 , −F1 , −F2 }.

An Equation for the Spherical Ellipse
Abbreviate α := dist(X, F1 ), β := dist(X, F2 ). The definition of a spherical ellipse says:cos(2a) = cos(α + β) = cos α cos β − sin α sin β.
with cos α = hX, F1 i, cos β = hX, F2 i.
We want to write the equation in terms of the scalar products which are linear in X. Therefore we replace sin2 =1 − cos2 to get:(1 − cos2 α)(1 − cos2 β) = (cos α cos β − cos(2a))2or
1 − cos2 α − cos2 β = −2 cos(2a) cos α cos β + cos2 (2a)or, by replacing the cosines by the scalar products:
sin2 (2a)hX, Xi − hX, F1 i2 − hX, F2 i2 =− 2 cos(2a) · hX, F1 i · hX, F2 i.
Observe that this is a homogenous quadratic equation in
X = (x, y, z). In other words: Our spherical ellipse is
the intersection of the unit sphere with a quadratic cone
whose vertex is at the midpoint of the sphere. So we get
the surprisingly simple result: If one projects a sphericalellipse from the midpoint of the sphere onto some planethen one obtains a (planar) conic section.
H.K.

Space Curve TOC

Space Curves of Constant Curvature on Cylinders*
These Curves are special cases of the ones described inSpace Curves of Constant Curvature on Tori,
but the situation simplifies so much that they deserve special attention.
First we roll the plane onto a cylinder of radius R = 1/bb:


 
x
x
F :
7→  R cos(y/r)  .
y
R sin(y/R)
In the plane we describeR a curve by its rotation anglesagainst the x-axis, α(s) = 0 κg (σ)dσ, where κg is the curvature of the plane curve, or its geodesic curvature whenrolled onto the cylinder:Z scos(α(s))
c0 (s) :=, c(s) :=c0 (σ)dσ.
sin(α(s))
0The cylinder has normal curvature 0 in the x-direction and
1/R in the y-direction. The space curvature κ of F ◦ c istherefore given byκ2 = sin4 (α(s))/R2 + κ2g (s) = sin4 (α(s))/R2 + (α0 (s))2 .
This is a first order ODE for α(s), if we want κ = const =dd. The default morph of 3D-XplorMath varies dd.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

This ODE is harmless, if we look for curves with κ > 1/R:q
α0 (s) = + κ2 − sin4 (α(s))/R2 > 0.
The solution curves are, in the plane, convex curves. Theyreach α = π/2 in finite time. They are closed because thenormals at α = 0 and at α = π/2 are lines of reflectional
symmetry.
To discuss curves with κ ≤ 1/R, we differentiate the squareof the ODE and cancel 2α0 (s):α00 (s) = −2 sin3 (α(s)) cos(α(s))/R2 .
This is a Lipschitz-ODE with unique solutions for any
given initial data.
If we choose κ < 1/R, then the second order ODE forces
α0 (s) to change sign when α(s) reaches αmax given bysin2 (αmax ) = κ/R < 1. The solution curves oscillate
around a parallel to the x-axis and look a bit like sincurves.
If we choose κ = 1/R, then αmax = π/2. We see thatα(s) := π/2 is a solution of the second order ODE. Hence,
any solution which starts with α(0) < π/2 cannot reach
π/2 in finite time, but converges to π/2 asymptotically.
The corresponding curve F ◦ c therefore spirals towards
one of the circle-latitudes of the cylinder - unexpectedly?
H.K.

Space Curve TOC

Space Curves of Constant Curvature on Tori *These are the simplest non-planar closed constant curvature space curves that we have so far met. Their existence
proof depends only on symmetry arguments. Example:

The program 3D-XplorMath allows to switch (in the Action Menu) between such curves on three surface families with rotation symmetry and equator mirror symmetry: namely on tori, ellipsoids and cylinders. The meridian curves of the tori and ellipsoids and the cross sectionsof the horizontal cylinders are ellipses with vertical axis
cc and horizontal axis bb. The midpoints of these ellipses,in the torus case, lie on a circle of radius aa (> bb). The* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

cylinders, viewed as limits of the tori, have their lengths
controlled by aa and the rotation symmetry degenerated
to translation symmetry. The ellipsoids are described as
tori with aa = 0.
Space curves of constant curvature k = dd, which lie on
a given surface, can easily be computed via the ODE below, if the desired space curvature is chosen larger than all
the normal curvatures of the surface. (In fact, only thosenormal curvatures which the curve meets, do matter.) We
are interested in curves which are symmetric with respect
to the equator plane and with respect to some meridianplane - such curves are made up of four congruent arcs
and are automatically closed. The initial point is therefore
chosen on the equator and the initial direction is vertical.
The ODE is such that the angle, with which the meridians
are intersected, increases until 90◦ is reached and we haveobtained the quarter piece which gives our closed curve.
One can have the initial point on the inner or the outerequator of the torus by switching the sign of the parameter bb.
The angle between the initial direction and the vertical
meridian can be set. It is π ∗ ee. All these curves haveselfintersections, but they give some feeling, how constantcurvature space curves wind around on the given surface.
With more care we can allow constant space curvature
which is smaller than the maximum of the normal curvatures of the surface. Below we will find closed ones also
among these. – First we turn to the ODE.

The ODE for Constant Space Curvature
For every tangential unit vector ~v surfaces have a normal
curvature b(~v , ~v ), where b( , ) is the second fundamental
form of the surface. Here we describe surfaces as the levelsof a function f : R3 7→ R, where the ‘level’ is the set ofpoints where the function f has the value 0. (This constantcan be changed with the parameter ff .) This description
as a level of f allows to compute the normal curvature asκn (~v ) = hd~v grad f, ~v i/|grad f |.
A curve on the surface with tangent vector ~v will havespace curvature dd at that point if the tangential curvature(also called geodesic curvature)ispκg (~v ) = dd2 − κn (~v )2 .
Note that N := grad f /|grad f | is the prefered unit normal field of the torus. The desired curve on the surface istherefore determined by the ODE:
c00 (s) = κn (c0 (s)) · N (c(s)) + κg (c0 (s)) · c0 (s) × N (c(s)).
Any solution of this ODE withf (c(0)) = 0 and c0 (0) ⊥ grad f (c(0))
stays on the level {f = 0}, i.e. on the given surface, andis a space curve of constant curvature k = dd.
To force such curves, with simple arguments, to close up
we need to employ symmetries of the surface. Therefore
we use this ODE on surfaces of revolution which, in addition, have a reflection symmetry orthogonal to the axis ofrotation. The previous argument works on such surfaces.

More Closed Constant Curvature Curves.
The idea is to look for other symmetries of the curves.
Our surfaces allow 180◦ rotations around normals at any
equator point. Such symmetries rotate an arc of positive
geodesic curvature κg into an arc with κg < 0, i.e. weneed κg = 0 where the curve crosses the equator. The
only choice for the space curvature thereforeis:pk = dd := ±κn (c0 (0)), hence κg := ± dd2 − κ2n .
We choose κg > 0 above the equator, κg < 0 below it.
Note that on the cylinder there are helices with these initial conditions. They solve our ODE. On the other hand,on the torus and on the ellipsoid of revolution the latitudes
have smaller radius than the equator. Angular momentum
conservation therefore requires that the solution curves increase their angle against the meridians. Exactly as in thesimpler case above they turn until they intersect a meridian
orthogonally and then continue reflection symmetric (withrespect to the plane of the meridian). This symmetry implies that they reach the equator again when their geodesic
curvature is zero. Therefore they can cross the equator assmooth curves and the continuation agrees with the 180◦
normal rotation symmetry! And so on at all further crossings until the solution comes around the surface and to thevicinity of the initial point. In general it will not close. We
can vary the size of the equator (aa for the torus, bb for theellipsoid) until the solution hits the initial point. There, it
is either half a period off or, because of the angular momentum, it reaches the initial point with the same tangent,as a smoothly closed curve! This constructs many closed
constant curvature space curves, because we have the parameters aa, bb, cc, ee to play with. – Numerically we canuse the morphing feature of 3D-XplorMath and appeal tothe intermediate value theorem to find solutions.

Surprisingly, we can find these oscillating curves also on
circular cylinders. We start the integration where we expect the reflection symmetry: tangential to a straight line
and with geodesic curvature κg (c(0)) ∈ (0, max κn ). For
the solution the angle against the straight lines will increase and the geodesic curvature decrease until it becomeszero. If we call the straight line through that point equatorof the cylinder, then we have on the cylinder the same kindof curve that we obtained before on tori and ellipsoids.
In the Action Menu of 3D-XplorMath one can switch between the described two kinds of symmetries of the curves.

Some Numerical Remarks
The helices on the cylinders show that we should expect
trouble when we try to solve our ODE numerically withinitial value κg (c(0)) = 0. Recall that the Runge-Kutta
method needs to make four first order trial steps before
the high accuracy Runge-Kutta step is obtained as an average of those four trial steps. These trial steps cannot
always be computed because dd2 − κ2n < 0 at the endpointof some trial step. Currently I do not know c000 (0) for
the theoretically constructed curves (the even derivatives
vanish because of the 180◦ symmetry). Therefore I cannot construct a numerical method which avoids the above
problem.
Instead I solve a slightly wrong equation by defining aslightly too large space curvature:k = dd := κn (c(0), c0 (0)) + 0.00001.
This error is big enough for Runge-Kutta to proceed andsmall enough so that the osculating circles, while drawing
the evolute, show no discontinuity. (The evolute increases
errors of the curve very much.)
Warning: If the selection in the Action Menu is such thatthe curves with two orthogonal reflection symmetries arecomputed then the user may set the space curvature k = dd
arbitrarily. To avoid crashes caused byp
square roots of negative numbers the program computes max(0, dd2 − κ2n ).
The computed curves in such situations are geodesics, notcurves with constant space curvature.
H.K.

Space Curve TOC

Space Curves of Constant Curvature *2 - 11 Torus Knot of
constant curvature.
See also:
About Spherical Curves
Definition via Differential Equations. Space Curves that
3DXM can exhibit are mostly given in terms of explicit formulas or explicit geometric constructions. The differential
geometric treatment of curves starts from such examples
and defines geometric properties, i.e., properties which do
not change when the curve is transformed by an isometry(= distance preserving map, also called a rigid motion) of
Euclidean space R3 . The most important such properties
are the curvature function κ and the torsion function τ .
Once they have been defined one proves the Fundamental
Theorem of Space Curves, which states that for any given
continuous functions κ, τ there is a space curve with thesecurvature and torsion functions, and, that this curve is
uniquely determined up to a rigid motion. In 3DXM weobtain closed constant curvature space curves from:
κ(t) := aa,
τ (t) := bb + cc · sin(t) + dd · sin(2t) + ee · sin(3t).
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

We recall the definition of curvature and how the ODE
shows up. At each point of a parametrized space curvec(t) there is a parametrized circle γ(t) withc(t0 ) = γ(t0 ), ċ(t0 ) = γ̇(t0 ), c̈(t0 ) = γ̈(t0 ).
This circle – which may degenerate to a straight line – iscalled the osculating circle at t0 , its radius is called curvature radius at t0 and the inverse of the radius is called
the curvature at t0 , κ(t0 ). The computation of curvature
is simpler if the curve is parametrized by arc length, i.e. if
the length of all tangent vectors is one, |ċ(t)| = 1. One
gets κ(t) = |c̈(t)|. Check this for the circle (arclengthparametrization) c(t) := r · (cos(t/r), sin(t/r)). The mostcommon way to proceed is to assume that κ(t) > 0. This
allows to define the Frenet basis along the curve:
e1 (t) := ċ(t),e2 (t) := c̈(t)/κ(t),e3 (t) := e1 (t) × e2 (t).
The Frenet basis defines three curves t 7→ ej (t) on theunit sphere. To emphasize the fact that ej (t) are to beconsidered as vectors, not as points, one calls the length oftheir derivative, |ėj (t)|, angular velocity or rotation speed
and not just velocity. For example, the formula c̈(t) =κ(t)e2 (t) says that κ(t) is the rotation speed of ċ(t). We getfrom ė1 (t) ∼ e2 (t) that the derivative ė3 (t) = e1 (t) × ė2 (t)is proportional to e2 (t). The proportionality factor, therotation speed of e3 (t), is called the torsion function τ (t)of the curve c(t). In formulas: τ (t) := hė3 (t), e2 (t)i.
Having made these definitions, one can change the pointof view and consider the two functions κ, τ as given. Thisturns the equations that were originally definitions of κand τ into differential equations for the curve, the famous
Frenet-Serret Equations:ė1 (t) =κ(t) · e2 (t),ė2 (t) = −κ(t) · e1 (t) − τ (t) · e3 (t),ė3 (t) =τ (t) · e2 (t),or, more compactly with the angular velocity vectorω~ (t) := −τ (t) · e1 (t) + κ(t) · e3 (t),ėj (t) = ω~ (t) × ej (t), j = 1, 2, 3.
Finallyċ(t) := e1 (t).
For given continuous functions κ, τ these differential equations have — for given orthonormal initial values — uniqueorthonormal solutions {e1 (t), e2 (t), e3 (t)}.
Rte1 (s)ds is then parametrized by arc
The curve c(t) :=length and has the given curvature functions κ, τ .
To find closed curves of constant curvature, we employ
symmetries. The simplest case is reflection symmetry innormal planes of the curve. When crossing such symmetry
planes the torsion, in our caseτ (t) := bb + cc · sin(t) + dd · sin(2t) + ee · sin(3t),changes sign and is skew symmetric w.r.t. those points.
Therefore we need bb = 0. The symmetry points then are
at t = n · π, n ∈ Z and all symmetry planes pass throughthe intersection line of two neighboring ones. If and only if
the planes intersect in rational angles do the curves close.

Therefore we find many closed curves in any 1-parameterfamily (if bb = 0), see e.g. the default morph.
With the τ -function that we chose, there is another typeof symmetry: If we set dd = 0 then τ is even w.r.t. theparameters t = π/2 + n · π, n ∈ Z and this implies that180◦ rotation around e2 (t) is a symmetry of the curve.
Closed curves with this symmetry can be obtained from theAction Menu in the Submenu Other Closed Curves. If
one chooses in the Action Menu Add Symmetry Elements
then these normal rotation axes are added to the picture.
The closing condition is easy to see:(i) The symmetry normals have all to be in thesame plane and therefore all will intersect in one
point. (ii) The intersection angle between two
adjacent normals has to be rational.
For curves with this normal symmetry the parameter bb,
the mean value of the torsion τ , has a very nice property.
To see it, select one of the curves with normal symmetry,
then Add Symmetry Elements. Next go to Set Morphing,the 2nd entry of the Settings Menu, click the button Init
to Current Parameters and finally increase the upper
morphing bound b1 by about 0.2. Preferably choose oneof the stereo options and start morphing: The symmetry
normals will move apart but remain orthogonal to a common, helicoidal, symmetry axis.
This regular behavior allowed us to program the morphbb Keeps SymLines Colinear, the last entry in the Animate Menu. Start with a closed curve with normal symme-

tries (from the submenu of the Action Menu). In Set Morphing click again the button Init to Current Parameters, but this time increase the upper bound c1 by about
0.2. Click OK and click the last entry in the Animate Menu.
In the resulting deformation the program adjusts the meanvalue bb of the torsion in such a way that the symmetry
normals remain collinear!!
This feature allows the user to find other closed curves:
Watch the morph until the symmetry normals appear toform again a rational angle, stop the morph by clicking
the mouse. The parameters at which you stopped are now
the current parameters. Click Init to Current Parameters in Set Morphing and now increase or decrease c1
by about 0.02, do OK and run the last-entry-morph again.
With few such steps the new curve can be visibly closed.
One last time Init to Current Parameters and now doin the Animate Menu aa, bb Keep Curves closed. This
entry is there to deform closed curves as closed curves, but
in the current use it will improve the visibly closed curve tobeing high precision closed. – Some of the closed examples
which the submenu offers, have been obtained with thismethod. The last example, the 11-2-knot, needed such alarge value of the mean value bb of τ (t) that the torsion ofthis example has no sign changes! It is easier to imagine
examples like 6 helices, which look more as one would
expect: being made up of left winding and right winding
pieces of helices, joined by pieces of circles.
See the tutorial movie on constant curvature space curves

at http://3d-xplormath.org/Movies/index.html
Do not miss to select Show Osculating Circles & Evolute. The constant radius of the osculating circles shows
the constant curvature and the rotating motion of the radius shows size and sign of the torsion.
In 3DXM one can choose in the Action Menu Parallel
Frame. This frame is designed to rotate as little as possible
along the curve, in R3 . This property is more obvious when
one looks at the torus knots than at the constant curvature
curves. For further details see curves of constant torsion.
The main advantage of these parallel frames is that theyneither make it neccessary to assume more than continuityof the second derivative c̈, nor that κ > 0 everywhere, evenstraight lines are not exceptional curves if one works withthese frames. Their differential equation is also simple:
Frenet-Serret Equations for Parallel Frames:ė1 (t) :=a(t) · e2 (t) + b(t) · e3 (t),ė2 (t) := −a(t) · e1 (t),ė3 (t) := −b(t) · e1 (t).
With an antiderivative T (t) of the torsion τ (t) = T 0 (t) we
can of course write the twodimensional curvature vector(a(t), b(t)) in terms of κ(t), τ (t), namely:(a(t), b(t)) := κ(t) · cos(T (t)), sin(T (t)) .
H.K.

Space Curve TOC

Autoevolutes*
Or: Closed constant curvature space curves
which are their own evolutes
2 - 3 - Autoevolutewith some osculating
circles
See also Closed Space Curvesof Constant Curvature
The suggestion to look for these spectacular curves is from
Ekkehard Tjaden, the details are joint work.
Assume that the curve c is parametrized by arc length s,has constant curvature κ and its Frenet frame is{T (s), N (s), B(s)}. Its evolute isc̃(s) = c(s) + κ1 · N (s), c̃ 0 (s) = τ (s)κ · B(s).
The velocity of the evolute is therefore |c̃ 0 (s)| = |τ (s)|κand its Frenet frame isT̃ (s) = sign(τ )B(s), Ñ (s) = −N (s), B̃(s) = sign(τ )T (s).
That the evolute has the same constant curvature κ follows
from: Ñ 0 (s) = −N 0 (s) = κ·T (s)−|τ (s)|· T̃ (s). For the torsion τ̃ (s) of the evolute differentiate B̃(s) with respect toarclength of the evolute: sign(τ )B̃ 0 (s) = T 0 (s) = −κÑ (s),hence τ̃ (s) = sign(τ )κ/|c̃ 0 (s)| = κ2 /τ (s).
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

The two curves c, c̃ are geodesics on the canal surface whichenvelops the spheres of radius 1/2κ with midpoints on the
curve m(s) = (c(s) + c̃(s))/2, because their principle normals N, Ñ are orthogonal to the canal surface. This canalsurface is, in the case of the example above, a wobbly torus- maybe that helps the visualization.
Constant curvature curves in general do not have congruent evolutes and even if c, c̃ are congruent, this is difficult
to check in case c is parametrized by arc length. One may
look for curves with non-constant velocity v(t) = |ċ(t)|,choose τ (t) in terms of v(t) and try to arrange things sothat the first half of the curve is congruent to the evoluteof the second half and vice versa. This is achieved by thefollowing version of the Frenet equations which Ekkehardsuggested. It assumes t ∈ [0, 2π] and v(t + π) = 1/v(t).
Ṫ (t) =+κ · v(t) · N (t),Ṅ (t) = −κ · v(t) · T (t) + κ/v(t) · B(t),Ḃ(t) =
−κ/v(t) · N (t).
In 3DXM we choose
p first h(t) = bb · (cc sin(t) + ee sin(3t))
and then v(t) = 1 + h(t)2 −h(t) or also v(t) = exp(h(t));
as with the other constant curvature curves κ = aa. Comparison with the standard Frenet equations showsτ (t) = κ/v(t)2 .
Note that v(t) and τ (t) are, relative to t∗ = π/2 + nπ, evenfunctions. This implies that the principle normals at thesepoints are symmetry normals, saying that 180◦ rotation
around these normals map the curve onto itself. Closed

examples can therefore be found by solving a 2-parameter
problem:a) get the symmetry normals to lie in a plane – they thenautomatically intersect in one point.
b) get neighbouring symmetry lines to intersect with a rational angle - preferably angles like π/2, π/3, 2π/3 . . ..
This looks similar to the case of single constant curvature
curves. However in that case the constant Fourier termof the torsion is a parameter that allows to solve problema) for all choices of other parameters. This allows to deal
with problem b) assuming that a) is already solved.
For the autoevolutes we have not found such a simplifying
parameter. By trial and error one has to close the curve topixel accuracy. Only then does the automatic solution of
the 2-paramter problem work and produce a high accuracy
solution.
Note that the curve has large
torsion – i.e. twists rapidly
and moves slowly – opposite to the points with small
torsion, where the curve isfairly circular and its velocity is large.

H.K.

Space Curve TOC

About Space Curves of Constant Torsion *See: About Closed Space Curves of Constant Curvature

Definition via Differential Equations
Most Space Curves that 3DXM can exhibit are given interms of explicit formulas or explicit geometric constructions. In “About Space Curves of Constant Curvature”
we explain how curvature and torsion of a space curve
are defined. The definition immediately translates into a
construction of the curve from curvature and torsion via
the following differential equations, the famous
Frenet-Serret Equations:ė1 (t) :=κ(t) · e2 (t),ė2 (t) := −κ(t) · e1 (t) − τ (t) · e3 (t),ė3 (t) :=τ (t) · e2 (t).

For given continuous functions κ, τ these differential equations have — for given orthonormal initial values — uniqueorthonormal solutions {e1 (t), e2 (t), e3 (t)}. The curve c(t) :=Rt
e1 (s)ds is then parametrized by arc length and has thegiven curvature functions κ, τ .
The simplest curves in the plane are straight lines and circles, curves of constant curvature. It is therefore natural todiscuss also space curves of constant curvature. In 3DXM
we illustrate these by finding closed examples in the following family:* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

κ(t) := aa,
τ (t) := bb + cc · sin(t) + dd · sin(2t) + ee · sin(3t).
To understand the Frenet-Serret equations better one canalso study other special cases. Experimentation shows thatthe following curves of constant torsion
κ(t) := bb + cc · cos(ff · t) + dd · cos(2ff · t)+ee · cos(3ff · t)
τ (t) = aa
have an amusingly strong change of shape as one changes
the parameters. Again we look for closed examples withthe help of symmetries. Note that 180◦ rotations aroundthe principal normals e2 (t) at t/ff = kπ, k ∈ Z are isometries of the curves. At t/ff = π/2 + kπ, k ∈ Z the 180◦
rotations around the other normal vector of the frame,e3 (t), are also isometries of the space curve. This allows
us to formulate the closing condition:
If the normals e2 (0) at c(0), e3 (π ·ff /2) at c(π ·ff /2) intersect and if their angle is a rational multiple of π thenthe space curve closes up. Numerically one can use theparameter cc to keep the angle constant, e.g. at π/3, π/4,
and then use aa to let the normals intersect. There aremany closed solutions. Typically they look like a collection of bed springs which are joint by fairly straight pieces.
If one allows these bed springs to have many turns thenthe closing values of aa and cc are almost equidistant. The
default morph of 3DXM shows this, it contains two closed
and three approximately closed curves which are made ofthree bed springs with an increasing number of turns. It iseasy to extend this family to springs with more turns, but
one can also find all the small values, down to just one half
turn for each spring. — We found no closed curves madeof only two springs.
Here is a list of numerically closed curves:
Curves with 3-fold symmetry, ff = 0.208,
aa,
cc,
aa,
cc,
aa,
cc,

0.178632213, 0.284031845, 0.417033334,
0.2874008,
0.90658882, 2.19234962,
0.513441035, 0.59263462,
0.628044,
3.489480574, 4.7901189,
5.4411264,
0.661324546, 0.69281176,
0.7227614
6.09244336,
6.7440016,
7.39575343

Curves with 4-fold symmetry, ff = 0.23,
aa, 0.2137654757,
cc, 0.234123448,
aa,
0.56642393,
cc,2.30473675,
aa,
0.76871766,
cc,
4.449136,

0.3704887,
0.479019355,
0.89640923,
1.59595534,
0.6414483533, 0.7081321561,
3.01756515691, 3.732639742,
0.8246012,
0.87671763
5.1666082,
5.8847911

Curve with 5-fold symmetry, ff = 0.2324,
aa = 0.73855871446286, cc = 2.96466
If ones does not begin with the differential equation but
starts from the curve, then one cannot define the torsion
at points where the curvature vanishes. This problem iscaused by the use of the Frenet frame. Another frame
is suggested by a mechanical consideration: If a massive
sphere would move along the space curve (imagine thespace curve as a wire and the sphere with a hole throughwhich the wire slides without friction) then inertia would
make the sphere avoid unnecessary rotations around thewire. In other words: A frame which is attached to thesphere so that it is normal to the wire remains normal
and the derivatives of the normal vectors have no normal components. Such frames are called “parallel as normal vectors”, or simply “parallel frames”. In 3DXM one
can choose Parallel Frame in the Action Menu . Now
Show Curve as Tube illustrates the behaviour of the chosen frame. In particular the torus knots show how theparallel frames avoid “unnecessary” rotations which theFrenet frames must make.
An advantage of such parallel frames is that they neither
require to assume more than two continuous derivatives of
the curve nor that κ never vanishes—even straight lines
are not exceptional curves if one works with these frames.
Let φ(t) be an antiderivative of the torsion function, i.e.,φ̇(t) = τ (t). Then the differential equation that determines
this frame has the following simple form:
Frenet-Serret Equations for Parallel Frames:ė1 (t) := κ(t) cos(φ(t)) · e2 (t) + κ(t) sin(φ(t)) · e3 (t)ė2 (t) := −κ(t) cos(φ(t)) · e1 (t)ė3 (t) := −κ(t) sin(φ(t)) · e1 (t).
H.K.

Space Curve TOC

Free Rotational Motion of Rigid Bodies *Part I: Angular Velocity and Rigid Motion
In this first part we will not yet consider solid objects with
their inertial properties, but only so-called rigid body kinematics, i.e., the study of rotational motions of space. The(simpler) particle mechanics analogue of the question that
we will discuss is the following: knowing the velocity curvev(t) of a point how can we reconstruct the travel path c(t)?
Since c0 (t) = v(t), c(t) is an antiderivative of v(t) and we
can find it easily by integration. (Historically v(t) was
called the hodograph of the motion.)Things to try in 3D-XplorMath
The last three entries of the Action Menu of Space Curves
show demos that illustrate the present discussion. The
first of these Actions, Use Curve as Hodograph, interpretes
the space curves of 3D-XplorMath as velocity curves of aparticle and reconstructs the path. The demo emphasizes
that the tangent vector of the constructed path is (parallel to) the position vector of the selected space curve, thehodograph.
The second of these Actions, Use Curve as Angular Velocity ω~ (t), reconstructs the rotational motion which has* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

the given space curve as given angular velocity function.
The visualization of the motion uses a sphere with random
dots and shows several consecutive points of the orbit ofeach random dot. One sees large orbit velocities near theequator of the rotation and small velocities near the axis of
the rotation at each moment. – More details are explained
below.
The third of these Actions, Use Curve as Components ofω~ (t) in the Moving System, again reconstructs that rotational motion that has its angular velocity given in themoving system by the selected space curve. The spacecurve therefore rotates with the motion. It leaves a trace
behind which shows the corresponding angular velocity
curve in the observer’s space. In the second Action thiscurve was the given one.
Finally, there is one very special space curve, Solid Body
(Euler’s Polhode). If this space curve is selected for thethird Action above then the resulting motion is the physical motion around the center of mass of a rigid body, taken
to be a brick with edge lengths aa ≥ bb ≥ cc and initial
components of the angular momentum dd, ee, f f , see the
ATO of Solid Body.
Angular Velocity given in the Observer’s Space
Mathematicians and Physicists have slightly different pictures of a motion in their minds. A physicist sees a solid
object moving in space, the movement is differentiable andall points ~xi (t) of the moving object have their orbit ve-locities ~xi 0 (t). So far these functions could also describe amass of moving air. The word rigid motion means that the
pairwise distances |~xi (t) − ~xj (t)| remain constant in time –the points ~xi (t) could be the atoms of a stone. For a mathematician on the other hand the primary concept is thatof a distance preserving map of space, and a motion is a
1-parameter family of such maps. For physicists and mathematicians it is important to understand the velocity fields~xi 0 (t) of all the particles. Physicists begin by studying rotations around fixed axes with constant angular velocities.
In such a situation one can compute all the velocities ~xi 0 (t)from one vector ω~ that is parallel to the rotation axis andfrom the particle positions ~xi (t) as follows:~xi 0 (t) = ω~ × ~xi (t).
It is now a mathematical fact that differentiable families of
distance preserving maps have a very similar formula forthe velocities of the particles: For each time t there exists
a vector ω~ (t) such that we have:~xi 0 (t) = ω~ (t) × ~xi (t).
And vice versa, if such a relation between the velocities andthe positions holds then all pairwise distances between theparticles are constant in time. Therefore mathematicians
and physicists agree that a differentiable rigid motion ischaracterized by this relation between particle positions
and particle velocities.

Now, a natural question is: If ω
~ (t) is a given vector function
in R3 , how can one reconstruct the rotational motion? Weanswer this question by constructing a so called moving
frame {~ex (t), ~ey (t), ~ez (t)}, a time dependent orthonormal basis. To do this we have to solve the following three
ODEs:~ex 0 (t) = ω~ (t) × ~ex (t), ~ex (0) = (1, 0, 0)~ey 0 (t) = ω~ (t) × ~ey (t), ~ey (0) = (0, 1, 0)~ez 0 (t) = ω~ (t) × ~ez (t), ~ez (0) = (0, 0, 1).
Next we observe that all linear combinations with constantcoefficients, i.e.
~x(t) := x · ~ex (t) + y · ~ey (t) + z · ~ez (t) satisfy~x 0 (t) = ω~ (t) × ~x(t) and are therefore orbits of the rotational motion defined by the angular velocity ω~ (t).
To visualize this motion observe that for each fixed t thevelocity field ~v (~x) := ω~ (t) × ~x is the velocity field of theordinary rotation around the axis ω~ (t)R with constant angular velocity |~ω (t)|.
Angular Velocity given in the Moving Space
What could it mean to give the angular velocity of a motion in moving space? We saw in the previous discussion that we can describe the motion of space by giving a moving frame {~ex (t), ~ey (t), ~ez (t)}. The particles
of moving objects have position vectors that have constant components ax , ay , az relative to this frame: ~xi (t) =ax~ex (t)+ay ~ey (t)+az ~ez (t). Similarly we can prescribe ω~ (t)by giving its components relative to the moving frame:{ωx (t), ωy (t), ωz (t)}.
There is again a natural question: can we again reconstructa corresponding rotational motion for any vector functionω~ (t) that is given in this way?
The answer is almost the same as for the first question,
except that the three ODEs are no longer separate but arecoupled by the fifth line:~ex 0 (t) = ω~ (t) × ~ex (t), ~ex (0) = (1, 0, 0)~ey 0 (t) = ω~ (t) × ~ey (t), ~ey (0) = (0, 1, 0)~ez 0 (t) = ω~ (t) × ~ez (t), ~ez (0) = (0, 0, 1)
withω~ (t) = ωx (t) · ~ex (t) + ωy (t) · ~ey (t) + ωz (t) · ~ez (t).
Historical note: The given curve {ωx (t), ωy (t), ωz (t)} inthe moving system is called the polhode of the motion andthe corresponding curve ω~ (t) = ωx (t) ·~ex (t) + ωy (t) ·~ey (t) +ωz (t) · ~ez (t) in the inertial space is called the herpolhode.
The moving polhode and the fixed herpolhode touch eachother at each time t with tangents of equal length – because
the points on the momentary axis of rotation, ω~ (t)R, haveat time t the rotational velocity field ω~ (t) × ω
~ (t) = ~0 in
R3 . A visual interpretation of this fact is that the moving
polhode rolls without slipping along the fixed herpolhode.
(This description actually determines the rotational motion because the origin is fixed so that the polhode has nofreedom to rotate around the common tangent with the
herpolhode, there is only one way to roll along.)
H.K.

Space Curve TOC

Free Rotational Motion of Rigid Bodies *What is to observe in the 3D-XplorMath exhibit
Solid Body (Euler’s Polhode) ?
A brick – in the program of edge lengths aa ≥ bb ≥ cc ≥ 0– is a good example of a solid (also: rigid) body. The
program illustrates the free rotational movement of such abrick (i.e. gravity is ignored):
Select Solid Body (Euler’s Polhode), stop the alternation
between two pictures by a mouse click and select Do Poinsot
Construction From Polhode at the bottom of the Action
Menu. The resulting animation shows a freely tumbling
brick. By changing aa, bb, or cc one may watch other bricks
tumbling.
There are three other input parameters, dd, ee, f f . These
are initial conditions for the tumbling motion. If one uses
(dd, ee, f f ) ≈ (1, 0.1, 0.1) or (dd, ee, f f ) ≈ (0.1, 0.1, 1) thenthere is not much tumbling. These motions are almost rotations around the longest axis (aa) of the brick, respectively the shortest axis (cc) . The fact that these rotation
axes stay close to their initial position is expressed by saying: the rotations around the longest and the shortest axis
are stable. Now look again at the default initial conditions(dd, ee, f f ) ≈ (0.1, 1, 0.1). One observes that the momen* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

tary axis of rotation moves almost to the direction opposite
to the initial direction and then returns back. One says:
the rotation around the middle axis of the brick is unstable.
– By putting a tape around a book and trying to throw
it so that it rotates around one of the three axes one canexperimentally test these theoretical predictions.
The explanation of this behaviour has a mathematical partand a physical part. The physical part is contained in theinitial picture, the mathematical part is the connection
between the initial picture and the annimation. We explain
the mathematical part inPart I: From Angular Velocity to Rigid Motion
It is available in the Topics part of the Documentation.
This mathematical part has no physical limitations, anyof the space curves in the program can be used as angularvelocity curve and in the Action Menu one can select animations that show the resulting motions.
The physical part requires in addition to angular velocity
the physical notions tensor of inertia and angular momentum. These are explained below. What can one say before
this theory about the initial picture of the program? We
see two space curves. The one on the sphere is the angularmomentum as a function of time in the coordinate systemof the brick. The other one is the angular velocity curve(called Polhode). Both are intersections of quadratic surfaces, represented by dots in the picture. The two curves

are related by a fixed linear map – given by the tensor ofinertia. To emphasize this linear map the quadratic surfaces alternate between the domain and the range of thismap. Finally, these two curves together determine Euler’s differential equation for either of them. For example
the derivative of the angular momentum curve is the crossproduct of the corresponding position vectors of the angular momentum curve and the angular velocity curve, in
formulas: ~`0 (t) = ~`(t)×~
ω (t). The Action Menu entry Show
Repère Mobile and ODE illustrates this connection. The
dotted curves on the sphere are solutions for other initialconditions dd, ee, f f with the same value dd2 + ee2 + f f 2 .
The default morph varies bb between aa and cc, it illustrates how the family of polhodes depends on the shape ofthe brick.
And here is the theory:
Part II: Tensor of Inertia and Angular Momentum
The tensor of inertia is a map that transforms angular velocity into angular momentum.
Historical note: The word tensor is a generic word that
describes objects from linear algebra that can be given bycomponents (indices!) with respect to a base. The tensor
of inertia is a linear map from the 3-dim vector space ofangular velocities to the 3-dim vector space of angular momenta. What we need below is that for each solid bodythere exists an orthonormal frame {~ex (t), ~ey (t), ~ez (t)} in
the rest space of the body (i.e. moving with the body) sothat the tensor of inertia Θ is a diagonal map:
angular momentum = Θ(~ω (t)) =ωx (t) · Θx~ex (t) + ωy (t) · Θy ~ey (t) + ωz (t) · Θz ~ez (t).
Θx , Θy , Θz are called principal moments of inertia.
We now explain the tensor of inertia in some more detail. The result of the explanation will be the above formula for the angular momentum. We view a solid bodyas a collection of points of mass mi and position vector~xi (t); the pairwise distances between these points are constant.
P The origin is the center of mass of these points,
i.e.
xi (t) = ~0. For each mass point we have the foli mi ~lowing definitions, the corresponding notions for the solid
body are obtained by summation:
linear momentum: p~i (t) := mi ~xi 0 (t)angular momentum with respect to the origin:~`i (t) := ~xi (t) × p~i (t)kinetic energy: Ei (t) := 21 mi h~xi 0 (t), ~xi 0 (t)i.
The body is rigid, i.e. the distances between the points
are constant, therefore there is an angular velocity function ω~ (t) that relates the positions and velocities:
rotational motion: ~xi 0 (t) = ω~ (t) × ~xi (t).
angular momentum: ~`i (t) = ~xi (t) × (~
ω (t) × ~xi (t)).
=: Θi (~
ω (t)).
This tensor of inertia is most easily understood if we use
the relation between cross-product and matrix-product andinsert it into the above definitions. We obtain the expressions for angular momentum and kinetic energy in termsof the tensor of inertia and the angular velocity as follows:  0−ωz ωyxω~ × ~x =  ωz0−ωx  ·  y z−ωy ωx
0

  
0
z −y
ωx
=  −z 0x  ·  ωy 
y −x 0ωz

We obtain~`i (t) =








0
zi −yi0−zi yiωx
mi  −zi0xi   zi
0−xi   ωy 
yi −xi0−yi xi
0
ωz
 2
 
2
yi + zi−xi yi−xi ziωx= mi  −xi yi yi2 + zi2 −yi zi   ωy 
−xi zi−yi zi x2i + yi2ωz= Θi (~ω ) (Note the symmetry of the matrix of Θi ).
1ω ), ω
~ i.
Ei (t) = hΘi (~2
P

The symmetry of Θ :=i Θi implies that we have an
orthonormal eigen basis for Θ. The corresponding eigen
values are the principal moments of inertia, Θx , Θy , Θz .

Finally, we will derive Euler’s equations, a first order ODE
for ω~ (t). Together with part I this determines the motionof a solid body that rotates without exterior forces. We
will always take the eigen basis of Θ as the moving frameof part I.
Newton’s laws imply that the total angular momentum isconstant in situations that are more general than the force
free rotation of a solid body. We omit this general theory
and show only that the conservation of angular momentum
is equivalent to Euler’s equations.
XX~`i (t) = Θ(~~`(t) :=
ω (t)) =ωξ (t)Θξ ~eξ (t)iimpliesd~`(t) =dt X

ξ∈{x,y,z}0ωξ (t) Θξ ~eξ (t) +ξ∈{x,y,z}Xωξ (t)Θξ ~eξ 0 (t).

ξ∈{x,y,z}Insert ~eξ 0 (t) = ω~ (t) × ~eξ (t) to getXωξ (t)Θξ ~eξ 0 (t) = ω~ (t) × ~`(t),ξ∈{x,y,z}

next compute the cross product in the base given by themoving frame:   ωx`xX ωy  ×  `y  · ~eξ (t),ω~ (t) × ~`(t) =ωz`zξ∈{x,y,z}ξfinally compare coefficients to get Euler’s equations:


0



0









`x
Θx ω xωx`x
 `y  =  Θy ωy  = −  ωy  ×  `y  ,
`z
Θz ωz
ωz
`z
where the physics is contained in the relation
between ω and ` :`x = Θx ωx , `y = Θy ωy , `z = Θz ωz .
Considered as differential equation for the ω-components
these are Euler’s equations. This ODE-system implies immediately that the two quadratic functions|~` |2 = `2x + `2y + `2z = Θ2x ωx2 + Θ2y ωy2 + Θ2z ωz2and2E = `x ωx + `y ωy + `z ωz = Θx ωx2 + Θy ωy2 + Θz ωz2
are constant along solution curves. The solutions are therefore intersections of two ellipsoids. If one considers the
ODE-system as differential equations for the `-components
then one of the ellipsoids is a sphere and the solutions(`x (t), `y (t), `z (t)) are spherical curves. The choice of the`-components as the functions to be determined therefore
simplifies the visualization and also leads to a slightly simpler ODE-system, since the tensor of inertia enters only on
the right side, linearly, into the equations.
H.K.

Space Curve TOC

User Curves By Curvature And Torsion *The exhibit allows to create examples for the standard
Frenet theory of space curves. The initial dialogue allows
to input user choices for curvature and torsion as functionsof arc length, κ(s), τ (s).
The solution curves are programmed as if they were explicitly parametrized. Therefore all the Action Menu entries
for parametrized curves are also available for these ODEdefined curves.
The differential equations in question are the famous
Frenet-Serret Equations:ė1 (t) :=κ(t) · e2 (t),ė2 (t) := −κ(t) · e1 (t) − τ (t) · e3 (t),ė3 (t) :=τ (t) · e2 (t).

For given continuous functions κ, τ these differential equations have — for given orthonormal initial values — uniqueorthonormal solutions {e1 (t), e2 (t), e3 (t)}.
Rt
The curve c(t) :=e1 (s)ds is then parametrized by arc
length and has the given curvature functions κ, τ .
H.K.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

Userdefined Parametrized Space Curves *These exhibits allow to input userdefined explicitly parametrized space curves in three different ways:
1.) User Cartesian: The three Cartesian coordinate functions x(t), y(t), z(t) can be entered (Of course t does nothave to be arc length.)
2.) User Polar: The coordinate functions can be enteredin spherical polar coordinates r(t), θ(t), ϕ(t). In particular,
this allows to enter spherical curves. As usual:x = r · sin θ · cos ϕ, y = r · sin θ · sin ϕ, z = r · cos θ.
3.) User Cylindrical: The coordinate functions can beentered in cylindrical coordinates r(t), θ(t), z(t), with theusual convention x = r · cos θ, y = r · sin θ, z = z.
Since Cylinders are isometric to the plane, this allows tocreate space curves that are given on all the cylinders
r = const by the same intrinsic geodesic curvature dataκg (s).
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Space Curve TOC

About the Unduloid *What are the different shapes that a soap film can take,or to put it somewhat differently, what can we say about
the differential geometry of a mathematical surface thatapproximates a soap film?
An important physical characteristic of the soap film is its
surface tension T . This depends only on the chemical composition of the liquid from which it is made, and so it is
the same at each point of the film. The difference in airpressure between the two sides of the film is an environmental variable that is also clearly the same at all pointsof the film. Now it follows from physical principles (thatwe will take for granted here) that the mean curvature Hof the soap film at any point is equal to P/T , and so we
see that a soap film is always represented by a surface thathas constant mean curvature.
For a soap film that we get by dipping a closed loop of wire
into soapy water, the air pressure on both sides is clearly
the same, so such a soap film must have mean curvaturezero. Such surfaces are called minimal surfaces, since it
can be shown that if we draw any small closed curve on
the surface, the area of the part of the surface inside thecurve is less than or equal to the area of any other surface
bounded by the curve.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/ Surfaces of Revolution

We consider minimal surfaces in considerable detail elsewhere, and here we shall be interested in the case of soap
bubbles. These are soap films that (perhaps together withsome other surfaces) enclose a bounded region of space (the“inside” of the bubble). For bubbles the pressure will beslightly greater on the inside than on the outside, so thatthe surface is what is called a CMC surface, that is it has
non-zero constant mean curvature (and of course for thefloating type it is often just a sphere).
If one blows a soap bubble between two parallel glass plates
then one can obtain CMC surfaces that are surfaces ofrevolution, and such CMC surfaces are called Unduloids.
Consider a curve in the x-y-plane, given parametricallyby x = x(t), y = y(t), or as a graph (x, f (x)) of a function f . If one rotates this curve about the x-axis, it iseasy to compute an expression for the mean curvature Hof the resulting surface of revolution in terms of the firstand second derivative of x(t) and y(t) (or, in the graphdescription, the derivatives of f ). If this expression is set
equal to a positive constant H, one gets differential equations for the functions x(t) and y(t) (respectively for thefunction f ), and solving these ODE provides a method forfinding all CMC surfaces of revolution. Delaunay studied
this problem in 1841, and being an expert on the theoryof roulettes (i.e., a locus traced out by a point attached tocurve as that curve rolls on a line), he recognized that thesolutions of this differential equation could be identified
with the roulettes traced out by a focus of a conic section

as it rolls along the x-axis. The special case that the conic
is an ellipse gives the Unduloid. In 3D-XplorMath, theUnduloid is literally constructed by this double process offirst rolling an ellipse and tracking one of its foci and thenrotating the resulting curve around the x-axis.
The default morph shows a family of unduloids that starts
with a cylinder and deforms towards a chain of spheres.
With the rolling construction of the Unduloid, we cannot reach the chain of spheres because the parameter lines
become concentrated near the narrowing necks of the surfaces. However, if one resizes these necks so they haveconstant waist size, then the necks converge to (minimal)
Catenoids. This fact was very important in the construction of very general examples by Kapouleas.
H.K.

Surfaces of Revolution

Curvature Properties of Surfaces *Any curvature discussion of surfaces assumes some knowledge about curvature properties of curves.
Planar Curves have, at each point c(s), only one kindof curvature. Consider a circle through c(s) that has thesame first and second derivative as the curve at c(s). Sucha circle is called osculating circle, it approximates the curve
better than any other circle and it can easily be recognized
if c000 (s) =/ 0 : The circle has the same tangent as thecurve and is on different sides of the curve before and afterc(s). See Osculating Circles in the Action Menu. The
radius of this circle is called curvature radius r(s), and thecurvature is defined as κ(s) := 1/r(s). If s is the arclengthparameter, i.e. |c0 (s)| = 1, then κ(s) = det(c0 (s), c00 (s)).
The fundamental theorem for planar curves states:
If a continuous curvature function κ(s) is given
then there exists, up to congruence, exactly one
planar curve with this curvature function, namely:Rs
define an antiderivative α(s) := α(0) + 0 κ(σ)dσ and with
R s cos α(σ)
it a curve c(s) := c(0) + 0 sin α(σ) dσ.
This curve c(s) is parametrized by arclength and has curvature function κ(s).
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/ Surface Organisation

Space Curves have the same definition of osculating circle and of curvature κ(s) as the plane curves. If s is arclength on c then c00 (s) is called principal curvature vector
and ~h(s) = c00 (s)/|c00 (s)| is called principal normal. The
Frenet frame along the curve is {c0 , ~h, ~b = c0 × ~h}.
In addition, space curves have a second kind of curvature,
the rotation speed of the binormal ~b, also called torsion
τ (s) := −h~b0 (s), ~h(s)i = +h~h0 (s), ~b(s)i. The fundamental
theorem for space curves (roughly) states:
If continuous curvature functions κ(s), τ (s) are
given then there exists, up to congruence, exactly
one space curve with these curvature functions.
Surfaces have as their most visible curvature properties
their normal curvatures: Consider at a surface point p theintersection of the surface with all its normal planes at p.
The curvatures of these normal sections are the normal
curvatures. They can be computed as follows: Let N bea unit normal field along the surface and c(s), c(0) = p anormal section with c0 (0) = ~e, |~e | = 1. Then its curvature,
the normal curvature in the direction ~e, isκnormal (p, ~e) = hD~e N, ~e i = −hc00 (0), N i.
These normal curvatures have a minimum and a maximum,
called the principal curvatures κ1 , κ2 at p. The corresponding vectors ~e are called the principal directions ~e1 ⊥ ~e2 .
H := κ1 + κ2 and K := κ1 · κ2 are mean curvature andGauss curvature.

If the surface is given by an explicit parametrization, it isstraight forward to compute these data. If the surface isgiven by an implicit equation f (x, y, z) = 0, one chooses
N (x, y, z) := grad f /|grad f | and computes hD~e N, ~e i, as
before.
For these surfaces one finds in the Action Menu the entries:
Add Principal Curvature Fields,
Move Principal Curvature Circles.
They allow to view and move the above curvature objects.
Surface Organisation

Projected Sphere*
The elementary and differential geometric properties of triangles and curves on spheres are explained in About Spherical Curves in the documentation for Space Curves.
This exhibit has two goals:
1) The mechanical constructions of Ellipse, Cycloid andLemniscate in the plane are repeated on the sphere to givemechanical constructions of Spherical Ellipses, Spherical
Cycloids and Spherical Lemniscates. In the plane and on
the sphere these constructions also give the tangents of thecurves. The curves are chosen in the Action Menu of thesphere exhibit. They have the same parameters as in thequoted space curve exhibits.
2) Two famous maps are explained: The angle preserving Stereographic Projection from the sphere to a tangent
plane and Archimedes’ area preserving projection from acircumscribed cylinder to the sphere.
The sphere can be rendered with four different grids, selectable in the Action Menu. After standard (geographical) polar coordinates have been chosen, one can add (alsofrom the Action Menu) four different spherical curves orArchimedes’ projection. The spherical ellipses can also beviewed on a grid consisting of confocal ellipses and standard longitudes. Stereographic projection is best explainedon a grid whose parameter lines are two orthogonal families of circles through the projection center (”South Pole”);* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/ Surface Organisation

these circles are mapped to lines parallel to the x-axis or
the y-axis. The parameter lines of the last grid are two
families of Loxodromes (Constant Angle with Meridians);
with this grid one can see – via stereographic projection– that groups of Möbius transformations may look much
simpler on the Riemann sphere than in the Gaussian plane.
Which parameters control what?
Spherical Ellipses:
bb = distance of focal points,
aa > bb/2 = major axis. The default morph changes aa.
For the elliptic grid also: bb = distance of the focal points.
The construction demo runs automatically; if 0 < ff < 1
is taken as morphing parameter, the demo runs as movie.
Spherical Cycloids:
aa · π = radius of the fixed circle.
Integer ee = order of symmetry (determines the radius of
the rolling circle). Length of drawing stick = bb · radius ofrolling circle. Automatic demo, morphing parameter ff .
Spherical Lemniscates:
cc = length of the middle rod,
dd = length of the two outer rods. ff parameter of thedrawing pen, ff = 0.5 midpoint of middle rod. Automaticdemo, can be saved with morphing parameter 0 < ee < 1.
Loxodromes, Loxodromic grids: aa = slope parameter, withaa = 0 giving a longitude curve resp. the standard polar
grid. cc = parameter (in degrees) for rotating the spherearound the x-axis - if also stereographic projection is chosen, then its center and image plane stay fixed while thesphere rotates.

Explanation of Archimedes’ Projection
Let −π/2 ≤ u ≤ π/2 be the latitude parameter on a unit
sphere parametrized as (sin(u) cos(v), sin(u) sin(v), cos(u)).
Add the tangential cylinder around the equator (u = 0),parametrized as (cos(v), sin(v), cos(u)). Archimedes’ mapis horizontal radial projection, i.e. points with the same
parameters (u, v) on the cylinder and the sphere correspond to each other. This map shortens latitudes of thecylinder at height cos(u) by the factor sin(u). The verticallines on the cylinder are, locally at height cos(u), stretchedby the factor 1/ sin(u).
With formulas:cylinder:sphere:d(cos(v), sin(v), cos(u)) = sin(u),dud(sin(u) cos(v), sin(u) sin(v), cos(u)) = 1.
duThis shows that Archimedes’ map is area preserving. The
unit sphere has therefore the same surface area as the cylinder of height 2 and radius 1, namely 2 · 2π.
This also implies that the volume of the solid sphere is 4π/3
because we can partition the boundary sphere into verysmall domains and view the solid sphere as the (disjoint)
union of height= 1 pyramids which have as base the small
domains and their vertices are at the center of the sphere.

Explanation of Stereographic Projection
Geometrically stereographic projection is defined as the
1-1 central projection from a point on the sphere to thetangent plane at the opposite point and vice versa. The
same definition works for the 2-sheeted hyperboloid, which
gives a map of similar importance as in the spherical case.
Formulas are slightly simpler if one projects to the planethrough the center of the sphere which is parallel to thetangent plane opposite the center. Take S = (0, 0, −1) as
projection center, then:/ − 1 putLet (x, y, z) ∈ S2 , i.e. x2 + y 2 + z 2 = 1. For z =St(x, y, z) := (x, y, 0)/(1 + z).
The inverse map is used just as often:
Let (u, v, 0) ∈ R2 ⊂ R3 , then with r2 := 1 + u2 + v 2 put
St−1 (u, v, 0) := (2u, 2v, 2 − r2 )/r2 .
The most important property of stereographic projection:
This map preserves angles or shorter: it is conformal.
Recall: A map preserves angles if for any two intersecting
curves in the domain holds: Their intersection angle is thesame as the intersection angle of their image curves. Of
course, the intersection angle of curves is defined as the intersection angle of their tangents at the intersection point.
To prove that angles are preserved, it is therefore enoughto take as ’curves’ in the image plane just all straight lines.
Their stereographic preimages on the sphere are circles
through the projection center S. Any two such circles intersect at S and their other intersection point with thesame angle. For any such circle holds: Its tangent at S
and its image line in the opposite tangent plane are parallel. In other words: The image lines of two circles throughS intersect with the same angle as the two circles.
This proves that stereographic projection is conformal.
A second basic property of stereographic projection is:
Circles on the sphere which do not pass through S are
mapped to circles in the image plane. (We know alreadythat circles through S are mapped to straight lines.)
Proof. Take a circle on the sphere and consider all tangents to the sphere which are perpendicular to the circle.
They form the tangential cone which touches the spherealong the given circle. For each of these tangents consider
the plane spanned by the considered tangent and the projection center S. Intersect these planes with the sphere to
get circles through S which all intersect the given circle
orthogonally and all pass through one point, namely theintersection of the line through S and the vertex of thetangential cone with the sphere. The stereographic imagesof these circles are all straight lines which all pass throughone point and meet the image curve orthogonally, i.e. they
are like radii. The image curve is therefore a circle – either
by the uniqueness theorem for ODEs or because the conewith vertex S through the given circle is a quadratic coneso that the image curve is known to be an ellipse - which
in turn has to be a circle because of the orthogonal radii.
(Provable also by computation)
Surface Organisation

Surfaces of Revolution *in particular with Constant Gauß or Mean Curvature
Surfaces of revolution are the simplest family of surfaces.
They are usually described by giving its meridian curves 7→ (r(s), h(s)), which then is rotated:(x, y, z)(s, ϕ) := (r(s) cos ϕ, r(s) sin ϕ, h(s)).
The meridian curve may be given explicitly. However, any
kind of curvature condition can be expressed as an ordinary differential equation for the meridian curve. Therefore ODE-determined meridians are more interesting.
Formulas are particularly simple if the meridian is parametrized by arclength, i.e., r02 + h02 = 1. The curvatureof the latitude circles is 1/r(s) and the normal curvatureof the surface in their direction is a principal curvature,
namely κ2 (s) = 1/r · h0 . The other principal curvature isthe curvature of the meridian: −h0 r 00
κ1 (s) = h h00 , r0 i = −r00 h0 + h00 r0 = −r00 /h0 .
For given Gauss curvature
K(s) = κ1 (s) · κ2 (s) = −r00 (s)/r(s)the meridian is determined by the simple ODE (as long as|r0 s)| ≤ 1):
Z sp
r00 (s) + K(s) · r(s) = 0,h(s) =
1 − r0 (t)2 dt.
0* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/ Surface Organisation

In the case of K(s) := 0 each solution r(s) is a linear function. We obtain circular cylinders and cones.
Ferdinand Minding (1806 - 1885) studied the case of constant nonzero curvature. He determined the geodesics on
the Pseudosphere (K = −1), but we could not find whether
he was the first to describe this famous surface. He obtained the following formulas:
The examples for K = 1:
The Sphere:r(s) = sin(s),0≤s≤πWith cone points:r(s) = a sin(s), 0 ≤ s ≤ π, 0 < a < 1With two singularity curves:
r(s) = a cos(s), −b ≤ s ≤ b, a > 1, sin(b) = 1/a.
The examples for K = −1:
The Pseudosphere (with singular curve {s = 0}):r(s) = exp(−s), 0 ≤ s
With cone point and singular curve:r(s) = a sinh(s), 0 ≤ s ≤ b, 0 < a < 1, cosh(b) = 1/aWith two singular curves:
r(s) = a cosh(s), −b ≤ s ≤ b, 0 < a, sinh(b) = 1/a.
The Pseudosphere cannot be extended beyond the singularcurve {s = 0}, because r0 (s) > 1 for s < 0. But the metric

of the Pseudosphere,
ds2 + exp(−2s)dϕ2 ,extends to a complete metric on R × R with K = −1 andthe same is true for the metric of the last example:
ds2 + a2 cosh2 (s)dϕ2 .
The fact that such a simply connected complete Riemannian plane with K = −1 is isometric to the non-Euclidean
geometry of Bolyai and Lobachevsky was not yet established when Minding studied geodesic triangles on the Pseudosphere. Beltrami showed it in 1865.
Surfaces with K = −1 in R3 have another interesting
property: If the asymptote directions are described by
two vectorfields of constant length, then these vector fields
commute. For parametrizations of K = −1 surfaces withasymptote lines as parameter lines one therefore has: theedge lengths of the parameter quadrilaterals are all thesame. Such nets are called Tchebycheff nets. They are
determined by the angle between the asymptote lines. A
quite unexpected theory developed from here. It is outlined in Introduction to Pseudospherical Surfaces (or see
the Documentation Menu). The Pseudosphere turns out
to be the simplest example of that theory.
These Tchebycheff nets also play a crucial role in the proofof Hilbert’s theorem, stating that the hyperbolic planecannot be smoothly immersed isometrically into R3 . The
proof shows that the distance to a point, where the angle
between the asymptote lines is 0 or π, is finite.

The Catenoid (H = 0)
For the mean curvature we haveH(s) = κ1 (s) + κ2 (s) = −r00 /h0 + h0 /r
p0with h (s) = 1 − r0 (s)2 .
Meusnier proved in 1776 that the Catenoid and the Helicoid are solutions of the minimal surface equation which
was discovered by Lagrange. The Catenoid is the surfaceof revolution where the meridian curve is the rotated graph(cosh(x), x). Reparametrization to arclength parameter
givesph(s) = sinh−1 (s)r(s) = 1 + s2 ,pp00r (s) = s/ 1 + s2 , h (s) = 1/ 1 + s2
p00
r (s) = ( 1 + s2 )−3 ,which implies indeed H(s) = 0, so that the Catenoid is
a complete minimal surface of revolution, in fact the onlyone.
The Unduloid (H = 1)
The Unduloid is a surface of revolution of constant mean
curvature H = 1. It was a crucial example for the development of the theory of constant mean curvature surfaces.
See The Unduloid.
H.K.

Surface Organisation

Explicit and Implicit Surfaces
Explicit versus Implicit Surfaces
Early Minimal Surfaces
Elementary Surfaces
Paraboloid
Ellipsoid
More About Quadratic Surfaces, includes:
1-Sheeted Hyperboloid, 2-Sheeted Hyperboloid
Hyperbolic Paraboloid, Cylinders and Cones
About Ruled Surfaces, includes:
Helicoid, Right Conoid, Whitney Umbrella
View Principal Curvature Directions on:
Monkey Saddle, Torus, Dupin Cyclides
Play With Formulas Lissajous Surface,
Double Helix, Column, Norm 1 Family
Snail Shell Surfaces are Families of Circles
Untangle Double Twist In SO(3)Dirac Belt and Feynman Plate Tricks
Surfaces of Constant Width
Stereographic Projections from S3
Clifford Tori
Hopf-fibered Linked Tori
Bianchi - Pinkall Tori
Nonorientable SurfacesMöbius Strip and Klein Bottle
Cross-Cap and Steiner’s Roman Surface
Boy Surfaces
Surface Organisation

Explicit versus Implicit Surfaces,*in particular Level Sets of Functions

Surfaces in R3 are either described as parametrized images
F : D2 → R3 or as implicit surfaces, i.e., as levels of functions f : R3 → R, as the set of points where f has somegiven value, i.e. {x ∈ R3 ; f (x) = given}. Graphs of functions h : R2 7→ R are both: F (u, v) := (u, v, h(u, v)) isa parametrization and f (x, y, z) := h(x, y) − z is a level
function, f = 0 the implicit equation.
For most simple surfaces one has both representations, examples in 3DXM: All Quadratic Surfaces, Tori, Cyclides,
Cross-Cap, Steiner Surface, Algebraic Boy Surface, Whitney Umbrella. In each case, the explicit and the implicit
version open the same Documentation.
One can more easily make images of parametrized surfaces
than of implicit surfaces, because every point p ∈ D can bemapped with the given function F to obtain ‘explicitly’ apoint F (p) of the surface. Note however that the opposite
problem: “Given a point in R3 , decide whether it lies on
the surface” does not have an easy answer. For an implicit* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

surface, on the other hand, it is easy to decide whether agiven point in R3 is on the surface (simply check f (x)), but
no point is given explicitly, one has to use some algorithm
to find points x ∈ R3 which satisfy f (x) = given value.
Even after one has found many points on the surface, howdoes one connect them, what is a good way to represent
the surface? The method of raytracing has been invented
as one solution. Choose some center point C, think of it tobe near the eyes of the viewer. Connect each pixel of thescreen with C by a line and decide whether this line meetsthe surface. If it does then of all the intersection points
on the line choose the one closest to C, compute the normal of the surface at this point x (i.e. compute grad f (x))
and decide with this information what light (from fixed
light sources) will be reflected by the surface at x towards
C. Color that pixel accordingly. In this way one produces
an image which presents the surface as if it were an illuminated object. The computation used to take very long, buttodays computers do such pictures while you wait, but notquite fast enough for real time rotations. These pictures
look very realistic, but of course they show only what isvisible from the viewer, in particular: farther away partsof the surface can be hidden by nearer parts.
In 3D-XplorMath a second method is offered. Imagine thatthe surface is intersected with random lines until around
10 000 points have been found on the implicit surface. Then
red-green stereo is used to project these points to the screen.
When viewing through stereo glasses we see all these points

in the correct position in space and our brain interpolates
them and lets us see a surface in space. This representation shows all parts of the surface (within some viewing
sphere), not just the front most portions. Since one canachieve fairly uniform distributions of points on level surfaces, one sees many points in the direction towards contours of the surface. This emphasis of the contour points
is so strong that one gets a fair impression of the surface
even if one does not look through red-green glases. Thismethod is fast enough for real time rotations.
Once an implicit surface has been drawn, one has solved
the problem of computing the 3D-data of surface points
selected by mouse on the screen. One can therefore moreeasily move geometric attributes, like curvature circles,around on an implicit surface than on a parametrized surface. See in both cases the Action Menu entry Move Principal Curvature Circles.
What surfaces can one see?
In addition to the simple surfaces already mentioned wehave two groups. Alebraic Surfaces which have been studied because of their singularities, these have establishednames and extensive literature. And Compact Surfacesof higher genus, these are added because such surfaces do
not come with explicit parametrizations. (Their names are
given in 3DXM and not known elsewhere.) Already fairly
simple functions may have level surfaces which are more
complicated than tori, they are called bretzel surfaces ofgenus g > 1.

How to find functions with compact levels of genus ≥ 2.
As an example, consider two circles of radius r = 1, in the
x-y-plane, with midpoints ±cc on the x-axis. These twocircles are described as the intersection of the x-y-plane:{g(x, y, z) := z = 0} with the zero set of the function((x − cc)2 + y 2 − 1) · ((x + cc)2 + y 2 − 1)h(x, y, z) :=.
2
2
1 + (1 + cc)(x + y )The denominator prevents the function from growing too
fast, the weight factor 1 + cc is experimental. Next definef (x, y, z) := h(x, y, z)2 + (1 + cc)g(x, y, z)2 .
Clearly, the zero set of f is the union of the two circles,
which is not a surface, because grad f vanishes along thiszero set. However, most of the levels {(x, y, z); f (x, y, z) =v > 0} are surfaces without singularities. If the two circles
intersect (0 < cc < 1), then for small v = ff the levels are
the boundary of a thickening of the two circles, i.e., surfacesof genus 3. As ff increases either the middle hole or thetwo outside holes close first (depending on cc). For large
ff the level surfaces are (not completely round) spheres.
Each time such a topological change occurs we observe one
special surface, it is not smooth like the other levels, but
has one or more cone like singularities.
If cc > 1 then, for small ff , the levels are disjoint tori.
As ff increases, either the tori grow together to a genus
2 surface, or the holes of the tori close first and later thetwo sphere-like surfaces grow together.
This family is called Pretzel in 3DXM.

Functions with compact levels in 3D-XplorMath
One should always experiment with the level value v of thefunction f . In 3DXM: v = ff . For small values of ff one
will see how the function was designed by guessing the degenerate level f = 0. The Default Morph often varies ff ,for example showing non-singular levels converging to the
singular one. In some cases other parameters are morphed,for example to get larger values of the genus g. Some cases
offer: Flow to Minimum Set {f = 0} (see Action Menu).
(Artificial looking denominators in the following preventthe function f from growing too fast.)
Note that the Action Menu has many decorations for implicit surfaces: Curvature line fields, net of curvature lines,
normal curvature circles, geodesics with mouse chosen initial data, geodesic nets.
Pretzel : See page 5 of Explicit versus Implicit Surfaces
The surface has genus 0,1,2 or 3, depending on parametervalues.
f (x, y, z) := h(x, y, z)2 + (1 + cc)z 2 with((x − cc)2 + y 2 − 1) · ((x + cc)2 + y 2 − 1)))h(x, y, z) :=1 + (1 + cc)(x2 + y 2 )Bretzel2 , a genus 2 tube around a figure 8, genus 0 forlarge ff :

2 2
2 2
2
((1 − x )x − y ) + z /2
f (x, y, z) :=.
(1 + bb(x2 + y 2 + z 2 ))Bretzel5 , a genus 5 tube around two intersecting ellipses:f (x, y, z) := ((x2 + y 2 /4 − 1) · (x2 /4 + y 2 − 1))2 + z 2 /2.

Pilz , a genus 3 tube around circle and orthogonal ellipse:
f (x, y, z) :=((x2 + y 2 − 1)2 + (z − 0.5)2 ) · (y 2 /aa2 + (z + cc)2 − 1)2 + x2 )− dd2 (1 + bb(z − 0.5)2 ).
Default Morph:0.03 ≤ cc ≤ 0.83.
Orthocircles, a genus 5 tube around three intersecting orthogonal circles (aa = 1, ff = 0.05) or a tube aroundthree Borromean ellipses (aa = 2.3, ff = 0.2) – choose in
the Action Menu.
f (x, y, z) :=((x2 /aa + y 2 − 1)2 + z 2 ) · ((y 2 /aa + z 2 − 1)2 + x2 ) ·((z 2 /aa + x2 − 1)2 + y 2 ).
Use: Flow to Minimum Set {f = 0} (from Action Menu).
DecoCube, tube around six circles of radius cc on the facesof a cube. Genus 5,13,17, depending on cc, ff :f (x, y, z) := ((x2 + y 2 − cc2 )2 + (z 2 − 1)2 ) ·((y 2 + z 2 − cc2 )2 + (x2 − 1)2 ) · ((z 2 + x2 − cc2 )2 + (y 2 − 1)2 ).
Default Morph: ff = 0.02, 0.25 ≤ cc ≤ 1.3 .
Use: Flow to Minimum Set {f = 0} (from Action Menu).
DecoTetrahedron has as its minimum set four circles on thefaces of a tetrahedron. The formula is similar but more
complicated than the previous one. cc changes the radiusof the circles, bb changes their distance from the origin, ff
selects the level. Use: Flow to Minimum Set to see thecircles used for the current image.
The Default Morph changes cc and with it the genus.

JoinTwoTori is a genus 2 surface such that the connection
between the two tori does not much distort them if ff issmall. (It is used for genus-2-knots in Space Curves.)
T orright := ((x − cc)2 + y 2 + z 2 − aa2 − bb2 )2+ 4aa2 (z 2 − bb2 )T orlef t := ((x + cc)2 + y 2 + z 2 − aa2 − bb2 )2+ 4aa2 (z 2 − bb2 )T orright · T orlef t.
f (x, y, z) :=22
2
2
1 + (x − cc) + (x + cc) + y + z /2
The Default Morph: 0.01 ≤ ff ≤ 2.5 joins the tori.
CubeOctahedron
The level surfaces of the function
fcube (x, y, z) := max(|x|, |y|, |z|)
The level surfaces of the function
focta (x, y, z) := |x| + |y| + |z|

are cubes.
are octahedra.

ã := min(2 · aa, 1), b̃ := 2 · min(bb, 1). These coefficientsfor the following linear combination allow an interesting
morph.
f (x, y, z) := max(ã · focta (x, y, z) + b̃ · fcube (x, y, z)).
Default: aa = 0.5, bb = 1, ff = 1. This truncated cube isArchimedes’ Cubeoctahedron.
Default Morph: aa = 23 → 31 , bb = 0.5 → 1.5, ff = 1.
This deformation from the octahedron to the cube passes
through three Archimedean solids.

Algebraic Functions with Singularities in 3DXM
CayleyCubic :f (x, y, z) := 4(x2 + y 2 + z 2 ) + 16 x y z − 1, ff = 0.
This cubic has 4 cone singularities at the vertices of a tetrahedron. The other surfaces in the Default ff -Morph arenonsingular.
ClebschCubic :f (x, y, z) :=81(x3 + y 3 + z 3 ) − 189(x2 (y + z) + y 2 (z + x) + z 2 (x + y)) +54xyz + 126(xy + yz + zx) − 9(x2 + x + y 2 + y + z 2 + z) + 1.
This cubic has no singularities but is famous for the 27lines that lie on it. The lines are shown in 3DXM. The
surface has tetrahedral symmetry.
DoublyPinchedCubic :f (x, y, z) := z(x2 + y 2 ) − x2 + y 2 .
This cubic has two pinch-point singularities at ±1 on thez-axis. The segment between the singularities lies on it.
The whole z-axis satisfies the equation; the Default Morphshows how an infinite spike converges to this line.
KummerQuartic :λ := (3aa2 − 1)/(3 − aa2 ),f (x, y, z) := (x2 + y 2 + z 2 − aa2 )2− λ((1 − z)2 − 2x2 )((1 + z)2 − 2y 2 ), aa = 1.3.
This quartic has 4+12 cone singularities and tetrahedral
symmetry. Six noncompact pieces, each with two conepoints, are connected by five compact pieces which look likecurved tetrahedra. The singularities survive small changes,
see the Default Morph : 1.05 ≤ aa ≤ 1.5, ff = 0.

BarthSextic√:√
c1 := (3 + 5)/2, c2 := 2 + 5
f (x, y, z) :=4(c1 x2 − y 2 )(c1 y 2 − z 2 )(c1 z 2 − x2 ) − c2 (x2 + y 2 + z 2 − 1)2 .
Barth’s Sextic has icosahedral symmetry. 20 tetrahedronlike compact pieces are placed over the vertices of a dodecahedron so that each tetrahedron has 3 of its vertices
at midpoints of dodecahedron edges. This accounts for 30of the cone singularities. Each of the 20 outward pointing
vertices of the tetrahedra is connected via a cone singularity to a cone-like noncompact piece of the Sextic. The
Default Morph embeds this singular surface in a family of
nonsingular sextics. Use Raytrace Rendering.
D4 :f (x, y, z) := 4x3 + (aa − 3x)(x2 + y 2 ) + bbz 2This family of cubics has a D4-singularity. At bb = 0 thefamily degenerates into three planes, intersecting along thez-axis.
UserDefined : Our example is the Cayley Cubic, see above.
H.K.

TOC

Early Implicit Minimal Surfaces*
See: Helicoid-Catenoid and Scherk, Catalan, Henneberg
Minimal Surfaces
The early minimal surfaces have also implicit representations:
Catenoid:x2 + y 2 − cosh(z)2 = 0

Helicoid:x cos(z) − y sin(z) = 0

Scherk 1-periodic: sinh(x) sinh(y) − sin(z) = 0
Scherk 2-periodic: exp(z) cos(x) − cos(y) = 0
After the Weierstrass representation was discovered they
were not much used – as far as we know. These implicit
equations are much less close to the geometry of the surfaces than the Weierstrass representation: other levels ofthese functions are not minimal and the associate family
cannot be obtained from this representation. Although theGauss map at solution points of the equations is of course
the gradient of the implicit function at the solution point(normalized to length 1), this does not give the Gauss map
as a meromorphic function.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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The Paraboloid *See in Documentation: About Quadratic Surfaces.

The Paraboloid in 3D-XplorMath is parametrized as
x = aa · u · cos(v),y = bb · u · sin(v),z = cc · u2 − dd,with the default aa = bb = 1, cc = 0.4, dd = 2. It is given
y 2
x 2
−
(
)
−
(
implicitly by f (x, y, z) := (z+dd)
ccaabb ) = 0.
The paraboloid is shown together with a few rays parallel to the z-axis, the axis of revolution symmetry of thissurface. These rays are reflected in the surface and continued until they meet in the focal point of this paraboloid.
This image looks somewhat like the reflector of a car headlight together with the rays from the light bulb, reflected
into parallel rays. The default Morph varies cc so thatthe image changes from a headlight reflector to a satellite
antenna, with incoming parallel rays concentrated on thereceiver at the focal point of the antenna.
The entry Remove Focal Rays in the Action Menu returns
to the standard rendering for surfaces. Only in Wireframe* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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Display can one switch on the focal rays in the Action
Menu.
For geometric arguments concerning the focal point see:Parabola in the Plane Curve category.
H.K.

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The Ellipsoid*
See in Documentation: About Quadratic Surfaces.
The Ellipsoid in 3D-XplorMath is parametrized as
x = aa · sin u · cos v,

y = bb · sin u · sin v,

z = cc · cos u,

with the default aa = bb = 1.5, cc = 2.0. It is given by the
Implicit Equation
f (x, y, z) := (x/aa)2 + (y/bb)2 + (z/cc)2 = 1.
In 3D-XplorMath the Ellipsoid is shown together with a
few rays which leave one focal point, are reflected in thesurface and come together again in the other focal point.
This illustrates the use of the Ellipsoid as a Whispering
Gallery. A whispering gallery may be realized by an Ellipsoid ceiling in a pub so that the conversations at one table
can be heard at another table. Whispering galleries werealso built in royal parks with some ellipsoid reflector neara table for visitors placed at one focal point and a hidden
chair for the listener at the other focal point.
The first default Morph varies the size of the Ellipsoid.
One can also select in the View Menu Morph Light Source
Of Rays to illustrate that the rays do not come together
at one point unless they start from a focal point.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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By selecting Remove Focal Rays in the Action Menu one
returns to the standard rendering of surfaces. One may
turn on the focal rays only if Wireframe Display is selected.
For geometric arguments concerning the focal points see:
Ellipse in the Plane Curve category.
H.K.

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About Quadratic Surfaces *Quadratic surfaces in R3 are the solution sets of quadratic
equations (see: Explicit versus Implicit Surfaces)h(x, y, z) := Ax2 + By 2 + Cz 2 + Dxy + Exz + F yz
+ Gx + Hy + Jz + K = 0.
Explicit parametrizations are given at the end.
There are poor examples, i.e.
with no solutions:x2 + y 2 + z 2 + 1 = 0,

solutions consisting of a point:
or solution sets consisting of a line:

x2 + y 2 + z 2 = 0,x2 + y 2 = 0,or solution sets like that of a linear function: x2 = 0.
But under mild assumptions, namely that the derivativeof h does not vanish on (most of) the solution set, we getmore interesting surfaces (possibly with singularities) as
solution sets.
For products of linear functions we have intersections of(or parallel) planes:(x − y + a)(x ± y + b) = 0,
we may have cylinders over quadratic curves, e.g. if theequation does not contain z:
elliptic cylinder

x2 + y 2 − 1 = 0,

hyperbolic cylinder

x2 − y 2 − 1 = 0,

parabolic cylinder

x2 − y = 0.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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There are various Cones, namely solution sets which contain for each solution (x, y, z) =/ (0, 0, 0) the whole liner · (x, y, z), r ∈ R through (0, 0, 0). For example, if we intersect the previous cylinders with the plane z = 1 and takethe cone with vertex at (0, 0, 0) then these are describedby the following (so called “homogenous”) equations.
Cones: x2 + y 2 − z 2 = 0, x2 − y 2 − z 2 = 0, x2 − yz = 0.
And finally we have the quadratic surfaces which have neither singular points nor are they cylinders:Ellipsoids
x2 /a2 + y 2 /b2 + z 2 /c2 − 1 = 0,
1-sheeted Hyperboloids x2 /a2 + y 2 /b2 − z 2 /c2 − 1 = 0,2-sheeted Hyperboloids x2 /a2 + y 2 /b2 − z 2 /c2 + 1 = 0,
Elliptic Paraboloids
x2 /a2 + y 2 /b2 − z = 0,
Hyperbolic Paraboloids
x2 /a2 − y 2 /b2 − z = 0.
All other quadratic surfaces are obtained via coordinatetransformations from these examples. Try theExperiment: Select Implicit from the Surface Menu andtype any quadratic equation into UserDefined. Compare
the displayed surface with those described above. (Notethat there may be no solutions.)
The 1-sheeted hyperboloids and the hyperbolic paraboloids
have an unexpected special property, they carry two families of straight lines, see also: About Ruled Surfaces .
The hyperbolic paraboloid: x2 − y 2 − z = 0 is cut by theparallel family x + y = const of planes in (disjoint) lines
and also by the parallel planes x − y = const.
The 1-sheeted hyperboloid: x2 + y 2 − z 2 − 1 = 0 is a surface

of revolution. Its tangent plane x = 1 intersects it in thepair of orthogonal lines (y +z)(y −z) = 0, x = 1. Rotation
around the z-axis gives two families of lines on the surface.
Each tangent plane cuts the surface in two lines, one fromeach family.
Explicit parametrizations
Ellipsoid:x = aa · sin u cos v, y = bb · sin u sin v, z = cc · cos u,
1-sheeted Hyperboloid:x = aa cosh u cos v, y = bb cosh u sin v, z = cc sinh u,
2-sheeted Hyperboloid (2nd sheet z → −z):x = aa sinh u cos v, y = bb sinh u sin v, z = cc cosh u,
Elliptic Paraboloid:x = aa · u cos v, y = bb · u sin v, z = cc · u2 ,Hyperbolic Paraboloid:x = aa · u, y = bb · v, z = cc · uv.
H.K.

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Ruled Surfaces *Cylinders, Cones, 1-sheeted Hyperboloid, Hyperbolic
Paraboloid, Helicoid, Right Conoid, Whitney Umbrella.
In other sections: Double Helix, Möbius Strip.
Informally speaking, a ruled surface is one that is a unionof straight lines (the rulings). To be more precise, it is asurface that can be represented parametrically in the form:x(u, v) = δ(u) + v ∗ λ(u)where δ is a regular space curve (i.e., δ 0 never vanishes)called the directrix and λ is a smooth curve that does
not pass through the origin. Without loss of generality, we
can assume that |λ(t)| = 1. For each fixed u we get a line
v 7→ δ(u) + v ∗ λ(u) lying in the surface, and these are therulings. (Some surfaces can be parameterized in the above
form in two essentially different ways, and such surfaces
are called doubly-ruled surfaces.)
A ruled surface is called a cylinder if the directrix lies ina plane P and λ(u) is a constant direction not parallel to
P , and it is called a cone if all the rulings pass through afixed point V (the vertex).
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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Two more interesting examples are quadratic surfaces:
The Hyperboloid of One Sheet:x2y2z2+ 2 − 2 = 1,2abcwhich is in fact doubly-ruled, since it can be given parametrically by:
x+ (u, v) = a(cos(u) − v sin(u)), b(sin(u) + v cos(u), cv)
andx− (u, v) = a(cos(u) + v sin(u)), b(sin(u) − v cos(u), −cv),
and the Hyperbolic Paraboloid:(x, y, z) = (a u, b v, c u v) = a(u, 0, 0) + v(0, b, c u).
Another interesting ruled surface is a minimal surface:
the Helicoid, aa = 0, (Catenoid, aa = π/2) in the family:
F (u, v) = bb sin(aa) cosh(v) cos(u), cosh(v) sin(u), v

+ bb cos(aa) sinh(v) sin(u), − sinh(v) cos(u), u= sin(aa) (0, 0, bb v) + bb cosh(v)(cos(u), sin(u), 0)+ cos(aa) (0, 0, bb u) + bb sinh(v)(sin(u), − cos(u), 0) .

A ruled surface is called a (generalized) right conoid if its
rulings are parallel to some plane, P , and all pass througha line L that is orthogonal to P . The Right Conoid isgiven by taking P to be the xy-plane and L the z-axis:
Parametrized: F (u, v) = (v cos u, v sin u, 2 sin u),x
4
Implicitly:( )2 − 2 = 1.
yzThis surface has at (sin(u) = ±1, v = 0) two pinch point
singularities. The default morph in 3DXM
deforms the Right Conoid to a Helicoid
so that the two stable pinch point singularities disappear,at the final moment, through two unstable singularities:
Faa (u, v) = ( v cos(u), v sin(u), 2aa sin(u) + (1 − aa)u ) .
Famous for such a singularity is the Whitney Umbrella,another right conoid with rulings parallel to the x-y-plane:
F (u, v) = (u · v, u, v · v), implicitly: x2 − y 2 z = 0.
Again the default morph emphasizes the visualization of
the singularity by embedding the Whitney Umbrella
into a family of ruled surfaces, which develop a second
pinch point singularity that closes the surface at the top:
u · (aa · v + (1 − aa) sin(πv))
.
Faa (u, v) = uaa · v 2 − (1 − aa) cos(πv)
R.S.P.

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Monkey Saddle, Torus, Dupin Cyclide *The Monkey Saddle is a saddle shaped surface with threedown valleys, allowing the two legs and the tail of the monkey to hang down. At its symmetry point both principal
curvatures are 0, and, this umbilic point is the simplest
singularity of a curvature line field. Choose in the Action
Menu: Add Principal Curvature Fields; in Wireframe
Display the parameter lines are omitted, the curvatureline fields (or one of them) represent the surface.
Its Parametrization as graph of a function is
FM onkey (u, v) = (aa · v, bb · u, cc · (u3 − 3uv 2 )).
In Geometry the word Torus usually implies a surface ofrevolution; often a circle in the x-z-plane ist rotated aroundthe z-axis. In 3DXM an ellipse with axes bb, cc is rotated,
its midpoint rotates in the x-y-plane on a circle of radius
aa. The following Parametrization is used:(aa + bb · cos u) cos v
FT orus (u, v) =  (aa + bb · cos u) sin v 
cc · sin u
Note that the parameter lines are principal curvature lines,
see Action Menu: Add Principal Curvature Fields.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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The Torus is also visualized among the Implicit Surfaces,
we derive its equation. In the x-z-plane we have two ellipses
and we multiply their equations:x − aa 2z 2
x + aa 2z 2(+− 1) · (
+
− 1)
bb
cc
bb
cc
x2 − aa2 2x2 + aa2  z 2
z 2
2
=
+
2
(
−
1)
+
(
−
1)
.
bb2
bb2
cc
cc
For thep rotation around the z-axis we have to replace x by
r = x2 + y 2 . The second expression avoids square roots.
Implicit Equation of the Torus:pfT orus (~x) = f (r, z) = 0 with r = x2 + y 2

andf (r, z) :=r2 − aa2 2
r2 + aa2  z 2
z 2
2
+
2
(
−
1)
+
(
−
1)
.
2
2
bb
bb
cc
cc
The Cyclides of Dupin are obtained by inverting theabove torus in a sphere. The sphere of inversion has its
center m~ = (dd, 0, ee) in the x-z-plane and has radius f f .
The Default Morph moves the center closer to the torus.
Note that inversions map curvature lines to curvature lines.
 
0
f f 2 (~x − m)~
The Inversion: ~x 7→ Inv(~x) :=+m~ +  0 ,
2|~x − m|~hhParametrization: FCyclide (~x) := Inv(FT orus (~x)),
Implicit Equation: fCyclide (~x) := fT orus (Inv−1 (~x)) = 0.
H.K.

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Lissajous, Double Helix, Column, Norm 1 Family *The French mathematician Jules Antoine Lissajous (18821880) studied vibrating objects by reflecting a spot of lightof them, so that the various modes of vibration gave rise to
Lissajous curves, see Plane Curve Category. Lissajous
Space Curves and Lissajous Surfaces are a natural mathematical generalization. We use the Parametrization:


sin u

FLissajous (u, v) = sin vsin((dd − aa u − bb v)/cc) .
The Default Morph joins a surface with tetrahedral symmetry and conical singularities and a surface with cubicalsymmetry and 12 pinch point singularities.
The Double Helix is a reminder of the famous double
helix from genetics. For playing purposes there are twomore parameters with default values dd = 0, ee = 0. We
use the parametrization:
AA := aa + dd u, α := (1 − ee u)u,
AA((1 − v) cos α + v cos(α + bb π))
FDblHelix (u, v) =  AA((1 − v) sin α + v sin(α + bb π)) cc u − 3.5.
This is a family of ruled surfaces: try the Default Morph,
it varies the limits of the parameter v. With bb = 1 we get* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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the Helicoid. The default parameters have been taken from
Watson-Crick and give a reasonably good representation ofmolecular DNA. Dick Palais’ biologist friend Chandler Fulton suggested the example and helped to get it right, manythanks.
We suggest to select Move Principal Curvature Circles
from the Action Menu; this is seen best in Point Cloud
Display from the View Menu.
Column Surface is used here in the sense of an architectural column, see the article by Marty Golubitzky and Ian
Melbourne at
http://www.mi.sanu.ac.rs/vismath/golub/index.html
The order of the rotational symmetry around the z-axis ischosen with the 3DXM-parameter ii. Additional symmetry types can be selected with the parameter hh = 1, . . . , 6;
for other values of hh the unsymmetrized column shapeis given by a formula that depends on the parameters
aa, . . . , gg, ii and on the coordinates (θ, z). The formula
is not determined by geometric properties, but is intended
for playing.
The Default Morph, with hh = 0, varies the shape only
mildly.
The Norm 1 Family is defined by the implicit equation:f (x, y, z) = (|x|p + |y|p + |z|p )1/p = 1,

0 < p < ∞.

We get at p = 1 an Octahedron, at p = 2 a Sphere andat p = ∞ a Cube. For 1 ≤ p ≤ ∞ these surfaces can beviewed as the unit sphere in R3 for a Banach metric determined by p.
We parametrize these surfaces by spherical polar coordinates:
xp := sin v cos u, yp := sin v sin u, zp := cos v,
x := sign(xp)|xp|e , y := sign(yp)|yp|e , z := sign(zp)|zp|e .
To obtain a reasonable family we set the exponent e interms of the default morphing parameter ee as follows:e := 1 + tan(ee), −π/4 < ee < π/2.
We obtain the Sphere at ee = 0,
the Octahedron at ee = π/4.
Numerical reasons prevent computation at ee = −π/4
(anyway a degenerate surface) and at ee = π/2, the Cube.
Already where we stop the computation the Pascal-valuesof sin near π had to be improved.
H.K.

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Snail Shell Surface *These snail-like surfaces are included for their entertaining
shapes. Try making one of your own. In spite of theircomplicated appearance, the snail surfaces are constructed
as one-parameter families of circles u 7→ Cv (u). First we
introduce two auxiliary variables. The surface parameterv is changed by a quadratic term that permits closing thesnails at the top. The parameter ee controls the size of the
opening of the snail (default ee = −2):
vv := v + (v + ee)2 /16.
The second variable controls the radius of the circles:s := exp(−cc · vv). (Note that s is a function of v.)
The circles u 7→ Cv (u) of radius s · bb lie in an r-y-plane:
r := s · aa + s · bb · cos(u),y := dd(1 − s) + s · bb · sin(u).
The parameter dd controls the length of the snail from top
to bottom. And the other two coordinates in R3 are
x := r cos(vv),z := r sin(vv),so that the plane of the circle Cv also rotates with v.
Advice: Make only small changes to cc and keep bb ≥ aa.
The Default Morph varies dd and adjusts bb a little.
T.K.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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Untangle Double Twist In SO(3) *The goal of this animation is first to visualize closed curves
in SO(3) and then morph these closed curves to visualize
homotopies. While a full rotation around a fixed axis is a
nontrivial loop in SO(3), doing it twice is a nullhomotopic
loop. It was made famous outside mathematics by Dirac’s
belt trick and Feynman’s plate trick.
Bob Palais has two descriptions of the homotopy in SO(3)which are much simpler than to look at a 2-parameter family of orthogonal matrices:(a) A rotation with axis vector ~a and rotation angle ϕ canbe obtained as composition of two 180◦ rotations aroundaxes ~b, ~c ⊥ ~a with angle(~b, ~c) = ϕ/2. To get the double
twist around the z-axis (the loop of the above tricks) choose~b := (1, 0, 0), ~c(s) := (sin(s), cos(s), 0), 0 ≤ s ≤ 2π.
It is obvious how to homotop the equator s 7→ ~c(s) into
the point loop ~b, hence homotop the double twist into id.
(b) A homotopy from one full twist to its inverse is given
by rotating the axis vector to its negative - simple enough.
Composition of this homotopy with the chosen full twist isa homotopy from a double twist to the id-loop.
In other words, the challenge is not to formulate such ahomotopy, but to visualize it. And, again, the challenge is
to visualize a loop in SO(3), because a homotopy is simply
a time dependent morph of a loop.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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Let A(s) ∈ SO(3) be a loop of rotations with A(0) =id = A(1). Consider a family of concentric spheres. They
should really be all the same spheres, but we use the radius parameter similar to the graph representation of a
1-dimensional function. Draw a spherical polygon P on
the innermost sphere of radius 1. Then draw the spherical polygons A(s)P on the spheres of radius 1 + s. By
following the position of the moving spherical polygon ass increases, one gets a good impression of the family A(s)of rotations.
We use two opposite spherical polygons instead of one, toemphasize the fixed innermost sphere.
The default morph 0 ≤ aa ≤ 1 unwraps the double twist,of course keeping the endpoints fixed.
The morph 0 ≤ bb ≤ 1 with aa = 0 creates the double
twist around the z-axis (a loop only if bb = 1).
The circum radius of the spherical polygons is cc.
One can choose with dd ∈ {3, 4, 5, 6} the number of verticesof the spherical polygon (dd = 5.2 is a pentagon star).
In the Action Menu is an entry intended for this demo:
Reflect tube in ImagePlaneYdirection
This allows to change the position of the spherical polygon.
The default position is around the z-axis. Moving it awayfrom the axis towards the equator creates rather different
images of the same family A(s) of rotations. Clicking thisentry a second time returns to the original position - unless
one has rotated the object with the mouse.
H.K.
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Dirac Belt and Feynman Plate Tricks *Dirac invented the famous belt trick to demonstrate aproperty of the motions of Euclidean Space that is indeed
difficult to imagine without visual help.
The trick is performed with a strip, or belt, that is initially parallel to the screen. The orthogonal projection of
the performance looks as follows: The left end of the belt
stays fixed, the right end moves around the left end in acircular motion. It is important that the moving end stays
parallel to the fixed end through the whole trick (parallelmeans: the final edge and the final normal each stay parallel to the initial edge and initial normal). One observeswith surprise:
After moving the right end once around the circle the belt
is twisted twice. After the second circular move the belt is
untwisted (as it was initially).
The trick is shown in stereo because it is impossible thatthe belt stays in its initial plane when the ends are moved
as described. It is important to visualize how the different
parts of the belt move vertically to the screen.
It will be no surprise to observe that the first circular movement – when looked at in 3D – is different from the second
circular movement. During the first circular movement the* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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middle part of the belt moves vertically to be in front of
the screen while the two ends stay on the screen. During
the second circular movement it is the other way round:the middle part of the belt moves vertically to be behindthe screen while the ends continue to be at their fixed vertical position. As soon as one can fix this image in one’s
mind it is obvious how the circular motion with parallelends produces the twist of the belt.
Another instance of the same property of the Group of Euclidean motions is the Feynman Plate Trick or Waiter’s
Cup Trick : It is possible to continuously rotate a cup
on ones horizontal hand in the same direction if during
the first rotation the hand is above elbow height, during
the second rotation below elbow height and so on, alternatingly above and below elbow height. Namely, imagine
that the shoulder is the fixed end of the belt and the always horizontal – but continuously rotating – middle partof the belt is the hand. Choose Do Plate Trick from theAnimation Menu to see half the belt performing the trick.
Some people cannot believe what they see. In such a case
one can switch to Monocular Vision and Orthographic
Projection in the View Menu to watch the belt – without
at all observing the motion vertical to the screen. Or try
Patch Display for a solid belt.
B.P.

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Surfaces of Constant Width*
The name of these surfaces derives from the fact that the
distance between opposite parallel tangent planes is constant. See first: Convex Curve in the planar curve category. There the default curve and the default morph show
curves of constant width and the ATO: On Curves Given
By Their Support Function explains how they are made.
Our surfaces of constant width are also described via theirSupport Function h : S2 7→ R as: x
F (x, y, z) := h(x, y, z) ·  y  + grad S2 h(x, y, z),zh(x, y, z) := aa + bb z 3 + cc xy 2 + dd yz 2 + ee xz 2
+ ff xyz + gg xy 2 z 2 > 0,
where

x2 + y 2 + z 2 = 1.

The constant aa has to be chosen large enough so thath(x, y, z) > 0. The default values are aa = 1, ff = 0.66, allothers = 0, for tetrahedral symmetry. With just aa, bb > 0
one gets surfaces of revolution, with aa, cc =/ 0 the surfaces
have 120◦ dihedral symmetry. Note that gg is the only
coefficient of a polynomial of degree 5.
Let c(t) = (x, y, z)(t) be a curve on S2 . One computes with* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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d
F (x, y, z)(t) ⊥ ~n, so that ~n(x, y, z) is~n := (x, y, z)t that dt
the normal of the surface parametrized by F . Therefore
h(x, y, z) is indeed the distance of the tangent plane at
F (x, y, z) from the origin. The origin is inside the surface
because h > 0. All terms defining h, except the constantaa, are odd. This gives h(x, y, z) + h(−x, −y, −z) = 2 · aa
and this is the distance between opposite tangent planes,i.e. the constant width.
Finally we compute the normal curvature, more precisely
the Weingarten map S.

dd~n(t) = ċ(t) =: S · F (c(t))
dtdtdF (x, y, z)(t) = hgrad S2 h, ċ(t)ic(t) + h · ċ(t)dt+ dċ(t) grad S2 htangential= h · ċ(t) + dċ(t) grad S2 h
= h · ċ(t) + Dċ(t) grad S2 h,
where Dċ , the tangential component of the Euclidean derivative dċ(t) , is the covariant derivative of S2 . We thus obtain
the Weingarten map S of the surface, computed in thedomain S2 of F :
−1 d
d
d
2
~n(t) = h · id + Dgrad S h· F (c(t)) = S · F (c(t)).
dt
dt
dt
H.K.

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Clifford Tori *a) Parametrized by Curvature Lines, b) Hopf-fibered
Clifford Tori are embeddings of the torus into the unit
sphere S3 of R4 , by (u, v) → F (u, v) := (w, x, y, z), where 
cos α cos uw cos α sin u x
FClifford (u, v) =  (u, v) = sin α cos v
ysin α sin vz
α := aa + bb sin(ee · 2 v), bb =/ 0 for Bianchi-Pinkall Tori.
(Note that this is, for bb = 0, the product of a circle inthe (w, x) plane with a second circle in the (y, z) plane,and so is clearly flat.) To get something that we can see in
R3 , we stereographically project S3 7→ R3 ; i.e., the Clifford
tori in R3 are the embeddings (u, v) → P (F (u, v)), where
P : S3 → R3 is Stereographic projection.
We take as the center of the stereographic projection map
the point (cos(cc π), 0, sin(cc π), 0). Varying cc deforms atorus of revolution through cyclides. The Default Morphvaries aa, hence changes the ratio of the two circles.
Another morph, Conformal Inside-Out Morph (also in
the Animation Menu), is in R4 a rotation (parametrizedby 0 ≤ ff ≤ 2π), that moves the torus through the centerof the stereographic projection. The image in R3 therefore
passes through infinity: we see a torus with one puncturethat has a flat end. It looks like a plane with a handle.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

The Clifford tori (in S3 ) are fibered by Great Circles, theHopf fibers, u + v = const. These Great Circles are ofcourse the asymptote lines on the tori. We show two versions of the stereographically projected Clifford tori: a)parameterized by curvature lines and b) by Hopf fibers.
(To get the explicit parametrization of the latter, takeF (u + v, u − v) in the above formulae.)
The classical Clifford Torus corresponds to α = aa = π/4.
It has maximal area among the family and divides S3 into
two congruent solid tori. But the other torus-leaves ofthe foliation, obtained by varying aa, are also interesting.
All of them are foliated by Clifford-parallel great circles
and hence flat. They are special cases of the flat BianchiPinkall Tori in S3 (visible after stereographic projection)and discussed in more detail in their ATO (“About This
Object...”), see the Documentation Menu.
Why is the ff -morph a Conformal Inside-Out Morph?
A compact surface divides R3 in two components and thebounded component is called the inside. One surface ofthe family, the once punctured torus that passes throughinfinity, divides R3 into two congruent unbounded components. This surface has no inside and at this moment in
the deformation inside and outside are interchanged. The
180 degree rotation in the rotation family is, on the π/4torus, a conformal anti-involution. It has a Hopf fiber as
connected fixed point set. Use in the Action Menu Surface Coloration and choose the default two-sided usercoloration which emphasizes the fixed fiber.
H.K.
TOC

Hopf Fibration and Clifford Translation *of the 3-sphere
See Clifford Tori
Most rotations of the 3-dimensional sphere S3 are quite
different from what we might expect from familiarity with
2-sphere rotations. To begin with, most of them have no
fixed points, and in fact, certain 1-parameter subgroups ofrotations of S3 resemble translations so much, that they
are referred to as Clifford translations. The description
by formulas looks nicer in complex notation. For this we
identify R2 with C, as usual, and multiplication
 by i in C
0 −1.
represented in R2 by matrix multiplication by
1 0
Then the unit sphere S3 in R4 is given by:
S3 := {p = (z1 , z2 ) ∈ C2 ; |z1 |2 + |z2 |2 = 1}
X4
∼ {(x1 , x2 , x3 , x4 ) ∈ R ;(xk )2 = 1}.
And for ϕ ∈ R we define the Clifford Translation Cϕ :S3 → S3 by Cϕ (z1 , z2 ) := (eiϕ z1 , eiϕ z2 ).
The orbits of the one-parameter group Cϕ are all great
circles, and they are equidistant from each other in analogy* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOCto a family of parallel lines; it is because of this behaviour
that the Cϕ are called Clifford translations.
But in another respect the behaviour of the Cϕ is quite
different from a translation – so different that it is difficult to imagine in R3 . At each point p ∈ S3 we have one
2-dimensional subspace of the tangent space of S3 which
is orthogonal to the great circle orbit through p. A Euclidean translation would simply translate these normal
spaces into each other, but a Clifford translation rotates
them so that the velocity of the translation along the orbit
is equal to the angular velocity of the rotation of the normal spaces. This normal rotation is responsible for a very
curious fact which is illustrated by the image in 3DXM:
Any two orbits are linked !
The fact that any two orbits are equidistant permits us tomake the set of orbits into a metric space., and one cancheck that this space is isometric to the sphere of radius
one-half in R3 . Therefore one can map h : S3 → S2 bymapping p ∈ S3 to its orbit, identified as a point of S2 ,and one can write this mapping in coordinates as:h(z1 , z2 ) = (|z1 |2 − |z2 |2 , Re(z1 z2 ), Im(z1 z2 ))This map h is called the Hopf map (or Hopf Fibration), andthe orbits, the fibres of this map, are called Hopf fibres. It
is named for Heinz Hopf, who studied it in detail, andfound the completely unexpected fact that this map could
not be deformed to a constant map.
The visualization in 3D-XplorMath shows four tori each ofwhich is made up of Hopf fibres. We emphasize this withthe coloration: each fibre has a constant colour and thecolour varies with the distance of the fibres. One can seethat any two of the four tori are linked, and one can also
see that any two fibres on any one such torus are linked.
Since 3D-XplorMath visualizes objects in R3 , not in S3 ,before rendering the tori, we first map them into R3 using
the following stereographic projection S3 → R3
‘T13’:(x1 , x2 , x3 , x4 ) → (x1 , x2 , x3 )/(1 + x4 ).
The use of morphing parameters:
The three cyclide-tori are made from the torus of revolution by quaternion multiplication in S3 with
Qt2 = (cos ee, 0, sin ee sin bb, sin ee cos bb),
Qt3 = (cos ff cos cc, sin ff, 0, cos ff sin cc),
Qt4 = (cos gg, 0, sin gg cos dd, sin gg sin dd).
One can therefore morph the first cyclide with the parameters bb, ee, the second with cc, f f and the third with dd, gg.
The Default Morph changes the first two cyclides slightly
while the third one performs a full rotation in S3 whichmoves it in R3 from its initial position through the torusof revolution and continues back to the initial position.
H.K.

TOC

Bianchi-Pinkall Flat Tori in S3 *See ‘Clifford’ and Hopf-fibered Linked Tori first.
1. Parameter Dependent Formulas in 3DXM
We can parametrize S3 , considered as a submanifold of C2 ,by:
F (u, α, v) = (cos(α)eiu eiv , sin(α)eiu e−iv ),where u ∈ [0, 2π), α ∈ [0, π/2], and v ∈ [0, π]. We will get
the Bianchi-Pinkall Tori first as flat tori in S3 by taking
α to be a function of v,
α := aa + bb sin(ee · 2v)(although the theory allows more general choices.) Next
we stereographically project S3 fromp = (cos(cc · π), 0, sin(cc · π), 0)
to get conformal images of the flat tori in S3 . The lines
v = const are circles, the stereographic images of the Hopf
circles u 7→ F (u, α, v).

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

The Default Morph chooses ee = 5 and changes the amplitude bb thus increasing the five ’ears’ of the torus. The
Range Morph starts with a narrow band between two Hopf
circles and widenes this band to the complete surface. Finally, the Conformal Inside-Out Morph, 0 ≤ ff ≤ 2π,
isometrically rotates S3 so that the Hopf circle v = 0 is therotation axis. The stereographic image of this rotation is a
conformal transformation of R3 ∪ {∞} which “rotates” R3around a circle on the pictured torus. In the case aa = π/4
we obtain for ff = 0 and ff = π the same torus, but inside and outside interchanged. This is best viewed withthe default Two Sided User Coloration, selectable fromthe Surface Coloration Submenu of the Action Menu.
2. Background and Explanations
The tori that we usually see are, from the point of viewof complex analysis, rectangular tori, meaning that they
have an orientation reversing symmetry and the set of fixed
points of this symmetry has two components. (The better known tori of revolution have isometric reflections withtwo circles as fixed point sets.) Of course one tries to deform these tori to obtain non-rectangular ones. Obviously
one can destroy the mirror symmetry, but this does not
imply that one gets tori with a non-rectangular complex
structure. The first proof, by Garcia, that one can embed
all tori in R3 was non-constructive and difficult.
A simpler and constructive way to get tori with arbitraryconformal type was found by Pinkall, whose idea was toconstruct tori that are flat in S3 (and hence have an easy
way to compute their conformal type from their flat geometry), and then stereographically project them to R3 .
While the resulting tori are no longer flat, this does preserve their conformal type.
The construction of flat surfaces in S3 goes back to 1894,
when Bianchi classified all flat immersions in S3 . In particular, he realized that the two families of asymptotic lines ofa flat surface in S3 are left translations of a pair of curves
that are either great circles or have constant torsion +1
and −1, respectively. The left translations arise by viewing S3 as the group of unit quaternions. An open problem
for Bianchi was to determine when his flat surfaces wereclosed.
The first case when one of the curves is a great circle is of
special interest for this problem. To explain why, we willneed the Hopf fibration. Thinking of S3 as being part of C2 ,we can multiply points of S3 by eiu , thus fibering S3 withcircles, the Hopf circles, and the set of all such circles forms
a metric space with distance being the distance between
the Hopf circles in S3 . As such it is isometric to a 2dimensional sphere of radius 1/2. We thus obtain a naturalprojection S3 → S2 , the Hopf map. It can be written
as (z1 , z2 ) 7→ z1 /z2 , where we interpret the range as the
Riemann sphere Ĉ. Moreover, Hopf circles are mapped to
Hopf circles by left translations.

Now suppose we have a flat surface in S3 where one of thegenerating curves is a great circle. We can arrange S3 sothat this great circle is part of the Hopf fibration, and thus
all curves of the same family of asymptotic lines are Hopf
circles. The surface in S3 is thus invariant under the Hopf
action and projects to a curve in S2 under the Hopf map.
Vice versa, the preimage of a curve in S2 under the Hopf
map yields a flat surface in S3 . In case the curve in S2 isclosed, the surface in S3 is a flat torus. (The explanation
so far is described in more detail in Spivak IV, p. 139ff.)
Pinkall found a simple way to determine the conformal
type of the flat torus in terms of the geometry of the curvein S2 — in particular it was then easy to see that all possible conformal types can occur.
3. Visualizing Parts of the Theoretic Description
We cannot visualize S3 in such a way that all distances arepreserved. We will use stereographic projection from p =(cos(cc · π), 0, sin(cc · π), 0) to map S3 − {p} one-to-one onto
R3 . Recall that: angles are not changed by stereographic
projection, circles are mapped to circles or straight lines,
and the images of great circles meet the equator sphere in
antipodal points, so many properties of S3 get represented
in geometrically comprehensible ways.
Our parametrization F of S3 emphasizes the Hopf fibration
since the great circles u 7→ F (u, α, v) are indeed the orbitsof the Hopf-action of S1 on S3 , given by (u, p) 7→ eiu p.
Each such “Hopf Fiber” lies in one of the parallel tori α =constant, and the great circles α 7→ F (u, α, v), meet these(α = constant)-tori orthogonally, so that α measures the
distance between them.
We get all the Hopf circles on each α-torus for 0 ≤ v ≤ π,
except that those tori degenerate to just one Hopf circle if
α = 0 or α = π. This makes it plausible that (α, 2v) arepolar coordinates on the metric space of Hopf circles, on
the image S2 of the Hopf map.
Pinkall has observed that the closed curves on this image
sphere, in polar coordinates given as: (α(s), 2v(s)), (with
α(s) never equal to 0 or π/2) allow one to write down
immersed tori in S3 as:(u, s) 7→ (F (u, α(s), v(s)).
For example taking α(s) = π/4 gives the “Clifford Torus”
in S3 , a minimal embedding of the square torus. For otherconstant α(s) in (0, π/2) one gets the above parallel familyof α-tori, the lengths of their two orthogonal generators are
2π cos(α) and 2π sin(α).
On all of these tori we still have that the parameter liness = constant are Hopf-Fibers, and since these are equidistant (as orbits of an isometric action) it follows thatthe metric is flat. Pinkall proved that length and area ofthe curve in S2 determine the conformal structure of thetorus in S3 , hence in R3 , and that all conformal structures
occur.
Observe that the usual tori of revolution in R3 are allrectangular, and most of the Bianchi-Pinkall tori shownby 3D-XplorMath are very different from these. The tori

with aa = π/4 are all rhombic, because they can be rotated into themselves by 180◦ rotations (in S3 , not in R3 )around any of the Hopf-Fibers on them. A cyclic morph
with 0 ≤ ff ≤ 2π rotates around the circle v = 0 (we seeof course the stereographic image of that rotation). For
ff = π we get an anti-involution of the torus with the circle as the (connected) fixed point set—only rhombic tori
have such anti-involutions. (The square torus is rectangular and rhombic.) In the rhombic case aa = π/4 we get for
ff = π/2 and ff = 3π/2 surfaces in S3 that pass throughp so that the stereographic images in R3 pass through ∞— otherwise we could not turn the torus inside out continuously.
The program takes α(v) := aa + bb sin(ee 2v) (with ee = 3
for the default image and ee = 5 for the default morph),
allowing rather different examples.
Again, these tori are shown in R3 by using the (conformal) stereographic projection of S3 \ {p} → R3 , wherep = (cos(cc · π), 0, sin(cc · π), 0) . Morphing cc therefore
gives other images of S3 , in particular other conformal images of these tori.
H.K.

TOC

Möbius Strip and Klein Bottle *Other non-orientable surfaces in 3DXM:
Cross-Cap, Steiner Surface, Boy Surfaces.
The Möbius Strip is the simplest of the non-orientablesurfaces. On all others one can find Möbius Strips. In
3DXM we show a family with ff halftwists (non-orientablefor odd ff , ff = 1 the standard strip). All of them are
ruled surfaces, their lines rotate around a central circle.
Möbius Strip Parametrization:aa(cos(v) + u cos(ff · v/2) cos(v))
FMöbius (u, v) =  aa(sin(v) + u cos(ff · v/2) sin(v)) aa u sin(ff · v/2).
Try from the View Menu: Distinguish Sides By Color.
You will see that the sides are not distinguished—because
there is only one: follow the band around.
We construct a Klein Bottle by curving the rulings of theMöbius Strip into figure eight curves, see the Klein Bottle
Parametrization below and its Range Morph in 3DXM.
w = f f · v/2

(aa + cos w sin u − sin w sin 2u) cos v
FKlein (u, v) =  (aa + cos w sin u − sin w sin 2u) sin v 
sin w sin u + cos w sin 2u.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

There are three different Klein Bottles which cannot be
deformed into each other through immersions. The best
known one has a reflectional symmetry and looks like aweird bottle. See formulas at the end. The other two
are mirror images of each other. Along the central circle
one of them is left-rotating the other right-rotating. See
the Default Morph of the Möbius Strip or of the Klein
Bottle: both morphs connect a left-rotating to a rightrotating surface.
On the Boy Surface one can see different Möbius Strips.
The Default Morph begins with an equator band which isa Möbius Strip with three halftwists. As the strip widens
during the deformation it first passes through the tripleintersection point and at the end closes the surfaces witha disk around the center of the polar coordinates.
Moreover, each meridian is the centerline of an ordinaryMöbius Strip. Our second morph, the Range Morph, rotates a meridian band around the polar center and covers
the surface with embedded Möbius Strips. - We suggest
to also view these morphs using Distinguish Sides By
Color from the View Menu.
On the Steiner Surface and the Cross-Cap the Möbius
Strips have self-intersections and are therefore more difficult to see. The Default Morph for the Steiner Surface
emphasizes this unusual Möbius Strip. - The Range Morphof the Cross-Cap shows a family of embedded disks, except
at the last moment, when opposite points of the boundary
are identified, covering the self-intersection segment twice.

The construction of the mirror symmetric Klein bottle
starts from a planar loop with parallel, touching ends andzero velocity at the ends (see the default morph):cx(u) = −cc · cos(u),0 ≤ u ≤ 2πcy(u) = ee · sin(u3 /π 2 ),0≤u≤πcy(u) = ee · sin((2π − u)3 /π 2 ),π ≤ u ≤ 2π(nx, ny)(u) is the unit normal,

0 < u ≤ 2π

The Klein bottle is a tube of varying radius rad(u) aroundthis curve. The function rad(u) is experimental:rad(u) = bb · (dd + sin(1.5π · (1 − cos3 ((u + 0.5)/2.85))));The parameters cc, ee, bb just control the amplitudes of thefunctions; dd > 1 allows to vary the ratio between maximum and minimum of rad(u).


cx(u) + rad(u) · nx(u) cos(v)FsymKlein (u, v) = aa ·  cy(u) + rad(u) · ny(u) cos(v) rad(u) · sin(v)
.
H.K.

TOC

The Cross-Cap and Steiner’s Roman Surface *Opposite boundary points ofthis embedded disk are identified along a segment of double points when the surface
is closed to make the CrossCap. Pinchpoint singularities
form at the endpoints of theselfintersection segment.
In the 19th century images of the projective plane where
found by restricting quadratic maps f : R3 7→ R3 to theunit sphere. For example a Cross-Cap is obtained withf (x, y, z) = (xz, yz, (z 2 − x2 )/2, and Steiner’s Roman
Surface with f (x, y, z) = (xy, yz, zx).
Parametrizations follow by restricting to a parametrized
sphere FSphere (u, v) = sin u cos v, sin u sin v, cos u .
Steiner’s surface has three self-intersection segments andsix pinchpoint singularities. The Default Morph emphasizes a (self-intersecting) Möbius Band on this surface.
Cross-caps occur naturally as a family by a differential geometric construction. Consider at a point p of positive
curvature of some surface the family of all the normal curvature circles at p. They form a cross-cap and the two* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

pinchpoint sigularities are the points opposite to p on thetwo principal curvature circles. The parameters aa, bb in
3DXM are the two principal curvature radii. The Default
Morph varies bb from bb = 0.4aa to bb = aa, a sphere. A
Range Morph starts by taking half of each normal curvature circle and slowly extends them to full circles.
To derive a parametrization of this family of cross-caps let
e1 , e2 be a principal curvature frame at p, let κ1 , κ2 be
the principal curvatures and r1 := 1/κ1 , r2 := 1/κ2 theprincipal curvature radii at p. The normal curvature in
the direction e(ϕ) := e1 cos ϕ + e2 sin ϕ isκ(ϕ) := κ1 cos2 ϕ + κ2 sin2 ϕ with r(ϕ) := 1/κ(ϕ).
In 3DXM we parametrize the circles by u ∈ [−π, π] anduse v = ϕ. Denoting the surface normal by n we get thefamily of normal circles asr(v) · (−n + n · cos u + e(v) · sin u), u ∈ [0, π], v ∈ [0, 2π].
Finaly we take {e1 , e2 , n} as the (x, y, z) coordinate frame
and allow translation by cc along the z-axis to get ourParametrization of the normal curvature Cross-cap
r(v) = r1 r2 /(r2 cos2 v + r1 sin2 v),
x = r(v) cos v sin u,
y = r(v) sin v sin u,z − cc = r(v)(−1 + cos u) = 2r(v) sin2 (u/2),u ∈ [0, π], v ∈ [0, 2π].

To also get an implicit equation we observe
y/x = tan v and z 2 /(x2 + y 2 ) = tan2 (u/2).
This leads tox2 /(x2 + y 2 ) = cos2 v,
y 2 /(x2 + y 2 ) = sin2 v,
z 2 /(x2 + y 2 + z 2 ) = sin2 u/2,
(x2 + y 2 )/(x2 + y 2 + z 2 ) = cos2 u/2.
The first two of these equations eliminate v from r(v). The
third one eliminates u from z/r(v) = 2 sin2 (u/2) and gives(with r1 = aa, r2 = bb) an
Implicit equation of the normal curvature Cross-cap
y2x2+ )(x2 + y 2 + z 2 ) + 2z(x2 + y 2 ) = 0.
(aabb
Finally, replacing z by z − cc will translate the cross-cap,
e.g. cc = 2bb puts the pinchpoint of the smaller curvaturecircle to the origin of R3 .
H.K.

TOC

Boy Surfaces,*
following Apery and Bryant-Kusner
See Möbius Strip and Cross-Cap first.
All the early images of the projective plane in R3 had singularities, the Boy Surface was the first immersion. Since
the projective plane is non-orientable, no embedding into
R3 exists and self-intersection curves have to occur on theimage. In fact, the self-intersection curve of the Boy surface is also not embedded, the surface has a triple point.
Boy discovered the surface while working for his PhD under Hilbert. Boy’s construction was differential topology
work, his surface has no special local geometry.
Apery found an Algebraic Boy Surface. Moreover, his
surface is covered by a 1-parameter family of ellipses. Thisis his Parametrization:√ 
cos2 (u) cos(2v)+sin(2u) cos(v)/ 2√
2−sin(2u) sin(3v) 2√  cos (u)√sin(2v)−sin(2u) sin(v)/ 2 FApery (u, v) = 
2−sin(2u)sin(3v)
√
2 cos2 (u)√( 2−sin(2u) sin(3v)).

This parametrization is obtained by restricting the following even map from R3 to R3 to the unit sphere:* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC√
denom := ( 2 − 6xz)(x2 + y 2 ) + 8x3 z√
2
2 2
Fx := ((x − y )z + 2xz(x2 + y 2 ))/denom√2
Fy := (2xyz − 2yz(x2 + y 2 ))/denomFz := z 2 (x2 + y 2 )/denom
The image of the unit sphere is also an image of the projective plane since (Fx , Fy , Fz )(−p) = (Fx , Fy , Fz )(p).
The Default Morph starts with a band around the equator, which is a Möbius Strip with three halftwists. The
complete surface is obtained by attaching a disk (centeredat the polar center). 3DXM supplies a second morph,
Range Morph in the Animation Menu. It starts with aband around a meridian, which is another Möbius Strip
with one halftwist. This Möbius Strip is moved over all
the meridians, covering the surface with embedded Möbius
Strips.
Bryant-Kusner Boy Surfaces are obtained by an inversion from a minimal surface in R3 . The minimal surface isan immersion of S2 − {6 points} such that antipodal points
have the same image in R3 , so that the minimal surface isan image of the projective plane minus three points. The
six punctures are three antipodal pairs, and the minimal
surface has so called planar ends at these punctures. Thisis the same as saying that the Weierstrass-integrand has
no residues, hence can be explicitly integrated. In this context it is important that the inversion of a planar end has apuncture that can be smoothly closed by adding one point.

The closing of the three pairs of antipodal ends thus gives
a triple point on the smoothly immersed surface which is
obtained by inverting the minimal surface.
As Default Morph and as Range Morph we took the samedeformations as in the algebraic case. The first emphasizes
the equator Möbius Strip with three halftwists, the second
covers the surface with embedded Möbius Strips that havemeridians as center lines.
A Parametrization is obtained by first describing the minimal surface as an image of the Gaussian plane, then invert
it in the unit sphere. Parameter lines come by taking polar
coordinates in the unit disk.
M inSurf (z) := Re(V (z)/a(z)) + (0, 0, 1/2), where√ 
3−3a(z) := z − z + 5 and
2−22−2 2i 3−3
V (z) := i(z + z ), z + z , 3 (z + z ) .
Finally the inversion:
Boy(z) :=M inSurf (z).
||M inSurf (z)||2

We add the description of an ODE which allows to computethe self-intersection curve as long as the two normals along
the self-intersection are not parallel – in other words, aslong as the self-intersection is transversal.
Let F : D2 7→ R3 be a parametrized surface with unit
normal field N : D2 7→ R3 . Let p, q ∈ D2 be any two
points with N (p) =/ ± N (q).

The vector T := N (p)×N (q) is then tangent to the surface
at F (p) and F (q).
Therefore we have unique vectors ṗ ∈ Tp D2 , q̇ ∈ Tq D2 suchthat T = dFp · ṗ = dFq · q̇.
This defines an ODE on D2 × D2 with singularities where
N (p) = ±N (q).
Claim: Along the pair of solution curves we haveF (p(t)) − F (q(t)) = const.
Clearly:·(F (p(t)) − F (q(t))) = dFp · ṗ − dFq · q̇ = T − T = 0.
In particular, if we start at a self-intersection point
F (p(0)) = F (q(0)),
then the solution curve runs along the self-intersection,
F (p(t)) = F (q(t)).
In case one does not have an explicit self-intersection point
one can start from nearby points, meaning the distance
between F (p) and F (q) is small. On the intersection of
the tangent planes at these points find the point C which
is closest to the segment F (p)F (q). Use C−F (p) = dFp ·∆pand C − F (q) = dFq · ∆q to get points p + ∆p, q + ∆q which
are much closer to a self-intersection point,
F (p + ∆p) ≈ F (q + ∆q).
Iteration of this procedure converges rapidly.
H.K.

TOC

Nonorientable Surfaces *All nonorientable surfaces in 3DXM are explained as subsetof the explicitly parametrized surfaces.
See TOC of explicitly parametrized or implicit surfaces.
Currently the menu offers seven examples:

Check the Animate Menu for special morphs, designed to
explain less obvious features of these surfaces.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/ Surface Organisation

Implicit Surfaces *All implicit surfaces in 3DXM are explained as subset ofthe collection Explicitly parametrized or Implicit Surfaces.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/ Surface Organisation

The Spherical Helicoids*
See also: K=1 Surfaces of Revolution
A general helicoid can be obtained by applying a screw
motion to a planar curve. Suppose the curve is given as
s 7→ (f (s), g(s)), then the helicoid can be written as
F (s, t) = (f (s) cos t, f (s) sin t, g(s) + ht)
with a non-zero constant h. The surface which is generally
called the helicoid arises as the special case f (s) = s andg(s) = 0. Our more general class of helicoids is useful
because it allows to construct concrete examples of surfacesthat are otherwise elusive.
For example, let us specialize a little by taking f (s) = s(so that the curve is a graph over the t-axis). Then theGauss curvature of these helicoids is−h2 + s3 g 0 (s) g 00 (s)K=22
h2 + s2 + s2 g 0 (s)This is just a first order ODE for g 0 (s), and it is easy tocheck that
s
1
h2
0
√
g (s) = −1 − 2 +sa − K s2
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

is a 1-parameter family of solutions in the case that K
is a (positive) constant. The family parameter is a. Observe that the additional integration constant we get byintegrating g 0 only amounts to a vertical translation of thesurface.
As all complete surfaces of positive constant curvatures areround spheres, any other example must necessarily develop
singularities. This was for a long time a perfectly goodreason to ignore them, and now it has been for a while aperfectly good reason to find them interesting.
For our spherical helicoids, the singularity arises as thecurve s = s0 for the value of s0 where the integrand g 0 (s)becomes 0, and is thus a horizontal circle.
The surface appears as an exercise in Eisenhart’s A Treatise On The Differential Geometry of Curves And Surfaces.
It was certainly well known much earlier.
M.W.

Alternative description: next page.

TOC

An alternative description
TOC
The velocity vector field X of a screw motion in R3 isX(x, y, z) = (−y, x, h). On a surface that is screw motion
invariant one finds that the unit speed curves γ orthogonal
D 0
to X are geodesics: The covariant derivative ds
γ is zero
because it is orthogonal to γ 0 and to X ◦ γ, namely:
D 0
0 = hγ 0 , γ 0 i0 = 2hγ 0 , ds
γi
D 0
0 = hγ 0 , X ◦ γi =⇒ 0 = h ds
γ , X ◦ γi + 0.
A Killing field restricts along a geodesic to a Jacobi field
J = X ◦ γ, and on a 2-dim surface we have (because ofγ 0 ⊥ J) that J/|J| is a parallel field. Therefore we getfrom the Jacobi equation|J|00 = −K · |J|, i.e., if K = 1 then |J(s)| = a · cos(s).
We write γ(s) = (γ1 (s), γ2 (s), γ3 (s)) and abbreviatep222r = x + y , r(s) := γ12 (s) + γ22 (s). This gives:p|J(s)| = r2 (s) + h2 = a · cos(s).
What remains is a first order ODE for γ(s).
We abbreviate the radial horizontal vector field as nr , i.e.,nr ◦ γ = (γ1 , γ2 , 0)/r, and we extend the orthonormal vectors {nr , X/|X|} to an orthonormal√ basis withnz (x, y, z) := (y · h/r, −x · h/r, r)/ r2 + h2 . Then00γ (s) = r (s) · nr ◦ γ(s) +p

1 − r0 (s)2 · nz ◦ γ(s).

One needs |r0 (s)| ≤ 1 and this condition givesssmax :=

h2
a4


+

1 − a22a2

2a2 − 1.
+2
2a

One can check that this condition implies
r2 (s) = a2 cos2 (s) − h2 ≥ 0.
The final family, with parameters a and h, is

γ1 (s) cos t − γ2 (s) sin t
F (s, t) =  γ1 (s) sin t + γ2 (s) cos t  .
γ3 (s) + h · t
If h = 0 one obtains surfaces of revolution with meridianγ(s), and a = 1 is the sphere.
H.K.
TOC

About the Sievert-Enneper Surface*.

The Sievert-Enneper Surface
This surface stems from the PhD thesis of Sievert which
he wrote under Enneper. It is a surface of constant Gauss
curvature K = 1. Such surfaces are locally isometric to theunit sphere and are therefore also called spherical surfaces.
Locally isometric means: if we make the Sievert-Enneper
Surface from a thin sheet of metal (which should suggestthat we can deform the surface by bending without stretching), we can take the surface by its tails and wrap it aroundthe equator of the unit sphere; it will bend into the shapeof the sphere, but it is not big enough to cover the whole
sphere. Choose extreme values, e.g. aa = 0.05, aa = 30; in* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

the latter case large portions of the surface are already very
spherically shaped, but the holes are always there. So one
may wonder why one does not extend Sievert’s surface beyond its boundary. A similar phenomenon can more easily
be explained for those K = 1 surfaces of revolution whichhave their equator longer than that of the unit sphere (andwhich are obtained in 3DXM by setting cc > 1 in the Set
Parameters Dialog). They, too, have holes instead of polar
regions. In this case one can see from the formulas thatthe meridians are more and more strongly curved as theirdistance from the equator increases; the holes cannot be
closed because the curvature of the meridians becomes ∞at the rim of each hole.
The Sievert-Enneper Surface has even more in common
with these K = 1 surfaces of revolution. As with them theparameter lines are principal curvature directions. Thismeans that the surface is least strongly, respectively moststrongly, curved in the directions of the two parameterlines through any point. On a surface of revolution theseprincipal curvature lines are planar curves, the meridians
and the latitudes. The infinitely long parameter lines of theSievert-Enneper surface are also planar curves, and theseplanes all contain the y-axis, the line which is tangent to
the two tails at infinity. (This is clear from the formulas
below because the function Φ depends only on u and not onv.) The second family of parameter lines on the SievertEnneper Surface (v = const) looks planar also, but thisis not the case. However, as aa becomes very large sothat the surface looks very spherical, one observes that thetwo families of parameter lines converge to two orthogonalfamilies of circles on the sphere. This explains why theylook almost planar.
The formulas for the Sievert-Enneper Surface do not lookvery appealing. We write them with the help of threeauxiliary functions:√√Φ(u) := −u/ aa + 1 + arctan( aa + 1 · tan u),a(u, v) := 2/(aa + 1 − aa sin2 v cos2 u),qr(u, v) := a(u, v) (1 + 1/aa)(1 + aa sin2 u) sin v.
The Sievert-Enneper Surfaces:x(u, v) := r(u, v) cos Φ(u),y(u, v) := r(u, v) sin Φ(u),log(tan(v/2)) + (aa + 1)a(u, v) cos v√.
z(u, v) :=
aa
M.W.

TOC

Surfaces of Gauss Curvature K = 1 *The 3DXM submenu of the Surface menu contains theseexamples:See first the second page of Surfaces of Revolution
Enneper-Sievert Surface
Spherical Helicoids
It is interesting to deform these surfaces by using the default morph in each case.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/ Surface Organisation

Pseudospherical Surfaces
Surface Organisation( Click the Names )1.) Introduction to Pseudospherical Surfaces
2.) Dini, Kuen and Breather Surfaces
3.) Surfaces of Revolution
4.) The Pseudosphere from SGE
5.) Surfaces from SGE solutions

About Pseudospherical Surfaces*
If X : M ⊂ R3 is a surface with Gaussian curvature
K = −1, then it is known that there exists a local asymptotic coordinate system (x, t) on M such that the first andsecond fundamental forms are
I = dx2 + dt2 + 2 cos q dx dt, II = 2 sin q dx dt,
where q is the angle between asymptotic lines (the x-curves
and t-curves). Such coordinates are called Tchebyshef coordinates. The Gauss-Codazzi equations for M in these coordinates become a single equation, the sine-Gordon equation (SGE)
qxt = sin q.
(SGE)
Surfaces in R3 having constant Gauusian curvature K
equal minus one are usually called pseudospherical surfaces(after the most well-known example, the pseudosphere)
and the so-called Fundamental Theorem of Surfaces gives
us a local correspondence between pseudospherical surfaces(up to rigid motion) and solutions of SGE. A general pseudospherical surface shares with the pseudosphere the factthat its intrinsic geometry is a portion of the hyperbolic
geometry of Lobachevsky.
Now classical results of Bäcklund and of Bianchi concerning pseudospherical surfaces provide methods to find many
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOCexplicit solutions of the SGE and construct the corresponding pseudospherical surfaces. In fact, the SGE is one of themodel soliton equations, and these classical methods giverise to all of the soliton solutions of SGE. We will describe
next a little of this very classical differential geometry.
For surfaces M, M ∗ in R3 , a diffeomorphism ` : M → M ∗is called a pseudospherical congruence with constant θ if:(i) the line joining p and p∗ = `(p) is tangent to both M
and M ∗ ,(ii) the angle between the normal of M at p and the normal of M ∗ at p∗ = `(p) is θ, and(iii) the distance from p to p∗ is sin θ for all p ∈ M .
The following result of Bäcklund is fundamental to thestudy of pseudospherical surfaces.
Bäcklund Theorem. Let M, M ∗ be two surfaces in R3 ,and ` : M → M ∗ a pseudospherical congruence with constant θ. Then(a) both M and M ∗ are pseudospherical surfaces,(b) the Tchebyshef coordinates x, t on M maps to the
Tchebyshef coordinates on M ∗ under `,(c) if q and q ∗ are the solutions of SGE corresponding to
M and M ∗ respectively, then q, q ∗ satisfies(∗
qx∗ = qx + 2s sin( q 2+q ),(BTθ )∗
qt∗ = −qt + 2s sin( q 2−q ),where s = tan θ2 .

Moreover, given q, system (BTθ ) is solvable for q ∗ if andonly if q is a solution of the SGE, and the solution q ∗ isagain a solution of the SGE.
We will call both ` and the transform from q to q ∗a Bäcklund transformation. This description of Bäcklundtransformations gives us an algorithm for generating families of solutions of the PDE by solving a pair of ordinary
differential equations. The procedure can be repeated,but the miracle is that after the first step, the procedure can be carried out algebraically. This is the Bianchi
Permutability Theorem. Given two pseudospherical congruences `i : M0 → Mi with angles θi respectively andsin θ12 6= sin θ22 , then there exist an algebraic constructionof a unique surface M3 , and pseudospherical congruences`˜1 : M2 → M3 and `˜2 : M1 → M3 with angles θ1 andθ2 respectively such that `˜2 `1 = `˜1 `2 . The analytic reformulation of this theorem is the following: Suppose q is asolution of the SGE and q1 , q2 are two solutions of system(BTθ ) with angles θ = θ1 , θ2 respectively. The Bianchi
permutability theorem gives a third local solution q3 to
the SGEq3 − qs1 + s2
q1 − q2
=
tan
,
tan
4
s1 − s24
where s1 = tan θ2i and s2 = tan θ22 .
To see how the scheme works, we start with the trivial
solution q = 0 of SGE, then (BTθ ) can be solved explicitly
to get∗−1 sx+ stq (x, t) = 4 tan (e),(1)

the 1-soliton solutions of SGE. (Here s = tan θ2 . ) An
application of the Permutability theorem then give the 2soliton solutions!
s1 x+ s1 t
s2 x+ s1 t
1 − e
2
s1 + s2 e−1q(x, t) = 4 tan. (2)s1 − s2 1 + e(s1 +s2 )x+( s11 + s12 )t
Repeated applications of the Permutability theorem givecomplicated but nevertheless explicit n-soliton solutions.
Note that the parameters s1 , s2 in the above formula for
2-solitons are real. But for s1 = ieiθ and s2 = −ie−iθ , eventhough q1 , q2 are not real-valued, nevertheless


sin
θ
sin(T
cos
θ)
(3)
q3 (x, t) = 4 tan−1
cos θ cosh(X sin θ)
is real and a solution of SGE, where X = x − t, and T =x + t are space-time coordinates. This solution is periodic
in T and is called a Breather .
The “surface” corresponding to q = 0 is degenerate and infact is a straight line. The surfaces corresponding to(i) 1-soliton ( formula (1)) with s = 1 is the Pseudosphere,(ii) 1-soliton (formula (1)) with s 6= 1 is a Dini Surface,(iii) 2-soliton (formula (2)) contains the Kuen Surface.
(iv) Breather solution (formula (3)) with cos θ a rational
number is a pseudospherical surface periodic in the T
direction.

Even though the breather solution q is periodic in T , thecorresponding pseudospherical surface may not be periodic in T . This is because when we use the Fundamental
Theorem of Surfaces to construct the surface from solution q of the SGE, we need to solve two compatible ODEs
whose coefficients are given by functions of q and qx . For
Breathers, the solutions of these ODEs are periodic in T
if cos θ is rational.
You will notice that all of the pseudospherical surfaces
shown in the program have obvious singularities. In fact, atheorem of Hilbert says that the hyperbolic plane can not
be isometrically immersed in R3 , and this implies that allcomplete pseudospherical surfaces must have singularities.
Although soliton solutions are smooth on the entire (x, t)plane, the corresponding pseudospherical surfaces have singularities where the induced metric becomes degenerate,i.e., where


1
cos q
det
= 0.
cos q
1
In other words, a surface corresponding to a global solution
q of SGE will have a singularities along the curves where
q is a multiple of π. Moreover, since the metric has rank 1
there, these surfaces have cusp singularities.
For an elementary and short introduction to soliton theory
and its relation to SGE, see the article by C. L. Terng
and K. Uhlenbeck: Geometry of Solitons, Notices of AMS,
47(2000), 17-25. One may also download the pdf file ofthis paper fromhttp://www.math.neu.edu/∼terng/MyPapers.html
CLT

TOC

Three Pseudospherical Surfaces,*
Dini Family, Kuen, Breather
The theory to obtain these surfaces of Gauss curvature
K = −1 is explained in
About Pseudospherical Surfaces.
The construction begins with explicit solutions of the SineGordon-Equation (SGE). A first and a second fundamental
form are written down in terms of an SGE solution. Thesesurface data satisfy the Gauss-Codazzi integrability conditions, hence determine immersed surfaces. For the earliest
examples this integration of the surface data could be carried out explicitly. See the quoted text above for moredetails.
Parametrization
Dini surfaces:   of the xu − t(u, v)
 y  (u, v) :=  r(u, v) cos(v)  , wherezr(u, v) sin(v)ψ := aa · π, aa ∈ [0.001, 0.999],
aa = 0.5 gives the Pseudosphere,g(u, v) := (u − v · cos(ψ))/ sin(ψ),
s(u, v) := exp(g(u, v)),r(u, v) := 2 sin(ψ)/(s(u, v) + 1/s(u, v)),t(u, v) := 0.5 · r(u, v) · (s(u, v) − 1/s(u, v)).
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

Parametrization of the Kuen Surface:
2 cosh(v)(cos(u) + u · sin(u))
cosh(v) · cosh(v) + u · u  2 cosh(v)(−u · cos(u) + sin(u))
xcosh(v) · cosh(v) + u · u y  (u, v) := z
2 sinh(v) cosh(v) v−
cosh(v) · cosh(v) + u · u















Parametrization of the Breather surfaces:
Parameter√ of the family is aa ∈ (0, 1).
If w := 1 − aa2 is rational then the surfaces are periodic.
denom := aa · ((w cosh(aa · u))2 + (aa sin(w · v))2 )2(1 − aa2 )x(u, v) := −u +cosh(aa · u) sin(aa · u)
denom
2w cosh(aa u)y(u, v) :=(−w cos(v) cos(w v) − sin(v) sin(w v))
denom
2w cosh(aa u)z(u, v) :=(−w sin(v) cos(w v) + cos(v) sin(w v))
denomH.K., R.S.P.

TOC

The Pseudosphere*
from a Sine-Gordon solution
The Pseudosphere was first found as a surface of revolution, with the Tractrix as meridian (see Planar Curves). It
has Gauss curvature K = −1. See:
Constant Curvature Surfaces of Revolution.
Later in the 19th century it was discovered that surfaces
with K = −1 can be constructed from soliton solutions of
the Sine-Gordon Equation (SGE). This is explained in:
About Pseudospherical Surfaces,
which can be obtained from the Documentation Menu.
At about the same time, in 1868, Beltrami proved thatthe axiomatically constructed non-Euclidean geometry of
Bolyai and Lobachevsky was the same as the simply connected 2-dimensional Riemannian geometry of Gauss curvature K = −1; for example the Riemannian metric of thePseudosphere, extended to the plane: du2 + exp(−2u)dv 2 .
Their common name today is Hyperbolic Geometry.
The meridians are examples of asymptotic geodesics, a key
notion in hyperbolic geometry. Curves, orthogonal to a
family of asymptotic geodesics are called horocycles in hyperbolic geometry. They have infinite length in the simply
connected case, on the Pseudosphere one sees finite por* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

tions as the latitude circles.
In the theory which relates SGE solutions to surface in R3
of Gauss curvature K = −1, one first writes down the firstand second fundamental forms in terms of such a solution
q(x, t) of SGE:
I = dx2 + dt2 + 2 cos q dx dt, II = 2 sin q dx dt,
The Gauss-Codazzi integrability conditions are satisfied,because q is an SGE solution. The Gauss curvature is
the quotient of the determinants of the two forms, i.e.
K = − sin2 (q)/ sin2 (q) = −1. One then obtains the firstparameter line of the surface by integrating an ODE andthe transversal other family by integrating a second ODE.
The first and second fundamental forms above are written
in asymptote coordinates, which means: the normal curvature of the surface in the direction of the parameter lines is
0. (Note that x and t are arc length parameters on the parameter lines. This leads to the Tchebycheff net mentionedin “About Pseudospherical Curves”.) Such parametrizations do not offer a good view of the surface. In 3DXM,
therefore, the integration first creates one curvature lineof the surface and secondly the orthogonal family of curvature lines with u = x + t, v = x − t. One can view the
integration before the surface is shown, with these parameter lines.
The SGE solution for the Pseudosphere is:q(x, t) := 4 arctan(exp(x+t)), qc(u, v) = 4 arctan(exp(u)).
H.K.

TOC

(K = −1) - Surfaces from SGE solitons *In 3DXM there are several explicit solutions q(u, v) of theSine-Gordon Equation (SGE: quu − qvv = sin(q(u, v)) ).
They use Parameters as follows:
Pseudosphere: No parameters, q(u, v) = 4 arctan(exp(u))
One-Soliton (or Dini family or Kink solutions):
0 < aa < 1, used for the so called Dini deformation:q(u, v) = 4 arctan(exp(u/ sin(πaa) + v/ tan(πaa)))
Two-Soliton: 0 < aa, bb < 1. The limit aa = bb is included, because it is a deformation of the Kuen surface.
Three-Soliton: 0 < aa, bb, cc < 1, no equal parameters.
Four-Soliton: 0 < aa, . . . , dd < 1, no equal parameters.
cc
Periodic SGE solutions: f 00 (u) = − aa
∗ sin(aa · f (u)),
aa = 1, cc = −1, f (0) = hh, f 0 (0) = bb, 0 < dd < ∞.
(Either bb = 0 for even, or hh = 0 for odd solutions.)(dd−1/dd)q(u, v) = f ( (dd+1/dd)
·
u
+
· v),
2
2
dd = 1 for rotational symmetry, try: hh = 1, dd = 1.35.
Breather: 0 < aa ≤ 1, bb = cc =√0, (aa = 1 allowed)
sin( (1−aa2 )·v)aaq(u, v) = 4 arctan( cosh(aa·u) · √(1−aa2 )If cc = 0 and bb > 0 integer, then 3DXM chooses aa so thatthe breather closes up with order bb rotation symmetry.
Finally cc > 0 applies the Dini deformation to this breather.
Double Breather: aa, cc =/ 0. Coded to do the complex
2-Soliton. Currently: z = aa + i · cc, z̄ = aa − i · cc.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

Each q(u, v) determines a pair of first and second fundamenu−v,
t=tal forms, in asymptote parameters x = u+v
2
2 :I = dx2 + dt2 + 2 cos q dx dt, II = 2 sin q dx dt,
for which the Gauss-Codazzi integrability conditions are
satisfied (because they simplify to q being an SGE solution qxt = sin(q(x, t)). This says:
There are parametrized surfaces in R3 with these first andsecond fundamental forms. They have Gauss curvature
K = −1 = det(II)/ det(I). The parameter lines are asymptote lines on these surfaces and x, t are arclengths on theparameter lines. For more details see:
About Pseudospherical Surfaces,
available from the Documentation Menu.
Since pictures of surfaces, drawn with asymptote line parametrization, do not give a good 3D-impression of the surface, 3DXM uses instead curvature line parametrizations,i.e. u = x + t, v = x − t with the corresponding SGE
solution qc(u, v) := q((u + v)/2, (u − v)/2). 3DXM firstcomputes one curvature line as solution of an ODE. Next
the family of orthogonal curvature lines is drawn. This
determines the parametrized surface. It is then renderedaccording to the viewer’s choice.
Recall the definition cosec := 1/ sin.
The One-Soliton solution of SGE (s+ 1s = 2cosec(aaπ)):q(x, t) = 4 arctan(exp( s · x + 1/s · t )), in curvature lines:
qc(u, v) = 4 arctan(exp( cosec(aa·π)·u+cotan(aa·π)·v )).
The first drawn curvature line is a planar curve.

For s = 1, aa = 1/2 the solution qc depends only on u, thecorresponding surface is the Pseudosphere. Vice versa, forevery surface of revolution the function qc depends only onu and the SGE reduces to the ODE f 00 (u) = + sin(f (u)).
The even solutions of this ODE correspond to the hyperbolic (K = −1) surfaces and the odd solutions give thesurfaces of revolution with finite cone singularities. The
only elementary solution is f (u) = 4 arctan(exp(u)) withf (0) = π, f 0 (0) = 2 and the cone singularity at infinity.
While this first step is obtained as explicit integration of
the ODE f 00 (u) = sin(f (u)), the next steps follow from a
result of Bianchi, reformulated for SGE solutions:
If a pair of functions q(x, t), Q(x, t) satisfy the pair ofODEs:
Qx (x, t) = +qx (x, t) + 2s · sin( Q+q
2 (x, t))Qt (x, t) = −qt (x, t) + 2/s · sin( Q−q2 (x, t))then, for given q(x, t), the system is solvable for Q(x, t)
iff q(x, t) is an SGE solution. And then Q(x, t) is also
SGE solution and the two functions are called Bäcklundtransformations of each other. Moreover, if, for s1 =/ s2
two such solutions Q1(x, t), Q2(x, t) are given, then a third
solution Qq(x, t) is marvelously obtained from
Q1−Q2
s1 +s2tan( Qq−q
(x,
t))
=
·
tan(
(x, t)).
4
s1 −s2
4
The 1-Soliton solution gives two such functions Q1, Q2
which are Bäcklund transformations of the trivial SGE solution q(x, t) = 0 and Qq(x, t) is the following 2-Soliton
solution(recall tan(α−β) = (tan(α)−tan(β))/(1+tan(α)·tan(β))):exp(s1 ·x+1/s1 ·t)−exp(s2 ·x+1/s2 ·t)+s2Qq(x, t) = 4 arctan( ss11 −s·
1+exp(s1 ·x+1/s1 ·t)·exp(s2 ·x+1/s2 ·t) )2
This Bianchi generalization also works for complex parameters and one can see directly that for s1 = z, s2 = z =/0
the new solution Qq(x, t) is again real. If |z| = 1, it is thebreather solution below. In 3DXM we use z = aa + i · cc.
The other (periodic) solutions of quu = sin(q(u)) are notBäcklund transformations of known solutions.
The curvature line version, which is needed for visualization, looks more complicated:
The Two-Soliton solution of SGE (parameters aa, bb):
Define a constant B and functions A1, A2 first.
B := (cos(bb π) − cos(aa π))/(cos((aa − bb)π) − 1),
A1(u, v) := cosec(aa π)u + cotan(aa π)v,
A2(u, v) := cosec(bb π)u + cotan(bb π)v, then put:)qc(u, v) := 4 arctan( B · exp(A1(u,v))−exp(A2(u,v))
1+exp( A1(u,v)+A2(u,v) ).

Now observe that the 2-Soliton solutions are Bäcklundtransformations of the two 1-Soliton solutions which wereused in the Bianchi construction!! Therefore the Bianchi
construction can be repeated to give the following:Three-Soliton solution of SGE (params aa, bb, cc):
Define three auxiliary functions E, F, H first.
E(ξ, u, v) := exp( cosec(ξ π)u + cotan(ξ π)v ),E(ξ1 ,u,v)−E(ξ2 ,u,v)
2 π)−cos(ξ1 π)
F (ξ1 , ξ2 , u, v) := cos(ξ·cos((ξ2 −ξ1 )π)−1
1+E(ξ1 ,u,v)·E(ξ2 ,u,v) ,F (ξ1 ,ξ2 ,u,v)−F (ξ1 ,ξ3 ,u,v)
3 π)−cos(ξ2 π)H(ξ1 , ξ2 , ξ3 , u, v) := cos(ξ·cos((ξ3 −ξ2 )π)−1 1+F (ξ1 ,ξ2 ,u,v)·F (ξ1 ,ξ3 ,u,v) ,qc(aa, bb, cc, u, v) := 4 arctan(E(aa, u, v))+ 4 arctan(H(aa, bb, cc, u, v)).
The Four-Soliton solution of SGE (params aa, bb, cc, dd)
we give in asymptote coordinates:
Define functions E, F, H (as before) and K first.
E(s, x, t) := exp( s · x + 1/s · t),E(s1 ,x,t)−E(s2 ,x,t)+s2·
F (s1 , s2 , x, t) := ss11 −s
1+E(s1 ,x,t)·E(s2 ,x,t) ,2

Recall that 4 arctan(F (s1 , s2 , x, t)) are 2-Soliton solutions.
F (s1 ,s2 ,x,t)−F (s1 ,s3 ,x,t)+s3
H((s1 , s2 , s3 , x, t) := sss2 −s·1+F (s1 ,s2 ,x,t)·F (s1 ,s3 ,x,t) ,3

K(s1 , s2 , s3 , s4 , x, t) :=H((s1 ,s2 ,s3 ,x,t)−H((s1 ,s2 ,s4 ,x,t)s3 +s4·s3 −s4 1+H((s1 ,s2 ,s3 ,x,t)·H((s1 ,s2 ,s4 ,x,t) ,Finallyq(s1 , s2 , s3 , s4 , x, t) :=4 arctan(F (s1 , s2 , x, t))+4 arctan(K(s1 , s2 , s3 , s4 , x, t)).
Recall as before qc(u, v) = q((u + v)/2, (u − v)/2).
The Breather (soliton)
solution of SGE (parameter aa):√
Abbreviate bb := 1 − aa2 , thenv)aaqc(bb, aa, u, v) := 4 arctan( sin(bb·bbcosh(aa u) ).

Recall from above that each of these solutions determines
a first and a second fundamental form which satisfy theGauss-Codazzi integrability conditions. Each parameterline gives a space curve via an ODE which is determinedby these fundamental forms. Because of the integrability

conditions these space curves fit together to form a surfaceof Gauss curvature K = −1.
H.K.

TOC

Minimal Surfaces
Surface Organisation
About Minimal Surfaces
Explicitly Parametrized Minimal Surfaces
Helicoid-Catenoid (1776)
Scherk, Catalan, Henneberg Surfaces (1835-1865)
Enneper Surfaces
Minimal Inverted Boy Surface5.) Kusner (Dihedral Symmetric)
Weierstrass’ Minimal Surface Representation
Surfaces parametrized by punctured spheres:ff -wavy k-fold Enneper Surface
Catenoid - k-fold Enneper
Planar k-fold Enneper
Double k-fold Enneper
Scherk Saddle Towers
Twisted Saddle Towers
Symmetric and Skew k-Noids
The Lopez-Ros No-Go-Theorem
Two Planar And Two Catenoid Ends
Half-Catenoids and Weierstrass Representation
Surfaces parametrized by punctured tori:
Chain of Half-Catenoids and Field of Half-Catenoids
Karcher jd Saddle Towers
Karcher je Saddle Towers
Riemann’s Minimal Surfaces
Continue next page

The Chen-Gackstatter Minimal Surface
Costa’s Minimal Surface
The Costa-Hoffman-Meeks Minimal Surfaces
Fence of Catenoids
Scherk’s Surface with a Handle
R. Schoen’s No-Go-Theorem
Embedded Triply Periodic Minimal Surfaces (genus ≥ 3):
The Conjugate Plateau Construction
Schwarz’ P-D-Families, with A. Schoen’s Gyroid
Schwarz’ H-Family, with Lidin’s Lidinoid
Four Triply Periodic Minimal Surfaces
The Fujimori-Weber Surfaces
Userdefined Minimal Surfaces
Bibliography
In most cases the program code follows the formulas in:[Ka1] H. Karcher, Embedded minimal surfaces derived from
Scherk’s examples, Manuscripta Math. 62 (1988) pp.
83–114.
or with more details in[Ka2] H. Karcher, Construction of minimal surfaces,
in “Surveys in Geometry”, Univ. of Tokyo, 1989, andLecture Notes No.12, SFB 256, Bonn, 1989, pp. 1–96.
For a discussion of techniques for creating minimal surfaces
with various qualitative features by appropriate choices ofWeierstrass data, see either:[KWH] H. Karcher, F. Wei, and D. Hoffman, The genus one
helicoid, and the minimal surfaces that led to its discovery, in “Global Analysis in Modern Mathematics,
A Symposium in Honor of Richard Palais’ Sixtieth
Birthday”, K. Uhlenbeck Editor, Publish or Perish
Press, 1993.
or pages 192 – 217 of:[DHKW] U. Dierkes, S. Hildebrand, A. Küster, and O. Wohlrab,
Minimal Surfaces I, Grundlehren der Math. Wiss. vol295, Springer-Verlag, 1991.
A survey with emphasis on theoretical advances is[HK] [Hoffman, Karcher] Complete embedded minimal surfaces of finite total curvature. Encyclopaedia of Mathematical Sciences, vol. 90. Geometry V (Ed. R. Osserman), pp. 5-93
The last two references have an extensive bibliography.
Minimal TOC

Surface Organisation

About Minimal Surfaces *Discovery and physical interpretation.

The question which surfaces locally minimize area led Lagrange in 1760 to the minimal surface equation for graphs.
By 1765 Meusnier had found that a geometric interpretation of this equation is: the mean curvature of the surface
vanishes. He discovered that the catenoid and the helicoid are nonplanar examples. It took until 1835 for the
next examples to appear, discovered by Scherk; his doubly periodic surface is a graph over the black squares of acheckerboard tesselation of the plane and his singly periodic surface is nowadays viewed as a desingularization of
two orthogonally intersecting planes. In the following years
complex analysis developed and by 1865 many examples
were known through the efforts of Riemann, Weierstraß,Enneper and in particular Schwarz.
Also in that period Plateau had made careful experimentswith soap films. He convinced people that soap films werea perfect physical realization of minimal surfaces, and heconvinced mathematicians that they should solve Plateau’s
Problem, i.e. prove that every continuous injective closed
curve in R3 spans a minimal surface. This problem wassolved in 1932 by Douglas and independently by Rado.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

On the way to this solution mathematicians had learnt a
lot about nonlinear elliptic partial differential equations.
In particular the importance of the maximum principle
had become clear, it implies for example that every compact minimal surface is contained in the convex hull of
its boundary and that boundary value problems are well
posed for the minimal surface equation.
On the other hand, although the Cauchy-Kowalewski theorem allows to solve locally initial value problems withanalytic data, there is no continuous dependence on thedata and no hope to obtain complete immersed examples
with this method. — But, already Weierstraß had established the close connection of minimal surfaces with complex analysis. In particular: the spherical Gauss map composed with stereographic projection is locally a holomorphic function G : M 2 → C. In terms of the 90◦ rotationof each tangent space of the minimal surface M 2 holomorphicity has the following intuitive interpretation: for eachv ∈ T M 2 we have the following version of the CauchyRiemann equationsdG(R90 ◦ v) = i · dG(v).
Moreover, the three component functions of the immersion(F 1 , F 2 , F 3 ) : M 2 → R3 are, locally, the real parts ofholomorphic functions, because the differential forms
ω j := −dF j ◦ R90
are closed for surfaces with mean curvature zero (Meusnier’s
above interpretation of “minimal”). This fact establishesthe Weierstraß representation:
Let G be the holomorphic Gauss map anddh := dF 3 − i · dF 3 ◦ R90
the (holomorphic) complexification of the differential of the height function F 3 then(F 1 , F 2 , F 3 ) = Re

Z 


1 1
i 1
( − G), ( + G), 1 dh.
2 G
2 G

The examples of the second half of the 19th century weremade with this representation. But results, achieved by
1960 by Huber and Osserman show, that all minimal surfaces of a certain kind can be obtained by a global application of this representation, namely:
Complete, immersed minimal surfaces of finite total curvature can be conformally compactified by closing finitely
many punctures; moreover, the Weierstraß data G, dh
extend meromorphically to this compact Riemann surface.
The wealth of examples, discovered since about 1980, relyon this theorem. To understand these examples better wenote the first and second fundamental forms (Riemannian
metric and, if |v| = 1, normal curvature)
1 1
I(v, v) = (+ |G|)2 |dh(v)|2
4 |G|
dG(v)
II(v, v) = Redh(v).
G

The points p on the Riemann surface which are poles of dh
do not correspond to points on the minimal surface. Every(differentiable) curve which runs into such a puncture p has
infinite length on the minimal surface. The same is true
if G has a zero or pole of higher order than the vanishingorder of dh. If these orders are the same then we simply
have a point with vertical normal on the minimal surface.
And at points where the vanishing order of dh is larger,
the metric becomes singular and the minimal surface has
a so called branch point, it is no longer an immersion.

Visualization of minimal surfaces.

The Weierstraß representation allows to write down a number of simple minimal surfaces which can be visualized likeany other surface for which an explicit parametrization is
given. Our parameter lines come from polar coordinateswith centers {0, ∞} or {1, +1}. Note that the zeros andpoles of G, dh fit together so that no branch points occur and so that the minimal surfaces are complete on the
punctured spheres mentioned in each case. The surfaces
are of finite total curvature, since the Gauss map is meromorphic, i.e., its image covers the Riemann sphere a finite
number of times.
First Examples,defined on C or C \ {0} or S2 \ {1, −1}:

Enneper Surface:z ∈ C, G(z) := z, dh := zdz
Polynomial Enneper:z ∈ C, G(z) := P (z), dh := P (z)dz
Rational Enneper:z ∈ C, G(z) := P (z)/Q(z), dh := P (z)Q(z)dz
P and Q are polynomials without common zeros.
Vertical Catenoid:z ∈ C \ {0}, G(z) := z, dh := dz/z,or G(z) := 1/z
Helicoid:z ∈ C, G(z) := exp(z), dh := idz = idGGHelicoid:z ∈ C \ {0}, G(z) := z, dh := idz/z
Planar to Enneper:z ∈ C \ {0}, G(z) := z k+1 , dh := z k−1 dz
Wavy Catenoid:z ∈ C \ {0}, G(z) := (1 +  · z k )/z, dh := G(z)dz
Wavy Plane:z ∈ C \ {0}, G(z) := z, dh := dz
Horizontal Catenoid:dz/zz ∈ S \ {1, −1}, G(z) := z, dh :=.
(z − 1/z)2
2

All of these simple minimal surfaces have symmetries: (i)straight lines on a minimal surface allow 180◦ rotations of
the minimal surface into itsself, and (ii) planar geodesics
on a minimal surface allow reflection (in the plane of thegeodesic) of the minimal surface into itsself. Since thesesymmetries become more important for understanding more
complicated surfaces one should learn how to recognize
them. The straight lines are geodesics with normal curvature II(c0 , c0 ) = 0 or dG(c0 )/G·dh(c0 ) ∈ i·R. In the present
context we recognize geodesics as fixed point sets of isometric involutions. The formula for the first fundamental form
is so simple that one can easily see in all of these examples
that the expected symmetry indeed does not change the
Riemannian arc length of curves. To recognize the planar
geodesics note that a geodesic on a surface is planar if it
is also a principal curvature line; in addition to seeing it
as the fixed point set of a length preserving involution wetherefore only need to check dG(c0 )/G · dh(c0 ) ∈ R, whichis also easy in these examples.
In 3D-XplorMath one can easily change (in the SettingsMenu) the range of the parametrization and also the symmetry of the surface. We recommend that the surfaces
are looked at from far away when a large range for theparametrization is chosen. We also recommend to look at
the default morphs of WavyEnneper and WavyCatenoid
since it is quite surprising how suddenly the perturbation
becomes visible. This should be taken as an illustrationthat the initial value problem for minimal surfaces is highlyunstable, it is ill posed and no numerical solution is possible.
More complicated spherical examples.

The sudden increase of the interest in minimal surfaces
after 1980 was largely caused by the discovery of a quite
unexpected embedded finite total curvature minimal surface by Costa with embeddedness discovered and proved by
Hoffman-Meeks. We are not yet close to such an example
because of the following
Theorem of Lopez-Ros. An embedded, minimal, finite total
curvature punctured sphere is a plane or a catenoid.
To practise using the Weierstraß representation we therefore have to be content with a few immersed punctured
spheres. We want to learn how to see the Gauss map when
one looks at the picture of such a minimal surface. The
main fact to use is: a meromorphic function on a compact
Riemann surface is determined up to a constant factor by
its zeros and poles. In the case of the Jorge-Meeks k-noids
one clearly sees a k-punctured sphere with a horizontal
symmetry plane. One observes only two points with vertical normal, one up, one down. The qualitative behaviourof the Gauss map along the horizontal symmetry line suggests a mapping degree k − 1. This leaves no choice but
G(z) = z k−1 . If we look back at the very simple examples then we can observe that, at a catenoid like puncture,
either Gdh or dh/G has precisely a double pole. This determines the dh below up to a constant factor.

The next two examples, the 4-noid with orthogonal ends ofdifferent size, and the double Enneper, have a quite different appearance, but they have the same Gauss map. The
vertical points are symmetric with respect to the origin andsymmetric with respect to the unit circle, and the degree of
the Gauss map is three; this determines the Blaschke product expression below. In the case of the 4-noid we need tocreate the four catenoid ends with double poles of dh andwe need to compensate the simple zeros and poles of G bysimple zeros of dh; then, if we also treat zero and infinity
symmetrically, the expression below is forced. In the caseof the double Enneper surface we just need to compensate
the simple zeros and poles of G (outside 0, ∞); symmetrictreatment of 0, ∞ gives the dh below (except for a constantfactor).
The last example illustrates in which way attempted counterexamples to the Lopez-Ros theorem fail. A residue computation for the Weierstraß integrands shows that closed
curves around the punctures ±1 on the sphere are notclosed curves on the minimal surface, if we want all limit
normals to be vertical. It is easy to close this so called
period when one allows tilted catenoid ends, but, as one
decreases the tilt, the distance between the half catenoids
increases, and they intersect the planar middle end if one
computes the surface far enough towards the punctures.
The k-noids of Jorge-Meeks:z ∈ S2 \ {e2πi·l/k ; 0 ≤ l < k},G(z) := z k−1 ,dh := (z k + z −k − 2)−1 · dz/z.

4-noids with two different orthogonal ends:
z+r
z−r· 1+rz,z ∈ C \ {0, −1, +1}, G(z) := z · 1−rz


2
−2
2
−2 −2
dh := 1 − zr2 +z·(z−z) · dz/z.
−2+r

Two Enneper ends joined by a catenoidal neck:z ∈ C \ {0},


2

−2

z+r
z−r
· 1+rz, dh := 1 − zr2 +z
G(z) := z · 1−rz
+r −2



· dz/z.

Three punctures, period closes for tilted ends:z ∈ C \ {−1, +1},
G(z) := ρ(z 2 − r2 ),

2

2

−r
dh := (zz2 −1)
2 dz.

Observe that the zeros and poles of the Gauss map which
are not in the list of punctures are compensated by zerosof dh. At the embedded ends, Gdh or dh/G have a double pole and at the Enneper ends they have higher orderpoles. — In this list we do not have simple poles of Gdh
and dh/G. If this Rhappens then the Weierstraß integral
behaves similar to dz/z: the unit disk, punctured at 0,
is mapped by log to an infinite number of half strips parallel to the negative real axis and of width 2π. Similarly,
the Weierstraß integral produces simply periodic embedded
minimal surfaces parametrized by punctured spheres.
Generalized Scherk Saddle Towers:z ∈ S2 \ {e±φ · e2πi·l/k ; 0 ≤ l < k},G(z) := z k−1 ,dh := (z k + z −k − 2 cos kφ)−1 · dz/z.

As in the simpler examples, observe that the symmetrylines can be seen from the Weierstraß data. We also notethat at this point an important decision has to be made. If
one represents the surfaces, as in all our examples, with parameter lines then each surface requires a special effort sothat the parameter lines on the one hand support the complex analytic background of the minimal surface and on theother hand suggest correctly how one should imagine how
the surface extends beyond what the picture shows. In 3DXplorMath this individual approach has been taken. The
other option is to spend considerably more general effort
by writing software which will create a suitable triangulation of the domain. David and Jim Hoffman have sucha program running. It requires much less individual work
to compute another minimal surface but it is harder to illustrate the complex analysis background of the computed
minimal surface.
The family of singly periodic embedded minimal surfaces
which resemble the above generalized Scherk Saddle towers
is much larger than the above explicit formulas suggest. So
far we have only talked about the real part of the Weierstraß integral. In fact, a 1-parameter (“associate”) familyof isometric (and in general not congruent) minimal surfaces are given by this integral because dh can be changedby the factor exp(−2πiϕ). In particular, the imaginary
part of the Weierstraß integral is the “conjugate” minimal
surface. In the case of the generalized Scherk saddle towers
we have that the conjugate minimal immersion maps theunit disk (with the punctures on the boundary) to a graphover a convex polygon; its edge lengths all agree. The minimal graph has over each edge the boundary value +∞ or−∞, alternatingly. — Jenkins-Serrin proved the converse:
every such infinite boundary value problem has a graphsolution, a minimal disk whose conjugate minimal surface
is the fundamental piece of an embedded singly periodic
saddle tower.
Having seen a good collection of minimal surfaces parametrized by punctured spheres we now turn to minimal surfaces parametrized by other Riemann surfaces. (To be
written)
H.K.

TOC

The Helicoid - Catenoid Family*
Lagrange found in 1762 the minimal surface equation as
Euler-Lagrange equation for the area minimization problem. Geometrically, this equation means that the mean
curvature is 0. He found no other solution than the plane.
In 1776 Meusnier discovered that the Helicoid and theCatenoid are also solutions. The next solutions were found
60 years later by Scherk.
The Helicoid (aa = 0) – Catenoid (aa = 1) Family:x(u, v) := cos(aa · π/2) sin(u) sinh(v)+sin(aa · π/2) cos(u) cosh(v))
y(u, v) := cos(aa · π/2) cos(u) sinh(v)+sin(aa · π/2) sin(u) cosh(v)z(u, v) := cos(aa · π/2) · u + sin(aa · π/2) · v
These formulas describe, for each aa, a minimal surface.
Moreover, these surfaces are all isometric to each other;
their Riemannian metrics, in the given parametrization, allhave the same expression. The parameter lines u = const
on the Helicoid (aa = 0) are straight lines and no otherminimal surface is such a ruled surface. The Catenoid(aa = 1) is the only minimal surface of revolution. These
two are the only embedded surfaces in the aa-family.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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The isometric aa-deformation is very fascinating, use Cyclic
Morph in the Animation Menu. The small parameter squares
do not change size or shape during the deformation, they
just get bent a little. If the two sides of the surfaces arecolored differently and aa varies from aa = 1 to aa = 3,
one can observe that the Catenoid is turned inside out.
The deformation period is 4 and really amazing to watch.
H.K.

TOC

Minimal Surfaces of Catalan, Henneberg, Scherk*
These surfaces are early discovered minimal surfaces. They
were found as explicitly parametrized surfaces, while soonafterwards variations of the Weierstrass representation became the main tool of description. This changed again in
the early 1930s when Douglas and Rado solved the Plateau
Problem with Functional Analysis methods. The Weierstrass representation had, after the work of Ossermann, acomeback in the 1980s.
Scherk’s doubly periodic minimal surface (1835):x(u, v) :=uv
1cos(v), y(u, v) :=z(u, v) :=
ln(
).
bb
bb
bb
cos(u)

Catalan’s minimal surface (1855), associate family:1x(u, v) := ( cos(aa · π)(u − sin(u) cosh(v))+
bbsin(aa · π)(v − cos(u) sinh(v)) − 4)1y(u, v) := ( cos(aa · π)(1 − cos(u) cosh(v))+
bbsin(aa · π) sin(u) sinh(v))
1z(u, v) := ( cos(aa · π)4 sin(u/2) sinh(v/2)+bbsin(aa · π)4 cos(u/2) cosh(v/2)).
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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Henneberg’s minimal surface (1875), associate family:
2 sin(aa · π)
2 cos(aa · π)
, saa :=
caa :=
0.01 + |bb|
0.01 + |bb|1
x(u, v) :=caa · (sinh(u) cos(v) − sinh(3u) cos(3v))
31+saa · (cosh(u) sin(v) − cosh(3u) sin(3v))
31y(u, v) :=caa · (sinh(u) sin(v) + sinh(3u) sin(3v))
314cc+saa·(− cosh(u) cos(v) − cosh(3u) cos(3v) +)33z(u, v) :=caa · (cosh(2u) cos(2v) − cc)+saa · sinh(2u) sin(2v)u ∈ R, v ∈ [−π, π], a cylinder domain.
The default value of the translation parameter, cc = 0,
puts the symmetry point of Henneberg’s surface at theorigin. If one wants to scale up a neighborhood of the
branch point at u = 0, v = 0, use cc = 1 to put thatbranch point at the origin and make use of the scaling
parameter bb in the denominator.
Scherk’s discovery of the above doubly periodic surface, of
its singly periodic conjugate surface and of three less spectacular ones was a sensation since the only other known
minimal surfaces, the catenoid and the helicoid, were already 50 years old. Scherk’s surfaces were destined to playa major role in the discovery period after 1980.

Catalan’s surface was next, 20 years later. This slow progress reflects the fact that no methods for the constructionof minimal surfaces were known. This changed with Riemann, Weierstrass, Enneper and finally Schwarz. Theirmethods built on complex analysis and allowed to write
down arbitrarily many examples. Therefore the emphasis
shifted to examples that had additional properties. In particular complete, embedded minimal surfaces were sought.
Quite a few triply periodic embedded ones were found,but it took another 100 years before the next embedded
finite total curvature example after the catenoid was constructed: Costa’s example, a minimal embedding of thethrice punctured square torus.
Henneberg’s minimal surface was studied a lot because of
its two branch point singularities (on the z-axis). Such singularities are difficult to imagine and Henneberg’s surface
is a simple example to exhibit them. The default image
in 3DXM shows a small neighborhood of the two branch
points. The segment between them is a selfintersection lineof the surface. The lines {x = ±y, z = 0} lie also on thesurface. 180◦ rotation around any of these straight lines
is a symmetry of the surface and the conjugate surface
has, correspondingly, the three planes which pass throughthe origin and are orthogonal to one of these three lines,
as symmetry planes. - Note: The conjugate surface has
twice the area of Henneberg’s surface. This is because
Henneberg’s surface is covered twice. For about 100 years
it was the only known non-orientable minimal surface.

The last entry in the Action Menu emphasizes a Möbius
band on Henneberg’s surface. The default morph is theassociate family morph.
The Range Morph (from the Animate Menu) shows larger
and larger pieces of the surface - scaled down to fit on thescreen. One can see still larger portions by increasing uMax
beyond 0.95. (Choose b1 > 1 to compensate for the growing size.)
A third morph expands a band around the two branch
points and moves it over the surface.
The parameter line u = 0, v ∈ [−π, π] is a symmetryline of the domain cylinder. It is mapped to the segment
between the branch points on the z-axis and 180◦ rotation
around it is a symmetry of the surface. The two lines v = 0
and v = π are also symmetry lines of the domain cylinder.
They are mapped to curves of reflectional symmetry, reflection in the x-z-plane. They end in the branch points
like a Neil parabola, {x3 = z 2 }.
H.K.

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About the Classical Enneper Surface *and some Polynomial Relatives
See also: About Minimal Surfaces
Definition with explicit formulas

The classical Enneper Surface is a minimal immersion of
the complex plane, C, into Euclidean space R3 . It is givenby the formula
F (z) := real({z 3 /3 − z, i · (z 3 /3 + z), z 2 }).
If one wants to see coordinate lines on the image one canuse Cartesian coordinates for the complex plane, z :=x + i · y, or polar coordinates z := r · (cos ϕ + i · sin ϕ),
and map those grids with F . The Cartesian choice is natural here since its parameter lines are principal curvaturelines. However the polar choice also has merits—namely,
rotations around the origin are isometries and the coordinate lines are orbits. Moreover all symmetry lines of thesurface are radial parameter lines. The Action Menu of
3DXM allows to switch between these parametrizations.
In 3DXM one can also deform this classical surface, but
we need to explain the significance of what one sees. See
the last page of this text.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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Early in the second half of the nineteenth century the
Enneper-Weierstraß representation of minimal surfaces was
discovered. Its main advantage is that it permits one towrite a formula for a minimal surface in terms of important
geometric quantities. Every surface in R3 can be mapped
to the 2-sphere S2 by sending each point on the surface
to the unit normal at this point; this map is called thegeometric Gauß map N . For minimal surfaces this map isangle preserving, but orientation reversing. Composition
of N with the orientation reversing stereographic projection therefore gives a map g from the surface into C which
is both and orientation preserving. Finally, if we interpret90◦ rotation on each tangent space of a surface in R3 as
multiplication of tangent vectors by i, then with this convention g becomes a meromorphic function, the meromorphic Gauß map of the minimal surface. This meromorphic
Gauß map is one-half of the Weierstraß data which areneeded to write down the Enneper-Weierstraß representation. The remaining part of these data is the differential
dF 3 of the third component of F , i.e., of the height function on the minimal surface. It might seem at first that wemust know a minimal surface rather well before we have its
Weierstraß data. However, on a large class of geometricallyimportant minimal surfaces the situation is simple indeed.
If a minimal surface is complete and has finite total curvature then the Gauß map g is determined—up to a constantfactor—by its zeros and poles, in other words by its verticalnormals. This important result extends to differentials, inparticular to the differential of the height function, after
we perform a small trick, namely extend the real valued
differential dF 3 to a complex valued one by putting forevery tangent vector v of the surface
dh(v) := dF 3 (v) − i · dF 3 (i · v):= dF 3 (v) − i · dF 3 (Rot90 (v)).
To make matters even simpler, observe that the points on
the surface, where the normal is vertical, are the same
points where the differential of the height function is zero.
More precisely, the zeros and poles of g on the minimal
surface are precisely the zeros of dh, even with the same
multiplicity. (To complete this discussion we would have to
study the situation at infinity, but we will omit this.) The
main point is to point out that very few, finitely many, data
about such minimal surfaces suffice to find their Weierstraßdata and therefore explicitly parametrize them. Here is
this famous formula:
Weierstraß Representation in terms of g, dh:Z z  
1
i
1
1
g−
dh,
g+
dh, dhF (z) := Re
2
g
2
g
∗
The classical Enneper surface is obtained if we put g(z) =z, dh = zdz.
This generalizes to polynomials P (z), put:g(z) = P (z), dh = P (z)dz.

3DXM allows P (z) := aa · z + bb · z 2 + cc · z 3 .
The pure powers have again the rotations around the origin
as metric isometry group and polar coordinates provide amuch better view of these surfaces and are recommended.
All surfaces of the associate family are, for g(z) = z k , congruent. There are straight lines on the surface, and if one
looks in the direction of the z-axis onto the surface, thenthe portion below these lines is drawn first. The default
morph deforms two such surfaces into each other.
H.K.

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A Minimally Immersed Projective Plane *See first: About Minimal Surfaces
This minimal surface is remarkable because its associatefamily contains an immersion of the triply punctured projective plane. It was discovered independently by Rob Kusner and Robert Bryant. It is also an example where theWeierstraß data can be explicitly integrated to a rationalparametrization of the surface:i(z 2 − z −2 )i(z 5 − z)√ =√W1 (z) =3−3
6z −z + 5
z − 1 + 5z 3
z5 + zz 2 + z −2√ =√W2 (z) =3−3
6z −z + 5
z − 1 + 5z 3


2i 3
−3
6
(z + z )
2i
2i
z +1
√ −
√
=
W3 (z) = 3
−1
3
−3
6
3
3
3 z − 1 + 5z
z −z + 5
The associate family is Fϕ (z) := Re(eiϕ W (z)).
Note the antipodal symmetry W (−1/z̄) = W (z), whichsays that F0 = Re(W ) = −Fπ is an immersion of the punctured projective plane. The poles√of W are at the six zerosof the denominator, with a3 = 3−2 5 , a ≈ 0.72556 they are(a, −1/a) · (1, e2πi/3 , e−2πi/3 ), three antipodal pairs.
The above formulas indeed represent minimal surfaces since
W10 (z) + W20 (z) + W30 (z) = 0.
This is expressed as: z 7→ W (z) is a holomorpic null curve.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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We also give the Weierstrass data:√
2 3
z (z − 5)g(z) = √, with g(−1/z) = −1/g(z),35z + 1√ √ 3
2 3
2iz (z − 5)( 5 z + 1)
√
dhz =
dz
6
3
2
(z − 1 + 5 z )For parameter lines we use conformal polar coordinatesz = exp(u + iv). A band −0.2 < u < 0, 0 ≤ v ≤ 2π ismapped to a minimal Möbius band, see the Action Menu.
Near the punctures the minimal surfaces Fϕ are asymptotic
to planes. One says: these minimal surfaces have planarends. At such planar ends g · dh or 1/g · dh must have adouble pole and g must have a branch point, i.e. g 0 (z) = 0
for finite z. Indeed, one can check that g 0 (z0 ) = 0 at thezeros z0 of the denominator of dh.
Planar ends have the following nice property: Invert theminimal surface in some sphere. Its planar ends invert tosurface pieces with one-point holes. These holes can besmoothly closed. This gives immersions of the complete
projective plane by inverting the minimal surface F0 andclosing the holes of its three inverted ends. The point at
infinity inverts to a triple point of such immersions. These
immersions are called Boy Surfaces since a topologicalversion was constructed in Boy’s PhD thesis under Hilbert.
It was the first immersion of the projective plane into R3 .
A model stands in front of the Oberwolfach Institute.
H.K.

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About Wavy Enneper *The surfaces Wavy Enneper, Catenoid Enneper, Planar
Enneper, and Double Enneper are finite total curvature
minimal immersions of the once or twice punctured sphere—
shown with standard polar coordinates. These surfaces illustrate how the different types of ends can be combinedin a simple way.
Morphing (0 ≤ bb ≤ 2) Wavy Enneper rotates a high order Enneper perturbation (amplitude = aa, frequency=ff)
over the ee tongues of a lower order Enneper surface. Note
how the perturbation decreases quickly with the distancefrom the boundary.
aa=0, ee=2 gives the standard Enneper surface.
Gauss map :Gauss(z) = z ee−1 (1 + aa exp(iπbb)z f f )/(1 + aa)
Differential:
dh = scaling · Gauss(z) dz
The pure Enneper surfaces (aa = 0) and the Planar Enneper surfaces have been re-discovered many times, because the members of the associate family are congruent* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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surfaces (as can be seen in an associate family morphing)
and the Weierstrass integrals integrate to polynomial (respectively) rational immersions. Double Enneper was oneof the early examples in which I joined two classical surfaces by a handle; we suggest to morph the size of thehandle or the rotational position of the top Enneper surface against the bottom Enneper surface.
H.K.

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About Catenoid Enneper *The surfaces Wavy Enneper, Catenoid Enneper, Planar
Enneper, and Double Enneper are finite total curvature
minimal immersions of the once or twice punctured sphere—
shown with standard polar coordinates. These surfaces illustrate how the different types of ends can be combinedin a simple way.
Here an Enneper perturbation with ee tongues (size adjusted with aa) crumples one rim of a catenoid in the suggested morphing. It is also interesting to choose aa=0.65,
ee=13, or so. The catenoid is aa=0.
eeGauss map : Gauss(z) = 4 − aa(1 + z ) /z
Differential: dh = (1 + aa2 /2) · Gauss(z) dz.
The pure Enneper surfaces (Gauss(z) = z k ) and the Planar Enneper surfaces have been re-discovered many times,
because the members of the associate family are congruent
surfaces (as can be seen in an associate family morphing)
and the Weierstrass integrals integrate to polynomial (respectively) rational immersions. Double Enneper was oneof the early examples in which I joined two classical surfaces by a handle; we suggest to morph the size of thehandle or the rotational position of the top Enneper surface against the bottom Enneper surface.
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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About Planar Enneper *The surfaces Wavy Enneper, Catenoid Enneper, Planar
Enneper, and Double Enneper are finite total curvature
minimal immersions of the once or twice punctured sphere—
shown with standard polar coordinates. These surfaces illustrate how the different types of ends can be combinedin a simple way.
The pure Enneper surfaces (aa = 0, dd = 0 ) and the Planar Enneper surfaces (aa = 0, dd = 2 ) with Weierstrass
data:
Gauss map : Gauss(z) = z ee+1 (1 + aa z f f )Differential: dh = scaling · Gauss(z)/z dd dz
have been re-discovered many times, because the membersof the associate family are congruent surfaces (as can beseen in the interesting associate family morphing!!) andthe Weierstrass integrals integrate to polynomial (respectively rational) immersions. For aa = 0, dd = 1 one does
not obtain a finite total curvature surface, but a periodic
surface that looks like a halfplane with periodically attached Enneper pieces. One obtains larger pieces of thisWavy Plane if one either increases ee or the range of v.
The members of the associate family are also congruent,
try Cyclic Associate Family Morph.
For small aa =/ 0 one has wavy perturbations of the Enneper end.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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For the (aa = 0)-examples ee is an integer valued parameter that determines the degree of dihedral symmetry of
the surfaces. Morphing the range of u helps to imagine
these surfaces by starting with a plane minus a disk andthen observe how an Enneper end is attached. In the Set
Morphing Dialog first press Initiate to current parameters,then choose umax0 = −0.5, umax1 = 0.6.
H.K.

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About Double Enneper *The surfaces Wavy Enneper, Catenoid Enneper, Planar
Enneper and Double Enneper are finite total curvature
minimal immersions of the once or twice punctured sphere—
shown with standard polar coordinates. These surfaces illustrate how the different types of ends can be combinedin a simple way.
The pure Enneper surfaces (Gauss(z) = z k , dh = Gauss(z) dz)
and the Planar Enneper surfaces(Gauss(z) = z k , dh = Gauss(z)/z 2 dz)
have been re-discovered many times, because the members of the associate family are congruent surfaces (as canbe seen in an associate family morphing) and the Weierstrass integrals integrate to polynomial (respectively) rational immersions. Double Enneper was one of the earlyexamples in which I joined two classical surfaces by a handle. The Weierstrass data are:Gauss map :Gauss(z) = z ee−1 (z ee − Aee )/(Aee z ee − 1)
Differential:
dh = eiϕ (1 − (z ee − z −ee )/(Aee − A−ee )) dz/z.
√
with A = aa · exp(iα), α = πbb/eeand (Aee + A−ee ) tan ϕ = −(Aee − A−ee ) tan 2α. The last
* This file is from the 3D-XplorMath project. Please see:
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equality is needed to avoid periods of the Weierstrass integral.
In this example the parameter aa controls the size of theneck between the top and bottom Enneper ends (it should
be kept in the range 3 < aa < 7). The parameter bb rotates the top and the bottom ends relative to each other,
The integer parameter ee = 2, 3,... determines the winding number of the Enneper ends and also the rotational
symmetry of the surface—the default is ee = 2. And umin
and umax control how far into the ends one computes.
Try a Cyclic Morph with the default parameters (this rotates the upper and lower ends in opposite directions). We
also suggest morphing the size (aa) of the handles. If theends intersect too much (e.g. for too large u-range or toolarge ee) one has to reduce the u-range. The Cyclic Morphis also interesting for large ee, for example ee = 12, umin =−1.45, umax = 1.5 and reduced scaling.
H.K.

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About the Saddle Tower Surface *These examples generalize Scherk’s conjugate pair of singlyperiodic/doubly-periodic minimal surfaces. The singlyperiodic examples stay embedded if the dihedral symmetry (and with it the number of punctures) is increased(Gauss(z) = z k , k = ee − 1). The most symmetric ones
(bb=0.5/ee) can be deformed by decreasing bb. See [K1],[K2] for more details.
These surfaces are parametrized by punctured spheres, but
the Weierstrass integrals have periods, a vertical one in the
singly periodic case, two horizontal ones for doubly periodic surfaces. The parameter lines extend polar coordinates around the punctures to the whole sphere—in thesecases giving level lines on the surfaces.
The degree of dihedral symmetry is, of course, a discrete
property, and it is controlled by the parameter ee. Thus,
ee should be set to an integer (the default is 2). For eachchoice of ee, changing bb gives a one-parameter family of
surfaces, of which the most symmetric member is obtainedby setting bb = 0.5/ee. Try setting ee to 3 and 4, andbb to 0.333 and 0.25 respectively. The wings of the singly
periodic saddle tower surfaces become parallel in pairs if(for ee > 2 ) one sets bb = 0.0825. These stay embedded* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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for ee=3 and ee=4.
We also recommend viewing the associate family morphing.
[K1] H. Karcher, Embedded minimal surfaces derived
from Scherk’s examples, Manuscripta Math. 62 (1988) pp.
83–114.
H.K.

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The Twisted Saddle Towers *The Twisted Scherk Saddle Towers are minimal surfacesthat were found in 1988 as deformations of the Scherk Saddle Towers. Therefore one should first look at these latter
simpler surfaces. One can imagine that one grips such asaddle tower at the top and the bottom and deforms thesurface by twisting it. Of course it is not clear whether
this can be done in such a way that the deformations stayminimal.
Fortunately, the most symmetric Scherk Saddle Towers
carry straight lines through their saddles, and it is easy
to imagine these lines staying on the twisted surface andremaining as lines of symmetry. Using these lines we canobtain the existence of the desired surfaces by solving thefollowing Plateau Problem.
Consider a pair of adjacent half-lines through one saddle
and another pair of half-lines, starting from the saddle
above the first and with one of its half-lines vertically above
the sector between the first two. These two “broken lines”
cut a simply connected strip out of the surface. (One cansee it by selecting “Don’t Show Reflections” from the Action Menu).
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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Now, vice versa, start with the two broken-lines and solve
the Plateau problem for this infinite boundary to find aminimal strip they bound. Finally use 180◦ rotations aroundthe half-lines to extend the Plateau strip to a complete
minimal surface, a Twisted Scherk Saddle Tower. Thesesurfaces played an important role in the development of
the theory of minimal surfaces since—except for the Helicoid itself—they were the first examples having helicoidal
ends,
The integer parameter ee controls the dihedral rotational
symmetry of the surface: the angle of each pair of halflines above is π/ee. The parameter aa controls the amountof twist, with aa = 0 giving the straight Scherk Saddle
Towers. We must keep aa < π/ee , since otherwise theexistence construction fails. Of course the default morph
varies aa.
In 3D-XplorMath minimal surfaces are computed via their
Weierstraß representation. The Scherk Saddle Towers areparametrized by punctured spheres and our twist deformation does not change this conformal type. However, due
to this twist, the Gauß map of the surface is not singlevalued: it is a rather a multivalued function on the punctured sphere, and this makes the computation more difficult than for the other spherical minimal surfaces, sinceduring the integration of the multivalued Weierstraß integrand, it is necessary to use analytic continuation in orderto guarantee that we always have the correct value of the

Gauß map.
The Weierstraß representation is given in:
Karcher, H., Embedded Minimal Surfaces derived from
Scherk’s Examples, Manuscripta math. 62 (1988), pp. 83- 114.
Recently Traizet and Weber have found a new construction of embedded singly periodic minimal surfaces thatcan be illustrated with the twisted Scherk saddle towers.
Choose aa close to its theoretical limit 1/ee and try to
see the resulting surfaces as made out of ee ordinary helicoids. Obviously one has to allow modifications of thehelicoids in the middle, but away from the middle one cansee these helicoids well. Traizet and Weber were able toturn this observation around. They found conditions how
to place a collection of helicoids so that they could provethe existence of a family of singly periodic embedded minimal surfaces which converged to the given helicoids in thesame sense as the twisted Scherk saddle towers converge
as aa → ±1/ee.
H.K.

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Symmetric and Skew-symmetric k-Noids *The symmetric and skew k-noids are parametrized by kpunctured spheres. The number k of ends is selected withthe 3DXM parameter ee. We use parameter lines that
extend polar coordinates around the punctures. Formulas are taken from [K2]. These formulas easily allow tochange the relative size of neighboring catenoid ends forthe symmetric 2k-noids, try the default morph which
varies the 3DXM parameter aa in [0, 1]. The symmetrick-noids, where the number k of ends is odd, cannot be deformed in 3DXM.
The ends of the skew-symmetric k-noids in 3DXM are
all of the same size. The default morph varies the angle
between neighboring ends, again by changing the 3DXM
parameter aa in [0, 1].
The size of the ends is always controlled using bb.
The intersection of the two families are the symmetric knoids with all ends of the same size, a family found by Jorge
and Meeks. Their k-noids are the first finite total curvature immersions for which the Weierstrass data were manufactured to fit a previously conceived qualitative global
picture of the surfaces.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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The default morph of the symmetric 2k-noids ends at symmetric k-noids with k punctures as aa goes to zero.
In the default morph of the skew-symmetric k-noids the angle between neighboring pairs of ends goes to zero. Near
the limit the 4-noid looks like a pair of parallel catenoids
that are joined by a very thin “catenoidal” handle. Infact the picture looked so suggestive that it convincedDavid Hoffman immediately that the idea of adding handles might be promising in a wide range of situations. Try
this default morph!
H.K.

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About the Lopez-Ros No-Go Theorem *The Lopez-Ros Theorem [LR] says that a complete, minimal embedding of a punctured sphere is either a Catenoid
or a plane. Our example is parametrized by a 3-punctured
sphere. Its Gauss map is Gauss(z) := cc(z − 1)(z + 1). The
differential dh = (z 2 − 1)/(z 2 − ee)2 puts the puncturesat +ee, −ee and ∞. Parameter lines on the sphere extend polar coordinates around the punctures at z = +ee,
z = −ee, z = ∞ in order to make the ends look nice.
A necessary condition for embeddedness is that the normals of all ends are parallel, i.e., ee = 1. In this casea residue computation shows that the period cannot be
closed, in agreement with the theorem of Lopez-Ros. If
ee > 1, then cc can be chosen to close the period, butthen the catenoid ends are tilted so that they intersect thethird (planar) end. The default morph in 3DXM shows
what happens when ee approaches 1 while the periods arealways closed (with a closing value of cc that grows to infinity). In a properly scaled picture the surface looks more
and more like two catenoids at larger and larger distance.
Since we also want to show morphs were the period opens
up, the program uses cc as follows: If the user sets cc = 0* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOCthen the program recomputes cc to the period-closing value
ccclose . Otherwise we only restrict cc by resetting it as
cc = max(0.5 · ccclose , min(cc, 2 · ccclose )).
To choose one’s own morph, note that the surface has a
gap if cc is larger than ccclose , and that it intersects itself
if cc is smaller than ccclose . As an example, compute firstwith ee = 1.01, cc = 0; then, in the Set Morphing Dialog,
click the button Initialize to current parameters, and finally morph ee from the current value ee0 = 1.01 to somelarger value, e.g. ee1 = 1.08.
[LR] F.J. Lopez and A. Ros, On embedded complete minimal surfaces of genus zero, Journal of Differential Geometry 33 (1), 1991, pp 293–300.
H.K.

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Two Planar And Two Catenoid Ends *This Minimal surface is parametrized by a sphere with fourpunctures. In the complex plane its Weierstrass data aregaus(z) :=

z−a
· z2,
az + 1dh :=z − 1/z − a + 1/a dz· .
2(z − 1/z − c + 1/c)z

The Gauss map has a simple zero and pole at a and −1/a.
These are finite points on the minimal surface since they
are cancelled by the simple zeros of dh at a, −1/a. And
the Gauss map has a double zero and pole at 0, ∞. These
give planar ends because dh is neither 0 nor ∞ there. The
double poles of dh at c, −1/c give catenoid ends because
the Gauss map has simple finite values there - see About
Minimal Surfaces in the Documentation Menu.
Catenoid ends usually come with a period. A residue computation shows that the choice a := c − 1/c makes theperiod of the real part of the Weierstrass integral zero.
Note that the Weierstrass data are symmetric with respect
to 180◦ rotation around i, −i. This corresponds to a 180◦
rotation of the minimal surface around the y-axis in R3 .
Recall that the Lopez-Ros theorem states that the only
complete embedded minimal surfaces that are parametrizedby punctured spheres (and have finite total curvature) are
the plane and the catenoid.
One can view the shape of the current surface as madefrom two copies of our Lopez-Ros-No-Go example by stacking these above each other and deforming the two half
* This file is from the 3D-XplorMath project. Please see:
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catenoids between them into the handle of the current surface between its two planes. The new surface is therefore
another failing attempt of a counter example to the LopezRos theorem. View in the Animate Menu the morph:
Watch The closed Catenoids Tilt. It shows: The twocatenoid ends get more vertical as c approaches 0, but inthe limit the whole surface degenerates to a doubly covered
plane.
Compare the current surface also to the Riemann minimal
surface. Its shape can be viewed as an infinite stack of
copies of the current surface – again, of course, with thehalf-catenoids between adjacent copies deformed into handles which join the copies.
Minimal surfaces look much better when the grid lines in
the parameter domain behave like polar centers aroundthe punctures. With this in mind, the current surface wasparametrize by the following grid in C:
3

2

1

0

-1

-2

H.K.

-3
-5

-4

-3

-2

-1

0

1

2

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Half-Catenoids and Weierstrass Representation *In this text we try to explain, starting from scratch, minimal surfaces parametrized by tori.
We assume from complex analysis roughly the following:
The complex logarithm, log(z), is a multivalued function
with derivative log0 (z) = 1/z. One can compute the logarithm by integrating the differential form dz/z along a
curve c : [0, 1] 7→ C with c(0) = 1, c(1) = z:
Z
Z 1 0
1
c (t)
log(z) :=
dζ :=
dt
c ζ
0 c(t)
Of course, the integration path c has to avoid 0. Therefore one can reach z from 1 in many ways. A basic property of such complex line integrals is that their value does
not change if the path is deformed, but with fixed endpoints. Two paths which wind around 0 a different numberof times cannot be deformed into each other and, indeed,
the values of the integrals along the two paths differ by
an integer multiple of 2πi. This number, the value of theintegral from 1 once around 0 and back to 1, is called
the period of the differential form dz/z at its singularity 0.
The real part of this integral has no period, we have:Re(log(z)) = log |z| for all paths from 1 to z.
Instead of integrating one differential form we can integrate
three, so that the real part of such an integral maps (a pieceof) the complex plane into R3 . The image is a surface* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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without singularities if the integrand is never (0, 0, 0). If
the integrand has singularities, the integration path has to
avoid them and there may be periods. This means thatthe integral gives us not just one image of its domain, but
a periodic repetition of one image. The obtained surfaces
are in general not minimal surfaces, but one is close tothem.
Weierstrass derived the following representation:
Let D ⊂ C be some domain, g : D 7→ C a holomorphic
function and dh a holomorphic 1-form on D. Then:Fϕ (z) := Re

eiϕ ·

Z z
∗


i
1
(1/g − g), (1/g + g), 1 dh
22maps, for each ϕ, D to a minimal surface piece in R3 .
The length of the image of curves c in D can be computedwith the Riemannian metric:1
1
ds = |g(c(t))| +|dh(c0 (t))dt|.
2|g(c(t))|Beyond the metric we do not discuss the differential geometry here. See About Minimal Surfaces for a derivation of
the above Weierstrass representation that starts from theminimal surface definition. Instead we discuss Weierstrass’
formula.
Since the Riemannian metric does not depend on the parameter ϕ the above surface pieces are all isometric to eachother (almost never congruent). This is a rare phenomenon

in surface theory. – Zeros of the differential form dh lead tozeros of dFϕ . Such singularities are called branch points ofthe minimal surface. The Riemannian metric shows thatthey can be avoided if the function g has a zero or a poleof the same order as the zero of dh. View branch points
on the Henneberg surface ( Scherk, Catalan, Henneberg).
Stereographic projection between the Gaussian plane C
and the Riemann sphere S2 is an important tool in complex
analysis. Composition of the function g with stereographic
projection is2~
N (z) := 2Re(g), 2Im (g), |g| − 1 /(|g|2 + 1) ∈ S2 .
This vector is orthogonal to the integrand of the Weierstrass formula and therefore a unit normal vector field of
the surface. The Weierstrass representation is therefore
built from a unit normal field and the differential of the~ are both called Gauss
height function. The functions g, N
map of the surface. – We mention that a surface is minimal~ is anti-conformal. The ’anti’ comes in
if its Gauss map N
because a minimal surface has Gauss curvature K ≤ 0. Of
the two choices for stereographic projection we used theanticonformal one.
One of the simplest examples, the catenoid and its conjugate, the helicoid, are obtained from the following Weierstrass data (see Helicoid-Catenoid ):dz.
D := C \ {0}, g(z) = z, dh =z
The 3rd component of the Weierstrass integrand has theimaginary period of dz/z which we discussed. The first andsecond components have no periods because the singularterms dz/z 2 have the antiderivative −1/z (so that the period integrals around the singularity vanish). If we set theassociate family parameter ϕ = 0 we get a surface without
period, a conformal immersion of C \ {0}, the catenoid. If
we set ϕ = π/2 we obtain a surface with vertical translation period, an immersion of the universal covering of
C \ {0}, the helicoid.
Remark. If one wants to visualize a surface one usuallyneeds to choose parameter lines. In case of the catenoid:
Only if one chooses polar coordinates around {0, ∞}, does
the catenoid look familiar. For all other choices one canhardly recognize it. With the good choice, the parameterlines are the principal curvature lines. I believe these lines
give the eye the best clues to imagine the surface correctlyin space.
As in the case of the catenoid: Interesting Weierstrass data
have isolated singularities, usually zeros or poles. The following summary helps to find interesting examples:
Poles of dh are never on the surface and lines into thepuncture are infinitely long on the surface.
Zeros of dh have to be canceled by either a zero or else apole of g which have the same orders as the zeros of dh.
If a puncture is to be a half-catenoid, then the Gauss map
g must be single valued at the puncture. Furthermore, either g · dh or else dh/g must have a double pole (as in thecase of the catenoid).

Let us try to find the Weierstrass data for the Trinoid ofJorge-Meeks (see Symmetric and Skew k-Noids):

We have learnt from the catenoid that the huge openingof a catenoid is conformaly only a puncture, just one point
missing. The picture therefore shows a sphere with threepunctures, say on the equator. With this orientation there
are just two points with vertical normals, one pointing
up, the other down. This implies that the Gauss map
is some power of z. Along the equator we find, for eachhalf-catenoid, one more normal which points in the same
direction as its limiting normal. The degree of g is therefore 2, hence g(z) = const · z 2 , and |const| = 1 to havehorizontal normals along the equator. Next dh. It needs
double zeros at 0, ∞ to cancel the double zero and pole ofg and it needs double poles at the third roots of unity tocreate the half-catenoids. Since dz/z is, up to sign, invariant under inversion z 7→ 1/z, we write dh as a multiple ofdz/z (which has simple poles at 0, ∞).

This gives the Weierstrass data of the trinoid:z3dzg(z) := z , dh = 3·(z − 1)2 z
2To get good parameter lines, i.e. polar coordinates near thepunctures, we use for one third of the unit disk the gridshown to the left of the trinoid. Closed parameter lines(’circles’) in the domain around a puncture are mapped
to closed curves in space since two orthogonal planes ofsymmetry cut the space curves into four congruent arcs.
Next let us grow two half-catenoids out of the plane asfollows (see also The Lopez-Ros No-Go-Theorem):

Observe that the limiting normals of the two half-catenoids
point in the same direction, but not quite opposite to thenormal of the plane. On each half-catenoid one can locate
a finite point where the normal is opposite to the normalof the plane. The Gauss map has two simple zeros at thesepoints, say at ±1, and a double pole at ∞. Therefore wefound g(z) = c · (z − 1)(z + 1). The zeros of g have to becanceled by simple zeros of dh and dh needs double polesat points ±r to create half-catenoids there.
This gives us a 2-parameter family of Weierstrass data:z2 − 1g(z) := c · (z − 1)(z + 1), dh := 2
· dz.
2
2
(z − r )In this case there is only one vertical symmetry plane cutting the half-catenoids. Therefore closed curves around thepunctures are not by symmetry mapped to closed curvesin space, a period orthogonal to this plane can occur. The
parameter c can be used to make this period vanish, foreach r > 1. As r approaches 1 the two half-catenoids movefurther and further apart. In other words: one cannot growtwo half-catenoids with limiting normals orthogonal to theplane. The half-catenoids will therefore always intersect
the plane so that all these surfaces are only immersed,
none is embedded.
It is however possible to grow infinitely many equidistanthalf catenoids out of the plane such that all these halfcatenoids have the same limiting normal and this normal is opposite to the normal of the plane. The straightsymmetry line between neighboring half-catenoids – which
was present in the previous example – continues to exist.
Therefore one needs to compute the surface only in a strip
between neighboring symmetry lines.
The Weierstrass data of this Catenoid Chain are easy toguess:
dz,
g(z) := c · sin(z), dh :=sin(z)

where c ∈ R controls the size of the half-catenoids relative to their distance. It is more complicated to get polar
coordinates in a strip. Here is the result:

While sin has in the plane an essential singularity at ∞,in the strip there is convergence. The strip with edges
identified is a cylinder with two half-catenoid punctures– or a sphere with four punctures, the two half-catenoids
and the two ends of the flat strip. This last point of viewsuggests rational Weierstrass data:1dz· , c ∈ R.
g(z) := c · (z − 1/z), dh :=z − 1/z zBeyond the singly periodic trigonometric functions one has
the doubly periodic meromorphic elliptic functions. Doubly periodic means that we have two independent translational symmetries. The group of translational symmetries
is called a lattice Γ in C. Each lattice has a parallelogram
as fundamental domain and identification of opposite edges
makes the parallelogram into a torus. Meromorphic elliptic functions can therefore be viewed as maps from tori to
the Riemann sphere. We visualize an elliptic function (denoted J F in 3D-XplorMath)by drawing the preimage on
the torus of the usual polar grid on the Riemann sphere:

The two coordinate centers in the middle are zeros of thefunction, the two centers on the (identified) boundary arepoles. The rectangles around the polar centers (made ofsymmetry lines of the picture) are preimages of the unit
circle.
If we use this elliptic function in the same way as we usedthe sin-function before:dzg(z) := c · J F (z), dh :=,J F (z)then we obtain the following doubly periodic minimal surface:

The picture shows two copies of a fundamental domain for
the translational symmetries. The half-catenoids are at
the zeros of the function J F . At its poles are the zerosof dh and these cancel the poles of the Gauss map g. In
other words: the poles of J F are points on the surface
with vertical normal. – The parameter lines on the surface
clearly are the image under the Weierstrass integral of thegrid on the torus shown before.

The Coordinate Viewpoint
The minimal surface just discussed could have been computed in the described way, but it was not. So far we
looked at the torus as a quotient C/Γ and meromorphic
functions on a torus therefore were the same as meromorphic functions in C which had the translations τ ∈ Γ asperiods, meaning f (z + τ ) = f (z). And the Weierstrass
integral was evaluated on a fundamental parallelogram forthe lattice Γ.
Elliptic functions J : C/Γ 7→ C are, away from their branch
points, locally invertible. They can therefore be viewed ascoordinate functions on the torus. The Weierstrass dataof our doubly periodic surface are given in terms of sucha function J. We also need to express dz in terms of J,
namely: dz = dJ/J 0 . This is almost all what we need, toperform the Weierstrass integration not on the torus, but
in the range of the coordinate function J, i.e. in C. We stillneed to express J 0 in terms of J. The Jacobi-type elliptic
functions which we constructed in ’Symmetries of Elliptic
Functions’ all had four branch values: ±B, ±1/B. The twofunctions (J 0 )2 and (J 2 − B 2 ) · (J 2 − 1/B 2 ) have the same
four double zeros and the same two fourth order poles.
Since the domain torus is compact this implies that these
two functions are proportional. We ignore this multiplicative constant because it only determines the scaling-size ofthe fundamental domain. Thus we derived the . . .

Differential Equation for Jacobi-type Elliptic Functions:
J 0 (z)2 = P (J) = J 4 − (B 2 + B −2 ) · J 2 + 1,
J 00 (z) = P 0 (J)/2 = 2J 3 − (B 2 + B −2 ) · J.
And the above Weierstrass data take this form:g(J) = c · J, dh =1
dJ·p
, J ∈ C.
J
(P (J))

This looks much simpler than before and can be directly
integrated on a standard polar coordinate grid in C. The
remaining problem is the square root. It is a multivalued
function and we have to choose the correct branch. Since
we are doing path integrals along curves which avoid thebranch values (i.e. the zeros of the polynomial P ), such ananalytic continuation of the square root can be built into
the integration routine.
Remark 1. This coordinate view point is a first important

step towards the theory of Riemann surfaces. While fortori this view point is only a simplification, it becomes essential for minimal surfaces parametrized by higher genus
surfaces.
Remark 2. The above differential equation is an example
where the dominating Runge-Kutta method fails: If one
starts the integration at a zero of P then Runge-Kutta
produces a constant solution. There are 4th order methods which use J 00 and work fine.
The same is already true for the ODE (sin0 )2 = 1 − sin2 .

Half-Catenoids with opposite normals
How can one find a Weierstrass representation for this surface? (See also Symmetric and Skew k-Noids.)

The surface looks like two catenoids joined by a handle, inother words: a sphere with four half-catenoid punctures.
We place the surface so that the four limiting normals of
the half-catenoids lie in the horizontal plane and form small
angles ±ϕ with the y-axis. On the handle, and also on thewaist of each catenoid, one sees one point with the normal
pointing vertically up, one point with the normal down.
The gauss map therefore has three zeros and three poleson the real axis: g(z) = z(z 2 − r2 )/(1 − r2 z 2 ). They are
positioned symmetric to the y-axis and to the unit circle.
The differential dh must have 6 zeros at these points tomake them into finite points on the surface, and it must

have four double poles on the unit circle to create the horizontal half-catenoids. This gives a 2-parameter family ofWeierstrass data:z 2 − r2,
g(z) := z
1 − r2 z 2
z 2 + z −2
z 2 + z −2−2 dz
dh := (1 − 2
)(
−
1)
.
−2
2ϕ
−2ϕ
r +r
e +e
z
For most choices of parameters the half-catenoid punctures
have vertical periods. A residue computation shows thatthese periods vanish if
e2ϕ + e−2ϕ = 2 cos 2ϕ = 4r2 /(1 + r4 ).
Historically this is the first surface that showed David Hoffman and me a handle which connected two minimal surfaces, deforming them slightly.
If we imagine such handles to grow in two opposite directions out of the waist of the catenoid, then symmetry
would allow the catenoid to stay straight. Therefore we canthink of making a fence of parallel catenoids, see Fence ofCatenoids. Parallel translation from the symmetry planeof one handle to the symmetry plane of the next handle
would be a congruence map of this surface. The quotientby this translation group is a torus with two half-catenoid
punctures whose limiting normals point in opposite directions. The handle has two points with normals parallelto the limiting normals of the half-catenoids. The surface
in the following picture therefore has a Gauss map with

two zeros and two poles. One zero and one pole are halfcatenoid punctures, the other two are finite points on thehandle.

So first, what kind of a torus is this? There are two typesof tori which have involutive symmetries, the rectangular
tori and the rhombic tori. They are easy to distinguish:
the fixed point set of such an involution of a rectangular torus has two components, while such involutions ofrhombic tori have fixed point sets with only one component. (Note that the interruption of a fixed point set by apuncture is ignored in this count because the puncture is apoint on the torus, it is only missing on the image minimal
surface.) There are symmetry lines going from the handle
into each puncture: They lie in the same plane and are twocomponents of the fixed point set of the surface reflection

in this plane. The parametrizing torus is therefore rectangular. Each of the two symmetry arcs joins points withvertical normals in opposite directions. The following picture allows to visualize this elliptic function, it is again thepreimage of a standard polar grid on the Riemann sphere.

The left bottom and right top polar centers are zeros of
the function, the other two polar centers are first orderpoles. Because of this diagonal arrangement this function
is called J D in ’Symmetries of Elliptic Functions’. The
rectangles which are filled with a polar grid are preimagesof the unit circle. The branch points are the points wheremore than 2 parameter lines cross. For rectangular tori,
the branch values are all on the unit circle.
We will make the two top polar centers the punctures, thetwo bottom polar centers the finite points with verticalnormals (on the handle). Then the function J F , repre-

sented by the earlier grid, gives us the second part of theWeierstrass data:g(z) := J D(z), dh := J F (z)dz.
As before we can integrate in the range of an elliptic function of our choice. The functions J D, J F can be computed from each other by solving quadratic equations (Dis the branch value of J D in the first quadrant):
1
1
D + 1/DJ D+=(J F +
),
J D
2
J
F


1
−J D ,(J F )0 = J F · (J D)0 (0) ·J D
dz = d(J F )/(J F )0 .
(Recall: (J D)0 (0) controls the size of the fundamental domain.) Again square roots appear and the correct branch
has to be chosen. (For the last surface a more complicatedgrid than the standard polar grid on the sphere was used.)
So far we have explained how the Weierstrass representation works for minimal surfaces which are punctured tori.
But we have not touched an important question. The
Weierstrass representation is well suited to create immersions, and rather many easily. But it requires separate
considerations to decide whether the immersed surface is
without selfintersections, whether it is embedded. For 300
years the plane and the catenoid were the only complete
embedded minimal surfaces (CEMS) which have finite to-tal curvature. Finite total curvature means that the normal Gauss map from the surface to the unit sphere has
finite degree, i.e. there is a number d such that almost
all points of the sphere are hit d times by the gauss map.
Or put differently: that the Gauss map is a meromorphic
function.
Lopez-Ros have proved: The only CEMS of finite total
curvature which are conformally punctured spheres, are theplane and the catenoid. The Lopez-Ros No-Go-Theorem.
R. Schoen has proved: The only CEMS which have no more
infinities than two catenoid punctures are the catenoids
themselves. R. Schoen’s No-Go-Theorem.
There is an obvious attempt to make a torus with two
catenoid ends: Instead of growing handles to the outside which created the fence of catenoids above - one can growthe handles to the inside. It is easy to start with small
handles which are too short to meet in the middle:
The handles end in planar symmetry lines which - after translation! - fit together perfectly.
The only problem is that acurve, which is closed on thetorus, has a Weierstrass image in R3 that is not closed, it
has a nonzero period. Schoen’s
theorem forbids that one canmake the handles long enough.
So, how does the example fail?

Things go well for a while: the handle grows and the gap
narrows. But, as one changes the parameter further in thepromising direction, the mouth of the handle becomes very
elongated and the surface starts to look like two catenoids
whose far out portions are connected. Except that the gap
remains no matter how far apart the middle portions of
the two catenoids are pushed. The following picture shows
how close to a counterexample of Schoen’s theorem one
gets, or, what difficulties Schoen’s proof has to overcome.

Failing attempt to grow a handle through a catenoid.
It was a big surprise when Costa found a minimally immersed punctured torus which Hoffman and Meeks proved
to be embedded! The surface is a bit similar to our earlyexample where two half-catenoids grow out of a plane.
Therefore we first try to imagine a torus with a planar
end. This is a difficult request until one turns it aroundand asks for a plane with a handle:

A plane with a handle is a torus with a planar end.
This surface is a stereographic projection of one of the
Clifford Tori , the projection center is on the surface. All
its parameter lines are circles, except for the two straightlines which pass through the midpoint of this surface. It isa torus with one puncture. The torus is the square torus
because it is a) rectangular since the fixed point sets (onthe surface) of the reflections in two vertical planes have
two components and it is b) rhombic because 180◦ rotation
about the straight lines are symmetries whose fixed pointsets have only one component.
The strategy is: puncture the torus in the two points where
the normal to the two straight lines (= the intersection of
the symmetry planes) intersect the surface again and growhalf-catenoids at these points towards the outside (or awayfrom the straight lines).
This plan contains enough information to write down the
Weierstrass data of

Costa’s Minimal Surface:We draw the quadratic fundamental domain of our square
torus such that the straight lines on the cyclide surface
become the diagonals. Note that they are also one component fixed point sets of orientation reversing involutions.
The pair of horizontal symmetry lines (the parallel boundary segments are identified) are a two component fixed
point set of an involution, and the pair of vertical symmetry lines also. The points C1, C2 are the punctures for thehalf-catenoids. The point P is the (conformal) point at infinity of our plane with handle. The center is called S (forsaddle). The normals at S, C1, C2 point in one direction,
the normal at P in the opposite direction. We take these
directions as vertical so that the Gauss map has simple
zeros at S, C1, C2 and therefore a triple pole at P .
This forces the height differential dh to have simple polesat C1, C2 to make these points
catenoid punctures. At S weneed to compensate the simple pole of 1/g with a simple
zero. The remaining other simple zero would cause a branch
point everywhere except if it
is at P . These zeros and poles determine g up to a factor a exp(i · α) and dh up to a factor b exp(i · β). Clearlyb only scales the surface and α rotates it. And β is theassociate family parameter, it is chosen to make the pe-riods of the catenoid punctures imaginary. Under these
conditions the fixed point sets of the involutions turn out
to be straight lines (the diagonals) and planar geodesics.
This was not at all clear initially, but it follows immediately from the description of the functions in ’Symmetries
of Elliptic Functions’. With the parameter a we have at
this point the Weierstrass data of a 1-parameter family ofcandidates. With an intermediate value argument one canchoose a so that the dotted curve - which is closed on thetorus - also has a closed Weierstrass image in R3 .
Weierstrass Data for the Costa Surface:g(z) = a · (J E · J F )0 (z), dh =dz.
J D(z)Up to a multipicative constant the function J E · J F is
the Weierstrass ℘-function (here for the square torus).
H.K.

TOC

Chain of Half-Catenoids
and Field of Half-Catenoids *These surfaces played no role in the history of minimal
surfaces, but they explain very nicely how the Weierstrass
representation works, compare Half-Catenoids and Weierstrass Representation.
Weierstrass representation of a vertical catenoid:g(z) = z, dh =dz.
zWeierstrass representation of a singly periodic chain of vertical half-catenoids:g(z) = bb · sin(z), dh =dz.
sin(z)The simple zeros of the sine function create half-catenoid
punctures. These punctures have no real periods since
two orthogonal planes of mirror symmetry cut each halfcatenoid into four congruent pieces. Neighboring halfcatenoids are separated by straight lines. They are axes of180◦ rotation symmetry. One needs to compute the surface
only in a strip between two neighboring lines. The scaling* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

factor bb in the Gauss map controls the size of the halfcatenoids relative to the distance between them. Two such
strips are a fundamental domain for the group of translation symmetries.
Weierstrass representation of a doubly periodic field of vertical half-catenoids:g(z) = bb · JF (z), dh =dz.
JF (z)

The function JF is a doubly periodic function on C, in thiscase with a rectangular fundamental domain. In each fundamental domain JF has two simple zeros and two simple
poles, see ‘Symmetries of Elliptic Functions’. The zeros of
the Gauss map g together with the poles of dh create thehalf-catenoid punctures. The poles of g are cancelled by
the zeros of dh, they give the polar centers on the surface.
As in the previous examples we have orthogonal planes ofmirror symmetry cutting each half-catenoid and we havestraight lines on the surface, running between the halfcatenoids. These symmetries of the minimal surface are aconsequence of the corresponding symmetries of the elliptic function which determines the Weierstrass data. The
scaling parameter bb of the Gauss map changes the size ofthe half-catenoids relative to their distance.
H.K.

TOC

The Doubly Periodic JD Minimal Surfaces *These families are parametrized by 4-punctured rectangular tori; they and their conjugates are embedded. We
therefore suggest the associate family morphing, and also
morphing of the modulus, bb, (0 < bb < 0.5) of the rectangular torus, which changes the size of the visible holes.
For the visual appearance of these surfaces it is particularly important that the punctures are centers of polar
coordinate lines. Formulas are taken from [K1] or [K2].
The Gauss maps for these surfaces are degree 2 elliptic
functions. The cases shown are particularly symmetric,
the zeros and poles of the Gauss map are half-period points
and the punctures are there. In the Jd-case the diagonal of
the rectangular fundamental domain joins the two zeros,and in the Je-case it joins a zero and a pole of the Gauss
map.
Under suitable choices of the modulus of the torus
these surfaces look like a fence of Scherk saddle towers with a vertical straight line (Je), respectively a planar symmetry line (Jd), separating these towers. The conjugate
surfaces look qualitatively the same in the Jd-cases andlike a checkerboard array of horizontal handles between
vertical planes in the Je-cases.
H.K.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

The Doubly Periodic JE Minimal Surfaces *These families are parametrized by 4-punctured rectangular tori; they and their conjugates are embedded. We
therefore suggest the associate family morphing, and also
morphing of the modulus (aa) of the rectangular torus,
which changes the size of the visible holes. (The images ofthis deformation looked best when the u-range was modulus dependent:
umax := (5-2*aa)+bb; umin := -umax;
The parameter bb lets the user overrule the above dependence.
For the visual appearance of these surfaces it is particularly important that the punctures are centers of polar
coordinate lines. Formulas are taken from [K1] or [K2].
The Gauss maps for these surfaces are degree 2 elliptic
functions. The cases shown are particularly symmetric,
the zeros and poles of the Gauss map are half-period points
and the punctures are there. In the Jd-case the diagonal of
the rectangular fundamental domain joins the two zeros,and in the Je-case it joins a zero and a pole of the Gauss
map.
Under suitable choices of the modulus of the torus
these surfaces look like a fence of Scherk saddle towers with a vertical straight line (Je), respectively a planar sym* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

metry line (Jd), separating these towers. The conjugate
surfaces look qualitatively the same in the Jd-cases andlike a checkerboard array of horizontal handles between
vertical planes in the Je-cases.
H.K.

TOC

Riemann’s Minimal Surfaces *This is the family of singly-periodic embedded minimal
surfaces found by Riemann. They are parametrized (aa)by rectangular tori. The Gauss map is the Weierstrass ℘function additively normalized to have a double zero at thebranch point diagonally opposite the double pole and multiplicatively normalized to have the values plus or minus iat the four midpoints (on the torus) between the zero andthe pole. The minimal surface has rotational symmetries
around the corresponding normals. This symmetry kills
the horizontal periods. The surface is parametrized by therange of the Gauss map with polar coordinates around thepunctures. The surfaces look like families of parallel planes
with one handle between adjacent planes. The associatefamily morphing joins two such embedded surfaces—they
are congruent for the square torus. The standard morphing (aa) changes the branch values of the Gauss map, i.e.
the tilt of the normal at the flat points (K = 0).
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

The Chen-Gackstatter Minimal Surface *and Higher Dihedral Symmetry Examples

This surface is the first finite total curvature immersion ofa Riemann surface of genus > 0 (here the square torus). It
looks like an Enneper surface with a handle added parallel
to its center saddle. This description determines the Gaussmap only up to a multiplicative constant (cc), which we
took as the morphing parameter. If this parameter is general then we get a doubly periodic minimal immersion of
the plane. The morphing indicates how the period can beclosed for one value of cc with the intermediate value theorem. The resemblance with the standard Enneper surface
is emphasized by using polar coordinates around the puncture. The dd=3 surface is an analogue which can be viewed
as a higher order (120 degree symmetric) Enneper surface
with a Y-shaped handle glued in. It was first published in
a 1988 Vieweg Calendar by Polthier and Wohlgemuth.
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

Costa’s Minimal Surface *This surface was responsible for the re-kindling of interestin minimal surfaces in 1982. It is a minimal embeddingof the 3-punctured square torus. Its planar symmetry lines
cut this surface into four conformal squares and the twostraight lines through the saddle are the diagonals of thesesquares.
The Gauss map of such a surface is determined by its qualitative properties only up to a multiplicative factor cc which
we suggest for the morphing (as in the Chen-Gackstattercase). It closes the period (at cc0) with an intermediate
value argument.
After Costa’s existence discovery, Hoffman-Meeks proved
embeddedness; they also found a deformation family throughrectangular tori, where the middle end deforms from a planar one to a catenoid end. They generalized this family to
any genus by increasing the dihedral symmetry.
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

About the Costa-Hoffman-Meeks *Minimal Surfaces
The original Costa surface was responsible for the rekindling of interest in minimal surfaces in 1982. It is a minimal embedding of the 3-punctured square torus. Its planar symmetry lines cut this surface into four conformal
squares and the two straight lines through the saddle are
the diagonals of these squares. Because of the emphasison the symmetries, our formulas are taken from [K2.] The
Costa-Hoffman-Meeks surfaces are generalizations of the
Costa surface; their genus grows as the dihedral symmetry(controlled by dd) is increased. The underlying Riemannsurfaces are tesselated by hyperbolic squares with anglesπk , (k = 2, 3, ...).
The Gauss map of such a surface is determined by its qualitative properties only up to a multiplicative factor cc which
we suggest for the morphing (as in the Chen-Gackstattercase). It closes the period (at cc0) by an intermediate value
argument.
As in Costa’s case, the qualitative picture determines theGauss map only up to a multiplicative factor. The stan* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

dard morph shows the dependence of the surfaces on thisfactor, closing the period at cc0.
H.K.

TOC

The Fence of Catenoids *These singly periodic surfaces are parametrized (aa) byrectangular tori; our lines extend polar coordinates aroundthe two punctures to the whole torus. The surfaces look
like a fence of catenoids, joined by handles; they weremade by Karcher and Hoffman, responding to the suggestive skew 4-noids. The morphing parameter aa is themodulus (a function of the length ratio) of the rectangulartorus. The surface is reflection symmetric with respect toplanes parallel to the coordinate planes; in each case has
the fixed point set, modulo translations, two components.
This proves that the parametrizing torus is rectangular.
(The fixed point set in the rhombic case has one component and the square torus has both symmetries.)
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

Scherk’s Surface with a Handle Added *This surface is a genus one version of Scherk’s doublyperiodic surface. Existence and embeddedness is proved in[KWH], and our formulas are from there.
The conjugate fundamental domain is bounded bystraight lines. This piece can be rotated to be a graphover a convex domain, and in this position the originalpiece is also a graph. For this we suggest the associatefamily morphing.
The surface has a period problem, because the position of the punctures is not determined on the square torus
by qualitative considerations. We suggest the (joint) position of the punctures as morphing parameter (ee), again
to illustrate the use of intermediate value arguments forkilling periods.
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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R. Schoen’s Catenoid Characterization *R. Schoen’s theorem says:
The only finite total curvature complete, embedded minimal surface having two ends, is the Catenoid.
Our example shows what happens if one tries—in spite ofthis theorem of Schoen—to add a handle to a Catenoid.
The fundamental piece is similar to that of the catenoid
fence, except that the handle does not go outward to theneighbouring catenoid but goes inward to meet its otherhalf. However a gap remains, and as one tries to close it(by morphing with the modulus, aa, of the underlying rectangular torus) the surface degenerates to look almost liketwo catenoids that move farther apart as one tries to close
the gap. We show this with the default morphing. The
deformation goes between rather extreme surfaces where
one has to adjust how far one computes into the end andthen also the size. While this animation is a bit jumpy, it
is instructive and therefore recommended.
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

The Conjugate Plateau Construction
Of Triply Periodic Minimal Surfaces *The Weierstrass representation for triply periodic minimal
surfaces is more difficult than what we have explained sofar. A simpler approach is the conjugate Plateau construction which, as we will see, can loosely be called the soap
film point of view. Without further explanation we use thefact from complex analysis that the derivative of the imaginary part of a function z 7→ f (z) can easily be computedfrom the derivative of the real part of f . This fact is known
as the ‘Cauchy-Riemann equations’. It implies that, on asimply connected domain, the real part of f determines theimaginary part up to a constant. Of course this holds forthe three components of the Weierstrass integral so thata simply connected minimal surface piece determines its
conjugate surface up to an R3 -constant.
The next step is the soap film part: The Plateau problem
arose from the soap film experiments of the 19th century
physicist Joseph A. F. Plateau. Its solution (1932) by Douglas and Rado states:
Every closed continuous injective curve in R3 is the boundary of at least one simply connected minimal surface.
This theorem expresses a fundamental property of the minimal surface equation: It is well suited for boundary value
problems. On the other hand, initial value problems usually have no solution. In particular, the minimal surface* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOCpiece of a Plateau solution usually cannot be extended beyond its boundary curve. So, how can it help to get complete solutions without boundary?
Here symmetries come to the rescue. If the boundary curve
contains a straight line segment then 180◦ rotation of a
Plateau solution around this segment extends the minimal
surface to twice as large a piece. If the boundary curve
is a polygon then extension by 180◦ rotation works for all
segments of the original polygon and all segments of therotated polygons. The resulting surfaces usually have singularities at the vertices of these polygons. However, if all
the angles of the polygon are of the form π/k, k ∈ N, thenrepeated rotation about the edges that start at one vertex
p results in a surface for which p is an interior, regularpoint. Such contours therefore lead to complete minimal
surfaces without boundary, but usually with lots of selfintersections. Only a few such minimal surfaces, which areembedded, are known. An example is the conjugate of the
Schwarz’ P-surface:

All angles of this hexagon are 90◦ , the z-axis divides it into
two 90◦ -pentagons.

The next step gives us a wealth of embedded examples.
We consider the conjugates of the just described Plateau
solutions with polygonal boundaries. A surprise happens
to the straight line segments of the boundary which aresymmetry lines on the extended surface with the symmetry being 180◦ -rotation:
The boundary of the conjugate piece consists also of symmetry arcs. These arcs are planar geodesics and the extended surface is reflection symmetric with respect to theseplanes.
The strategy of the conjugate Plateau construction is, tochoose the polygonal contour for the Plateau solution insuch a way that the extension of its conjugate piece byreflection in the planes of the boundary arcs give an embedded minimal surface.
This is helped by two geometric facts:(i) The Plateau solution very often is a graph over a convex
domain which is bounded by a suitable orthogonal projection of the boundary polygon. Therefore it is embedded.
Romain Krust proved that in this situation the conjugate
piece is also a graph, hence embedded.
(ii) The boundary polygon determines the angle by which
the normal of the Plateau solution rotates along each boundary segment. The Weierstrass representation shows thatthe normal of the conjugate piece rotates along each boundary arc by the same angle.
The Fujimori-Weber surfaces are very good examples forthe described construction. In the Action Menu one canselect Don’t Show Reflections to see the fundamental
piece. Also in the Action Menu one can switch between aminimal surface and its conjugate. In the View Menu one
can switch between WireFrame Display, Patch Display,
Point Cloud Display. In WireFrame and in Point Cloud
Display the Action Menu offers Emphasize Boundary, sothat one can easily see the polygonal contour for the Plateau
solution and its conjugate – which is the fundamental pieceof a Fujimori-Weber surface, bounded by planar symmetryarcs. The following pictures are an ff = 4 example:

The Plateau contour is a hexagon with four 90◦ and two
60◦ angles. The z-axis divides it into two pentagons. Observe that the conjugate piece has its normals parallel tothe normals of the Plateau piece – at corresponding pointsof course. The third picture is an assembly of twelve suchfundamental domains.
All of our Fujimori-Weber surfaces can be described in theabove way. Note that not only the angles of the hexagon
between adjacent edges are important, but also the following: Let a, b, c be three consecutive edges. The angle
between a plane normal to a and a plane normal to c has tobe either 0 or of the form π/k, k = 2, 3, 4, 6. Otherwise thegroup generated by the reflections in the symmetry planesof the conjugate of the Plateau solution is not discrete andthe extended surface cannot be embedded.
Several of the Fujimori-Weber surfaces agree with other examples in 3D-XplorMath: ff = 2 is the Schwarz P -surface,
ff = 5 is the Schwarz H-surface, ff = 8 is the A. Schoen
S-S-surface and ff = 7 is the same surface ’inside-out’,
that is, the assembled piece has the other side of the surface as its outside. If a minimal surface carries a straight
line, then the 180◦ symmetry rotation interchanges thetwo sides of the surface – therefore there is no geometric distinction between the two sides. Here this happens
if the pentagon-half of the Plateau hexagon has a reflection symmetry, as in the cases ff = 2, 5 above. The othercases come in such ’inside-out’ pairs, which look like different surfaces but are not, because the pentagon-halfs of thePlateau hexagon are the same polygon. The other pairs are
ff = 1, 3 (A. Schoen’s H-T –surface),
ff = 4, 6 (A. Schoen’s H-R–surface) and
ff = 9, 10 (A. Schoen’s T -R–surface).
Although the triply periodic surfaces in 3D-XplorMath canmost easily be understood by this conjugate Plateau construction, they are not computed in this way. It is anotherstory to explain the functions and their domains which areused in the Weierstrass representation to compute thesesurfaces.
In [Ka] a Weierstrass representation for several Plateau
problems as above is derived. Also fifteen 1-parameter
polygonal contours are discussed which give further examples because the obstructing period problem can be solvedby the intermediate value theorem.
In [FW] the Fujimori-Weber surfaces are obtained with anon-standard use of the Weierstrass representation. The
collection of examples described in this paper is much larger
than what is shown in 3D-XplorMath.
Bibliography[Ka] Karcher, H.: The triply Periodic Minimal Surfaces ofAlan Schoen And Their Constant Mean Curvature Companions. Manuscripta Math. 64, 291-357 (1989).
[FW] Fujimori S., Weber M.: Triply Periodic Minimal Surfaces Bounded By Vertical Symmetry Planes. Manuscripta
Math. 129, 29 - 53 (2009).
H.K.
TOC

About the Schwarz PD Family *This is a 2-parameter family of triply periodic genus 3
minimal surfaces. For all parameter choices are the original surface and the conjugate surface embedded. The
most symmetric example (with a cubic lattice) which is
obtained when cc = 0, dd = 0, was already constructedby H. A. Schwarz. When Alan Schoen found more triply
periodic surfaces around 1970 he named the two surfaces
which Schwarz found the P-surface (P for cubic primitive)
and the D-surface (D for diamond). He also found a thirdembedded(!) surface in the associate family of these, theGyroid (associate parameter 0.577 which is approx. 52 degrees). If dd=0 then a fundamental cell for the lattice isa prism with square base. In the morphing cc changes theheight of the prism.
K. Grosse-Brauckmann, M. Wohlgemuth: The Gyroid isembedded and has constant mean curvature companions.
To appear Calc. Var. 1996
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

About the Schwarz H family *This is a 1-parameter family of triply periodic surfaces.
The surfaces are made of pieces which one could call “triangular catenoids”; annular Plateau solutions bounded by
two parallel equilateral triangles. In the morphing aa changes
the height-to-edge length ratio of these triangular catenoids.
Observe that, as in the case of circles bounding catenoids,
the distance between the triangles has to be small enoughand then they bound a stable and an unstable triangular catenoid. In the PD-family with dd=0 one can observe
analogous “square catenoids”, except that our parametrization does not emphasize them.
When Alan Schoen found more triply periodic surfaces around 1970 he named these “Schwarz’ H surfaces”.
(Maybe Schwarz constructed only one member of the family.) Later, the Swedish chemist, Lidin, found another embedded example (now called the Lidinoid) in the same associate family, when aa is approximately 0.55, and the associate family parameter is 0.7139, (about 64.25 degrees).
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

TOC

Four Triply Periodic Minimal Surfaces *The four surfaces are:
· Neovius’ surface inside a cube (no deformations)· A. Schoen’s S-S-surface in a quadratic prism (deforms)· A. Schoen’s H-T-surface in a hexagonal prism (deforms)· A. Schoen’s T-W-surface in a hexagonal prism (deforms)
The images in 3D-XplorMath show a fundamental domain
for the group of translational symmetries. The curves
on the boundary are planar symmetry arcs, reflection in
their planes extends the fundamental domain to neighboring pieces. These planes, or rather their halfspaces that
contain the fundamental domain, intersect in a crystallographic cell.
Neovius was a student of H. A. Schwarz. A. Schoen worked
for NASA and he even made models of the surfaces he discovered. The T-W-surface is not in his list, but made in
his spirit. Initially his descriptions were not accepted bymathematicians. In [Ka] it was shown that their existence
follows easily with The Conjugate Plateau Constructionand Schoen was finally given credit for his discoveries.
What else should one observe in these pictures?
The images were not computed via the conjugate Plateau
construction, but by using the Weierstrass representation.
This representation requires the understanding of functions which are related to these minimal surfaces. To construct such functions is a mathematical challenge and tovisualize them is not easy either. In 3D-XplorMath complex functions are (mostly) visualized by showing a grid in* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

the domain and the image grid. We need functions thatare well defined on the whole minimal surface. Thus visualization seems to require that we put a grid on the wholesurface - which is impossible.
One such function is always the Gauss map which one cansort of ‘see’ by translating the normals of the surface to theorigin, thus mapping the surface to the Riemann sphere.
The four surface pictures are covered by patches whichlook like polar coordinates. This offers a different kind ofvisualization: The function on the surface which is responsible for these polar grids, maps the surface in such a way
to the Riemann sphere that our grid becomes the standard
polar grid on the sphere. One may compare this with theinterpretation of a hiking map with level lines.
This S-S-surface shows these levellines for the Gauss map: The two
big polar centers are a zero and apole (of order 3); the preimage ofa hemisphere (around 0 or ∞) in
most directions ends at a horizontal
symmetry line – i.e. at the preimage of the equator; but these horizontal arcs are connected by vertical (boundary) symmetryarcs, how is that possible? These vertical symmetry arcs
have inflection points in their middle, i.e. double points of
the Gauss map; the vertical symmetry arcs are therefore
preimages of slits in the upper (resp. lower) hemisphere.

In the other three cases the level lines belong to functions
which are obtained as quotient by a smmetry group. Although such a construction of a function is rather abstract,
the grid lines at least show how the surface is mapped tothe sphere.
Consider the T-W-surface first. Clearly
there is a 120◦ rotation symmetry andorthogonal to its axis are three 180◦ rotation axes. Identification with this order 6 symmetry group makes out of thetwelve polar grid patches just two, one
around the northpole the other aroundthe southpole of the quotient sphere.
In other words: the parameter lines on this surface are thelevel lines of the quotient function to the sphere.
The Neovius surface has its eight
polar centers on the space diagonals of its cubical crystallographic
cell. The group of 180◦ rotations
around the coordinate axes has order four. Identification by this symmetry group (modulo translations)
makes again a sphere out of the
Neovius surface and the 8 polar
patches are the preimages under the quotient map of standard polar coordinates on the quotient sphere. The preimage of the equator consists, for each polar patch, of thetwelve symmetry arcs which form the boundary of thesepolar patches. Our parameter lines, therefore, give an excellent impression of this quotient function.
Finally the H-T-surface. The fundamental piece itsself has a 60◦ symmetry rotation group. But one cansee that the neighboring polar centers along the top boundary should
be preimages of a northern and asouthern hemisphere, while the polar centers which lie vertically above
each other should be identified because there is a branch
point of the identification map between them. We therefore take the symmetry group as for the T-W-surface: the120◦ rotations around the vertical axis plus the three 180◦
rotations around the horizontal axes through the branch
points between the polar centers. Then again our parameter lines are the preimages of standard polar coordinates
on the quotient sphere.
It still takes more effort to understand the Weierstrass representation. But if these images help to see functions fromthese minimal surfaces to the Riemann sphere then they
have prepared the ground for the final step.
Bibliography[Ka] Karcher, H.: The triply Periodic Minimal Surfaces ofAlan Schoen And Their Constant Mean Curvature Companions. Manuscripta Math. 64, 291-357 (1989).
H.K.
TOC

The Fujimori-Weber Surfaces*
S. Fujimori and M. Weber derived in [FW] a Weierstrass
representation for a large collection of embedded triply periodic minimal surfaces. In 3D-XplorMath ten of theirfamilies are realized, choose ff = 1, . . . , 10. For each ff we
get a 1-parameter family, where the parameter cc controls
the length ratio between the horizontal and vertical closed
symmetry lines, see the default morph.
In the third part of About Minimal Surfaces (available
from the Documentation Menu) we explain a constructionof these surfaces with the help of minimal surface pieces
which are bounded by hexagons in R3 . To see these fundamental pieces, select in the Action Menu Don’t Show
Reflections and look also at the conjugate piece (whichis the one bounded by a hexagon). In Wire Frame and in
Point Cloud Display one can emphasize the boundary in
the Action Menu.
Some of these surfaces can also be seen with a different
parametrization in 3D-XplorMath. The parameter lines
on the Fujimori-Weber surfaces are level lines and lines ofsteepest descent for the height function in z-direction. For
the other triply periodic surfaces the level lines are pullbacks of polar coordinates under a complex differentiable
function defined on the minimal surface.
The Weber-Fujimori surfaces give forff = 2 the Schwarz P-surface,* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOCff = 5 the Schwarz H-surface,
ff = 8 the A Schoen S-S-surface,
ff = 7 the same surface ’inside-out’.
For minimal surfaces which carry straight lines there isno geometric distinction between the two sides, because
180◦ rotation around such a line maps the minimal surface onto itsself, but interchanging the two sides. If there
are no lines on the surface then the two sides may look so
different that they appear to be different surfaces. But inreality the fundamental pieces are only assembled in different ways – showing mainly one side of the surface in one
case and the other side in the other case (choose in the
View Menu: Distinguish Sides By Color). The othersuch pairs are
ff = 1, 3 (A. Schoen’s H-T-surface, also in 3DXM),
ff = 4, 6 (A. Schoen’s H-R-surface),
ff = 9, 10 (A. Schoen’s T-R-surface).
[FW] Fujimori S., Weber M.: Triply Periodic Minimal Surfaces Bounded By Vertical Symmetry Planes. Manuscripta
Math. 129, 29 - 53 (2009).
H.K.

TOC

User Defined Weierstrass Data *The close connection between minimal surfaces and complex variables was worked out in the second half of the
19th century and resulted in a flourishing of the theory ofminimal surfaces. One consequence of this new insight isthe so-called Weierstrass representation formula for minimal surfaces. Originally this representation was a local
one that only in exceptional cases allowed the representation of a complete surface. It was not until the workof Osserman (1962) that it became clear that the Weierstrass representation was in fact global. Unfortunately it
is considerably more difficult to explain the global interpretation of the Weierstrass representation than it is towrite down the local formula. Moreover, the input datafor global numerical computations are much more complicated than what is needed in order to draw just a local
piece of a surface. For these reasons, the dialog box in
3DXplorMath for user defined Weierstrass representaions
only allows for the making of a local patch, and it is only
the local formulation that we discuss below.
The input data for the local Weierstrass representation are
two complex differentiable functions f, g defined on a region U of the complex plane. A basic fact for the repre* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

TOC

sentation formula to work is that the integral of a complex differentiable function along a curve gives a constantresult if the curve is deformed keeping both end points
fixed. We can therefore define a differentiable map F fromthe simply-connected region U into R3 , by specifying athree-dimensional integrand in terms of f and g, and thenintegrating from a fixed base point ∗ ∈ U and taking thereal parts:
z 7→F (z) :=Z z 
Re(1 − g(z)2 ), i(1 + g(z)2 ), 2g(z) f (z)dz∗

The surface piece defined by this map is always a minimal surface piece, and this formula therefore allows a userof 3DXM to view as many minimal surface pieces as desired. A short computation shows that the function g has avery nice geometric interpretation. It is customary to associate a unit vector u in R3 to each complex numbers w, byconsidering the complex plane as the x-y-plane in R3 andtaking u to be the inverse image of w under stereographic
projection. (To see examples choose in the Conformal Category for any selected function in the Action Menu Show
Image On Riemann Sphere.) If one associates in this way aunit vector u to w = g(z), then one obtains for u a unit vector orthogonal to the surface at F (z). In other words: thefunction g composed with stereographic projection gives
the normal Gauss map of the surface. The normal Gauss
map is basic in the study of surfaces, and for example thevarious curvatures considered by differential geometers all
have simple expressions in terms of the normal Gauss map.
A geometric interpretation of the function f is much lessimmediate and this is perhaps one reason why it took solong for the above formula to be understood globally.
Sufficiently small pieces of any minimal surface are realistic models of soap films—provided the derivative of theparametrizing map F never vanishes. A point on a minimal
surface where the derivative of F vanishes is called a branch
point. Since soap films do not have branch points, one
wants to look at minimal surfaces without branch points,
and this can be decided from the Weierstrass data as follows:
If the function f has a zero at some point z0 then thederivative of F at z0 vanishes unless the other part of theintegrand becomes infinite. At such points, where g(z0 ) is
infinite, f needs to have a zero of twice the order as theinfinity (pole) of g at z0 . If the order of the zero of f islarger than this then one still has a branch point and if theorder of the zero is smaller, then F (z0 ) itself is infinite, so
we do not get a point on the surface. The default examples
allow one to contemplate these facts.
Note that we use polar conformal coordinates, that is,z = exp(u + iv), with umin ≤ u ≤ umax, 0 ≤ v ≤ 2π.
The unit circle is the image of u = 0.
The Catenoid data have a pole at z = 0, and indeed there
is no corresponding point on the catenoid. The situation

is similar at z = ∞ so that the catenoid is parametrized
by a sphere minus two points. – If f is changed to i · fthen one obtains the so called conjugate surface. In thiscase one obtains the helicoid, a singly periodic surface sinceintegration of the Weierstrass integrand once around z = 0
adds a period to the third coordinate function. To seelarger pieces of the helicoid, increase the range of v, i.e.,
integrate more than once around z = 0. – The ’Cyclic
Associated Family Morph’ turns the Catenoid inside out.
The Henneberg surface is a simple surface with branch
points. These are at the 4th roots of unity since f has
zeros there, but g is not infinite at those points. Notethat the branch points lie on the parameter line u = 0, on
the F-image of the unit circle. The Henneberg surface has
Enneper ends at z = 0 and at ∞. – The default Cartesian
grid touches only one branch point so that the image isless complicated than in the polar case.
The Enneper surfaces are parametrized by the full complex plane, and are therefore the simplest minimal surfaces to represent graphically. Note that a change of f
by a unitary factor, exp(iϕ) · f (z) gives in general an isometric but non-congruent surface of the associated family.
The Enneper surfaces are exceptional since all membersof the associated family are actually congruent. – For theclassical picture choose aa=2 and ’Cartesian Grid’. – For
the most general Enneper surface take f constant and g apolynomial.

The Trinoid, bb = 3, has poles of f at the 3rd roots ofunity. The Weierstrass map F is therefore not defined atthose points. It is not obvious to see what happens at thosepoints since the polar coordinates that the user definedsurfaces use are not adapted to the situation. Compare thecase ee = 3 of the Symmetric kNoids in the list of 3DXM
minimal surfaces. – Choosing the default ’Cartesian Grid’is most instructive for the Fournoid, our bb = 4 default.
As in all cases the ’Cyclic Associated Family Morph’ isinteresting.
Even the simplest surfaces can be given by Weierstrassdata from which one does not immediately recognize thesurface. Try:g(z) = (z k − 1)/(z k + 1),
f (z) = 0.8i(z k + 1)2 ,−1.8 ≤ u ≤ 0.2, 0 ≤ v ≤ 2π,(k = 0, 1, 2, 3, 4).
H.K.

TOC

About Ward Solitons *The Ward equation, also called the modified 2 + 1 chiralmodel is the following equation for a map
J : R2,1 → SU(n):(J −1 Jt )t − (J −1 Jx )x − (J −1 Jy )y − [J −1 Jt , J −1 Jy ] = 0.
This non-linear wave equation is obtained from a dimension reduction and a gauge fixing of the self-dual YangMills equation on R2,2 . The Ward equation is an integrable
system, and has a Lax pair. We explain this next.
Let11
u = (t + y), v = (t − y)22
be the light-cone coordinate system for the yt-plane, andgiven smooth maps A and B from R2,1 to su(n), we consider the following linear system for a map ψ : R2,1 × C →GL(n, C):(∗)(λ∂x − ∂u )ψ = Aψ,(λ∂v − ∂x )ψ = Bψ.

R. Ward showed that if ψ(x, u, v, λ) is a solution of (∗) andsatisfies the U(n)-reality condition
ψ(x, u, v, λ̄)∗ ψ(x, u, v, λ) = I,
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/ Surface Organisation

then J(x, u, v) = ψ(x, u, v, o)−1 is a solution of the Wardequation.
We call ψ an extended solution of J. The degree of ψis defined to be the number of poles for λ counted withmultiplicities. A solution J of the Ward equation is calleda Ward soliton if J has an extended solution ψ(x, u, v, λ)that is rational in λ, ψ(x, u, v, ∞) = I, and if for each t,
J is asymptotically constant as (x, y) approaches ∞. A
Ward soliton is called a k-soliton if the smallest degree ofextended solutions of J is k.
Ward used the solution of the Riemann-Hilbert problem
to write down all solitons whose extended solutions have
only simple poles. He also used a limiting method to constructed 2-solitons whose extended solution have one double pole, and showed that these 2-solitons have non-trivial
scattering. Anand and Ioannidou-Zakrzewski found Wardsolitons whose extended solutions have only one pole ofmultiplicities 2 and 3 (see the Anand-Ward Solitons submenu of the Surfaces Catergory). Dai and Terng usedBäcklund transformations and an order k limiting method
to construct all Ward solitons. Their method gives explicit formula for Ward solitons. In fact, given constant
z1 , . . . , zr ∈ C \ R, positive integers n1 , . . . , nr , and rational functions v0 , . . . , vk , they wrote down a k-soliton whose
extended solution has poles at λ = z1 , . . . , zr with multiplicities n1 , . . . , nr respectively. We call(z1 , . . . , zr , n1 , . . . , nr )the pole data of the Ward soliton. For detail of Dai andTerng’s construction, we refer the reader to their paper“Bäcklund transformations, Ward solitons, and unitons”.
This algorithm was used to write the code for the Wardsolitons in the 3D-XplorMath. The program shows thewave profile of the energy density
E(x, y, t) = ||J −1 Jx ||2 + ||J −1 Jy ||2 + ||J −1 Jt ||2 .
The default Morph shows a sequence of these profiles for
an increasing time ti for the SU (2) Ward solitons. A Wardsoliton with pole data (z1 , . . . , zr , n1 , . . . , nr ) represents theinteraction of r solitons with pole data (z1 , n1 ), . . ., (zr , nr )respectively. The shapes of these r solitons are preserved
after the interaction, but with possible phase shift. However, the solitons with pole data (z, k) do not share thiscommon phenomenon of soliton equations. In fact, thelocalized lumps scatter after interaction.
To use the program to see the wave profiles of the SU (2)Ward solitons, you should be in the Surface Catergory
and make a selection from the Ward Solitons submenu of
the Surfaces menu. The submenu MultiSolitonI means
Ward k-soliton whose
extended solution has a multiplic√
ity k pole at i= −1, 3-solitonI I Z means Ward 3-soliton
whose extended solution has poles i, i, Z, and ... etc.
After choosing the Ward Solitons submenu, then you canspecify the pole locations and rational maps from the submenu Ward Soliton Settings to construct the corresponding Ward solitons. For simplicity of writing the codes, weonly wrote code for k-solitons with 1 ≤ k ≤ 4 and therational maps from C to C2 are of the form (1, ai (w)),
where ai (w) is a polynomial in w of degree less than 5.
You may specify the coefficients of these ai ’s in the Ward
Soliton Settings. Since the maximum of the energy densityof these solitons is often large, you may choose a scaling
factor in the Settings menu. If you choose Energy Scale
Factor := 0.05, this means the wave profile you see fromthe program is for 0.05E(x, y, t). If you want to see some
Quicktime movies made from this program, you may go to
www.math.neu.edu/~terng/WardSolitonMovies.html.
CLT
Surface Organisation

About Anand-Ward Solitons*
Christopher Anand
Generally speaking, solitons are solutions to nonlinear wave
equations that exhibit particle-like behaviour, but the term
is frequently retricted further by demanding that the wave
equation be “integrable” in one of several technical sensesof that term. The earliest known soliton equations (Korteweg-DeVries, Sine-Gordon, Cubic Schroedinger, ...) were
all in one space dimension, and for a time it was even suspected that soliton-like behavior could not occur in higher
dimensions.
The solutions in this subcategory are all pure soliton solutions of a modified Chiral Model introduced by Richard
Ward. One ‘reason’ that this model is integrable is that
it is a reduction of the Yang-Mills equations, or the Bogomolny equations in an indefinite signature. It is thisrelationship which allows the solutions to be constructedusing twistor or inverse-scattering methods.
Twistor theory works by setting up a correspondence between analytic objects (e.g. solutions of differential equations) and certain geometric objects (e.g., complex analytic
functions with prescribed poles, holomorphic vector bun* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/ Surface Organisation

dles). In our case, the correspondence is between thosesolutions of Ward’s model that extend analytically to thecompactification of space-time (a highly restrictive ’boundary condition at infinity’) and holomorphic bundles on a
two-dimensional, compact, complex manifold. In analogy
with Liouville’s theorem, stating that a bounded, holomorphic function on the complex plane is constant, Serre’s
GAGA principle states that certain objects on compact,
complex spaces, known a priori only to be complex analytic, are necessarily algebraic. Think of this as an existence theory. The next step is to find an efficient way torepresent the associated vector bundles. It turns out that
there is a nice representation in terms of “monads” (which
are matrices satisfying certain relations), and one can evenwrite the solutions and energy density down explicitly—
without any integrals or derivatives, in terms of the monaddata. This is how the pre-programmed solutions here havebeen created, and it is how you can construct further solutions by entering monad data using the Setup AW solitons... dialogue.
To get solutions of Ward’s equations, the matrices mustsatisfy the relation
α1 · γ − γ · α1 + b · a = 0.
You are responsible for checking this equation yourself.
There is also an additional nondegeneracy condition to insure that the associated solution is not singular, and acondition that corresponds to the solution being static,

but the description of these conditions is too complicatedto include here.
Under the Set Parameters... menu, you can also change
the way the energy density is displayed. The parameters
bb and cc scale distance and energy respectively; aa is time;and dd is a cut-off value for the energy above which thevalue ee will be plotted (this is because some solutions canhave very tall ”spikes” at certain times).
The full story, including the derivation of all these conditions and the geometry behind the construction can befound in my preprints at:
http://www.maths.warwick.ac.uk/~anand/preprints.html
Please address any questions to me at:sendmail://anand@maths.warwick.ac.uk.
Surface Organisation

Hydrogen Probability Densities*for (n, l, m) = (1, 0, 0) to (n, l, m) = (3, 2, 2)
Point clouds, preferably in stereo, are used to visualize theelectron probability densities in a Hydrogen atom. Since
these densities are rotationally symmetric around the zaxis, only a wedge of the density cloud is shown.
The Action Menu offers several viewing options:
The entry Show Coordinate Slices plays a sequence of
coordinate slices, first in polar coordinates, then in Cartesian coordinates.
The entry
Show Constant Density Surface as Point Cloudshows that surface along which the density has its meanvalue. One can morph the constant density surfaces withthe parameter ff . In some cases these surfaces have several components inside each other. Therefore a stereo point
cloud gives a better impression than the raytrace version.
The entry Show Voxel Raytrace is the slowest option. It
is visually more impressive than the point cloud, but lessinformative.
The entry Create returns to the default volume cloud.
The electron densities are of course listed in Physics texts.
For comparison with our visualizations we repeat themhere:
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/ Surface Organisation

In 3DXM we use the parameter ii for the Bohr radius a0 .
(n,l,m)=(1,0,0):2r
density := exp(− a0 ) ,(n,l,m)=(2,0,0):2density := (2 − ar0 ) · exp( 2−ra0 ) ,(n,l,m)=(2,1,0):
2

−r
r
density := a0 · exp( 2 a0 ) · cos θ ,(n,l,m)=(2,1,1):2density := ar0 · exp( 2−r,a0 ) · sin θ(n,l,m)=(3,0,0):2density := (27 − 18 ar0 + 2( ar0 )2 ) · exp( 3−ra0 ) ,(n,l,m)=(3,1,0):2density := (6 − ar0 ) · ar0 · exp( 3−r,a0 ) · cos θ(n,l,m)=(3,1,1):2density := (6 − ar0 ) · ar0 · exp( 3−r,a0 ) · sin θ(n,l,m)=(3,2,0):

2
r 2
−r
density := ( a0 ) · exp( 3 a0 ) · (3 cos2 θ − 1) ,(n,l,m)=(3,2,1):2density := ( ar0 )2 · exp( 3−r,a0 ) · cos θ · sin θ(n,l,m)=(3,2,2):

2
2
r 2
−r
density := ( a0 ) · exp( 3 a0 ) · sin θ ,H.K.

Surface Organisation

Part I: Platonic Solids*
Relations among them, Simple Truncations
Go To Page 1
Part II below: Archimedean Solids
includes: Cubeoctahedron, Icosidodecahedron, BuckyballPart III below: DualPolyhedra
There are five (and only five) Platonic solids. Three ofthem are easy to imagine — Cube, Octahedron and Tetrahedron — while the remaining two are more difficult: Icosahedron and Dodecahedron. The earliest known models
date from the Stone Age.

What to do in 3D-XplorMath?
First, the program shows how the other four Platonic solids
are obtained from the Cube: Select first one of the otherpolyhedra, then in the Action Menu:
Show Relation with Cube.
For the Octahedron one sees that its six vertices are the
midpoints of the faces of a cube; the Octahedron faces are
equilateral triangles. The Tetrahedron sits in the cube sothat its four vertices are vertices of the Cube, and the six
Tetrahedron edges are face diagonals of the Cube. The
Icosahedron can be placed inside a Cube so that its twelve
vertices lie on the six faces of the Cube: see the default
morph in 3DXM, preferably when Show Relation with
Cube is chosen. The Dodecahedron can be placed around* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/ Go To Platonics

a Cube so that its twelve pentagon faces rest on the twelve
edges of the cube. See again the default morph going fromthe Rhombic Dodecahedron via the Platonic Dodecahedron to a cube with subdivided faces.
Second, by cutting off appropriately the vertices or theedges of a Platonic solid one obtains the simpler ones of theArchimedean solids. These are polyhedra whose faces
are (several kinds of) regular polygons, whose edges all
have the same length and whose vertices all look the same.
Select in the Action Menu any of the three Truncations
and then do the associated default Morph. For example,
the Edge Truncations morph from one Platonic solid toanother one (which is called its dual).
Third, since our intuition handles two dimensions better
than three, it is interesting to project the Platonic solids
from their midpoints onto a circumscribed sphere. View
in Wire Frame and select in the Action Menu:
Show Central Projection to Sphere.
These two-dimensional spherical views of the Platonic solids
come very close to explaining why there are no other such
beautiful polyhedra.
Fourth, the Icosahedron and the Dodecahedron have very
beautiful Stellations, polyhedra that fascinated Kepler. Select (Action Menu): Create Stellated. Note that all thementioned views have their own default morphs. Kepler
imagined the stellated Dodecahedron as having pentagonstars as faces. One can also select it in the Polyhedra Menu

as Kepler’s Great Dodecahedron. In 3DXM it is drawn
as a (negative) stellation of the Icosahedron.
Fifth, for the Cube and the Icosahedron there are two special entries in the Action Menu when viewing these solids
in Wire Frame. For the Cube select Show Intersection
With Plane, preferably in one of the stereo modes. The
plane is represented by random dots and the dots inside
the Cube are deleted; the Cube can be rotated and moved
forward and backward, always showing its polygonal intersection with the plane. For the Icosahedron select in
the Action Menu Add Borromean Link, preferably in oneof the stereo modes. Note how the boundaries of the emphasized rectangles are intertwined or linked. The edge
lengths of each rectangle are equal to the lengths of anedge and a diagonal of a regular pentagon, thus showing
the relation of the Icosahedron inside the Cube with theGolden Ratio. Also, the default morph of this image isworth viewing.
Finally, stone objects with Platonic Symmetry were found,mainly in Scotland. They were dated 2500 B.C. They arecarved from bigger stones, but they look as if they wereconceived as collections of balls. Therefore we have added
the Action Menu entry: Show As Stone Balls. In Patch
Display the balls are fine triangulations of the Bucky Ball,in Wire Frame the balls are shown with random dots. One
can also view other sphere triangulations after one has
selected (in Patch Display) Create Subdivided: anotherentry appears: Triangulate Further.
Part II: Archimedean Solids
Here is the definition again: All faces are regular polygons(of up to three different kinds). All edges have the samelength. All vertices (with their outgoing edges) are congruent. In addition to the five Platonic solids there are
twelve of them.
We have already seen the simplest ones: Truncate the vertices of a Platonic solid; there are two possibilities, if some
portion of the edges remain this is called Standard Truncation, and if the truncation cuts go through the midpoints of the edges we have a Midpoint Truncation. One
can also truncate the edges; this deformation leads tothe same Archimedean solid if one starts from the Octahedron or the Cube, and also if one starts from the Icosahedron or the Dodecahedron. Two more are obtained if
one truncates the edges and the vertices; the deformation
is easier to observe if one uses the standard truncation oneither the Cubeoctahedron or the Icosidodecahedron. In
fact, not quite the standard truncation, because that would
make rectangles instead of squares from the truncated vertices.
Finally there are the Snub Polyhedra. We could not findwhat ’snub’ means in this context. We describe the construction and call it ’to snub’. Each face of a Platonic
solid is scaled down from its midpoint and also rotatedaround the midpoint. The ’snubbed’ polyhedron is theconvex hull of these deformed faces. This 2-parameter deformation can be adjusted to give a 1-parameter familyof polyhedra whose faces are either regular polygons or
isosceles triangles. In each family is an Archimedean solid.
A snubbed Tetrahedron is an Icosahedron, snubbed Cubeand Octahedron give the same Archimedean solid and alsosnubbed Dodecahedron and Icosahedron agree.
One can probably understand all these truncations better
if one selects in the Action Menu
Snub Or Truncate Polyhedron In Polyhedron.
This will add the original polyhedron (as wire frame) tothe truncation.
Adapted Morphs: Each selection in the Action Menu
will cause that the default deformation, Morph in the Animation Menu, is adjusted to the Action Menu selection.
Part III: Dual Polyhedra
Duality has the simplest definition for Platonic solids: Theconvex hull of the centers of the faces is the dual polyhedron. The following definition works for Platonic andArchimedean solids: Take the tangent planes to the circumscribed sphere at the vertices of the polyhedron; consider them as the boundaries of halfspaces which containthe polyhedron; the intersection of these halfspaces is thedual polyhedron. For example the rhombic dodecahedronis dual to the cubeoctahedron. Similarly, consider convex
polyhedra whose inscribed sphere touches all faces; theconvex hull of the contact points of the faces with the inscribed sphere is the dual polyhedron.
These definitions of duality are unique up to scaling. For ageneral convex polyhedron definitions are no longer unique
up to scaling. Choose an interior point and project thepolyhedron radially onto a sphere around the chosen point;
one obtains a tessalation of the sphere with the same combinatorial properties as the boundary of the polyhedron;consider on the sphere the Dirichlet or Voronoi domains of
the vertices (the points which are closer to one vertex thanto all others); these domains form the dual tessalation; any
polyhedron whose radial projection to the sphere has thesame combinatorics as this dual tessalation, is called a dual
polyhedron to the given one.
In 3D-Xplormath one can view all Archimedean solids inside their dual polyhedra: First select one of the Platonic
solids from the Polyhedra Menu. Then check in the Action
Menu the entry:
Show All Polyhedra Inside Their Duals.
Now, each of the provided truncations gives an Archimedeansolid, shown inside its dual polyhedron. These images canbe morphed, including the surrounding dual polyhedra.
H.K.

Go To Platonics

Complex Functions Or Conformal Maps

Go To Page 1
Visualisation of Conformal Maps
Elementary Functions (Click on Names)
1.) Complex Square z 7→ z 22.) Exponential Map z 7→ exp(z)3.) Complex Inverse z 7→ 1/z
4.) Confocal Ellipses from z 7→ z + 1/z
5.) Rolling Curves from Polynomials z 7→ z ee + ee · z6.) Fractional Linear Maps z 7→ (a · z + b)/(c · z + d)
(z+cc)
7.) Hyperbolic Translations z 7→ (1+cc·z)
8.) Nonconformal Complexp
Map z 7→ conj(z) + aa · z 2
9.) Complex Square Root(z 2 ) = ±z10.) Complex Logarithm log(exp(z)) = z + 2πi · Z11.) Complex Sine sin(z) = (exp(iz) − exp(−iz))/2i
12.) Complex Tangent Hyperbolicus
13.) Stereographic Projection and its Importance
Elliptic Functions
14.) Jacobi Type Elliptic Functions
15.) Symmetries Of Elliptic Functions

Visualization of Conformal Maps *Here we deal with complex numbers, C, and with complexvalued functions from domains D ⊂ C to C. The functions
are assumed to be differentiable in the complex sense. Thismeans that differences f (z + h) − f (z), z, h ∈ C can bewell approximated bycomplex linear maps: lz (h) = c(z) · h, c(z) ∈ C.
For each z the constant c(z) is called derivative of f at z
and written as c(z) = f 0 (z).
2
For visualization,CandR
are identified by writing the 1stbasis vector 10 of R2 as 1 and the second basis vector 01
as i. The point xy ∈ R2 means the complex number x+i·y.
cNonzero complex numbers can be written as c = |c| · |c|,cwhere |c| ∈ R is called the absolute value of c and u := |c|is a unitary complex number. The complex linear map
l(z) := |c|·z is in R2 ordinary scaling by the factor |c|. The
complex linear map l(z) := u · z, viewed in R2 , is distancepreserving: |l(w)−l(z)| = |u·(w−z)| = |u|·|w−z| = |w−z|.
For u =
/ 1 we have only the fixed point 0:l(z) = z ⇔ (u − 1) · z = 0 ⇔ z = 0.
Multiplication in C by a unitary complex number u =/ 1 istherefore a rotation around the origin. Together with thescaling by |c| we have: Complex linear maps l(z) := c · zpreserve angles, they are ‘conformal’. Multiplication by i,* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Conformal TOC

considered in R2 , is counterclockwise 90◦ rotation. Evidently this is a square root of multiplication by −1. Thus
the geometric interpretation removes all the mystery froma square root of −1. Computation with complex symbols
started in the 16th century, but it was done reluctantly untill, around 1800, the geometric interpretation was foundand accepted rapidly. In the next 50 years the field of complex analysis exploded to great sophistication.
If we map two intersecting curves with a complex differentiable function f then the tangents of these curves at theintersection point w are mapped by the derivative approximation of f . If f 0 (w) =/ 0 then the image curves intersectwith the same angle as the preimage curves. We say: thefunction f is a conformal map.
Conformality is very important because real differentiability plus preservation of oriented angles implies complex differentiability. The words complex function and conformal
map are therefore used interchangebly. Our visualizations
try to emphasize conformality as clearly as possible.
A linear map (in real dimension two) is already conformal,
if two non-parallel right angles are preserved, for example
the angles of the edges of a square and the angle of its
diagonals. Because of this fact our visualizations decorate
the domain with a square grid and we show this grid andits image. To make the image grid as large as possible, it
fills the window of 3D-XplorMath. For a comparison withthe preimage one has to press the Option and Commandkeys and then press and release the (left) Mouse Button.

Showing a domain grid and its image grid visualizes complex functions as maps. This is not very common, but occurs already in Maxwell’s work. The image grid shows veryclearly the points where the derivative vanishes. The Action Menu entry Show Derivative At Mouse Point shows
the linear approximation of the function at the mouse point- in other words it shows, how the derivative scales and rotates the vicinity of the preimage of the mouse point.
The conformal image of a square grid has grid meshes
which are “squares” with slightly curved edges. The exponential map, for example, transforms a cartesian squaregrid into a “conformal polar square grid”. Its grid meshes
look more and more like squares the more grid lines one
displays by choosing uResolution, vResolution larger.
For some functions, e.g. z 7→ z + 1/z, it is therefore better to choose such a conformal polar grid in the domain
instead of a Cartesian square grid whose edges are lines
parallel to the coordinate axes. – The non-conformal map
z 7→ z̄ + aa · z 2 shows that non-conformality can easily berecognized in the image grid of a cartesian grid with small
square grid meshes: The grid meshes of the image are in
general parallelograms.
To the domain grid one can, in the Action Menu, add lines,
intervals or circles. Their images appear with the image
grid. For example, if a circle passes through a point where
the derivative has a simple zero, then the interior 180◦ angle of the circle at that point is doubled to a 360◦ interior
angle of the image curve. Note that the image grids for allconformal maps look very similar in the immediate vicinityof a simple zero of the derivative of the map.
Many functions show their symmetries better when theirimage grid is viewed (via Stereographic Projection) on the
Riemann Sphere instead of the Gaussian Plane. The second entry of the Action Menu toggles between these two
representations. The Riemann Sphere can also be viewed
in anaglyph or in cross-eyed rendering.
For most functions one can switch between a cartesian
grid in the domain and a polar grid. The different grids
bring out different properties of the function. An exception are the elliptic functions: the domain is always one
half fundamental parallelogram of their torus and the gridlines are parallel to the edges of the parallelogram; the image grid covers the Riemann sphere once.
The Conformal Maps Menu offers the possibility to precompose or postcompose the selected complex function
with a choice of six other functions, thus increasing thecollection of immediately available complex maps.
H.K.

Conformal TOC

Complex Square z → z 2 *We use the parameter dependent mapping z 7→ aa(z −bb)ee + cc, with the default values aa = 1, bb = 0, cc = 0,and ee = 2. The default Morph joins z 2 to the identity,
varying ee ∈ [1, 2].
Look at the discussion in “About this Category” or Visualisation of Conformal Maps for what to look at, what toexpect, and what to do.
Just as the appearance of the graph of a real-valued function x 7→ f (x) is dominated by the critical points of f ,it is an important fact that so also, for a conformal map,z 7→ f (z), the overall appearance of an image grid is verymuch dominated by those points z where the derivative f 0
vanishes. Most obviously, near points a with f 0 (a) = 0 thegrid meshes get very small and, as a consequence, the gridlines usually are strongly curved. If one looks more closely
then one notices that the angle between curves througha is not the same as the angle between the image curves
through f (a) (recall: f 0 (a) = 0). We will find this generaldescription applicable to many examples.
One should first look at the behaviour of the simple quadratic function z → z 2 near z = 0, both in Cartesianand in Polar coordinates. One sees that a rectangle, which* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Conformal TOC

touches z = 0 from one side is folded around 0 with strongly curved parameter lines, and one also sees in Polar coordinates that the angle between rays from 0 gets doubled.
The image grid in the Cartesian case consists of two families of orthogonally intersecting parabolas.
One should return to this prototype picture after one has
seen others like z → z + 1/z, z → z 2 + 2z and even the
Elliptic functions and looked at the behaviour near theircritical points.
The first examples to look at, (using
Cartesian and Polar√
Grids) are z → z 2 , z → 1/z, z → z, z → ez .
H.K.

Conformal TOC

The Complex Exponential Map z 7→ ez *Our example is z 7→ exp(aa(z − bb)) + cc, its parameters
are set to aa = 1, bb = 0, cc = 0; aa gets morphed in C.
See the functions z 7→ z 2 , z 7→ 1/z and their ATOs first.
The complex exponential function z 7→ ez is one of themost marvellous functions around. It shares with the realfunction x 7→ exp(x) the differential equation exp0 = exp
and the functional equation exp(z + w) = exp(z) · exp(w).
This latter identity implies that one can understand thecomplex Exponential in terms of real functions, for if we
put z = x + i · y then we haveexp(x + i · y) = exp(x) · exp(i · y) =exp(x) · cos(y) + i · exp(x) · sin(y).
This says that a Cartesian Grid is mapped “conformally”(i.e., preserving angles) to a Polar Grid: the parallels to thereal axis are mapped to radial lines, and segments of length
2π that are parallel to the imaginary axis are mapped tocircles around 0. This function is therefore used to make,in the Action Menu, the Conformal Polar Grid. Observe
how justified it is to describe the image grid as “made outof curved small squares”.
If you have seen z 7→ ez and z 7→ z + 1/z then now look at
z 7→ sin(z).
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Conformal TOC

The Complex Map z 7→ 1/z *The actual mapping for this example is z 7→ aa/(z−bb)+cc,
with the default values aa = 1, bb = 0, and cc = 0.
Look at function and ATO Complex Square z 7→ z 2 first.
The function z 7→ 1/z should be looked at both in Cartesian and Polar Grids. The default Morph varies bb ∈ [0, 1].
Notice first:
1) The real axis, imaginary axis and unit circle aremapped into themselves,2) the upper half plane and the lower half plane are interchanged, and3) the inside of the unit circle and its outside are alsointerchanged.
This is best seen in the (default) Conformal Polar Grid. In
the Cartesian Grid one should in particular observe thatall straight parameter lines (in the domain) are mapped tocircles (some exceptions, like the real axis, remain lines).
The behaviour of these circles near zero can be looked at
as an image of the behaviour of the standard Cartesian
Grid near infinity. In fact all circles are mapped to circles
or lines.
Examples to look at after this are* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Conformal TOC

z 7→ (az + b)/(cz + d) and z 7→ (z + cc)/(1 + ccz),¯
both of which can be obtained from z → 1/z by composition with translations z → z+a or scaled rotations z → a·z.
Therefore all of these so-called “Möbius transformations”
map circles and lines to circles and lines.
H.K.

Conformal TOC

Complex Map z 7→ z + aa/z *See the functions z → z 2 , z → 1/z, z → z 2 + 2z, z → ez
and their ATOs first. Use the default Morph, aa ∈ [0, 1].
This function is best applied to a Conformal Polar Grid.
The image of the outside of the unit circle is the same as
the image of the inside of the unit circle, namely the fullplane minus the segment [−2, 2]. The unit circle is mapped
to this intervall, each interior point w = 2 · cos(φ) ∈ [−2, 2]
appears twice as image point, namely of z = exp(±iφ).
The default choice shows how the outside of the unit disk
is mapped to the outside of the interval [−2, 2]. If we notethat f 0 (±1) = 0 then we understand this behaviour: theinterior 180◦ angle at these critical points ±1 of the outsidedomain is again doubled to become the angle of the image
domain (outside [−2, 2]) at ±2.
A domain circle zR (φ) = R exp(iφ) is mapped to the image
ellipse (R + 1/R) cos(φ) + i(R − 1/R) sin(φ), and a domain
radius zφ (R) = R exp(iφ) is mapped to the Hyperbola (R+1/R) cos(φ) + i(R − 1/R) sin(φ). The image grid therefore
consists of a family of ellipses that intersect orthogonally afamily of hyperbolae, and all these Conic Sections (see thePlane Curves Category) are “confocal”, i.e., they have thesame Focal Points, namely at +2 and −2.
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Conformal TOC

Complex Map z 7→ aa · z ee + ee · z *(Default: z → z 2 + 2z)
Look at the functions z → z 2 , z → 1/z and their ATOs
first. The default Morph varies aa ∈ [0, 1.2] for ee = 4.
Of course, since z 2 + 2z + 1 = (z + 1)2 , this function is
not very different from the first example z → z 2 . But
the change puts the critical point to −1 on the unit circle(f 0 (−1) = 0). Therefore, if one looks what this map does
to a Polar Grid, one can study the behaviour near thecritical point z = −1 with a different grid picture thanin the first example. Circles outside the unit circle aremapped to Limaçons (Plane Curves Category, Cardioidand Limacon) which wind around −1 twice. The unit circle
is mapped to a Cardioid and one can see the interior angleof 180◦ of the unit circle at −1 mapped to the interior
angle of 360◦ of the Cardioid at −1. Also one can seethat a neigbourhood of −1 is strongly contracted by thisfunction.
See the function z → z + 1/z next.
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Conformal TOC

Fractional Linear Maps *or Möbius Transformations
z 7→ (a · z + b)/(c · z + d)
See the functions z 7→ z 2 , z → 1/z and their ATOs first.
The default Morph uses a conformal polar grid and variesa ∈ [1, 2], c ∈ [0, 1].
These functions are called fractional linear maps or Möbiustransformations. They differ from the map z 7→ 1/z bycomposition with a translation z 7→ z + a or scaled rotations z 7→ a · z. As discussed for z 7→ 1/z they transform
lines and circles to lines and circles.
The default special case is z 7→ (z − 1)/(z + 1). It is best
understood in the (default) Conformal Polar Grid. Since
it maps 0 to −1 and ∞ to +1, one can see the Polar coordinate centers moved from 0, ∞ to −1, +1. This pictureis the first step towards understanding the complex (or“Gaussian”) plane plus the point at infinity as the “Riemann Sphere”. Use the 2nd entry in the Action Menu:
Show Image On Riemann Sphere.
See also the other Möbius transformations from the Conformal Maps menu.
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Conformal TOC

The Möbius Transformation *z→(z + cc)of the unit disk.
(1 + cc · z)

Look at the Möbius transformation z → (a·z+b)(c·z+d) and its
ATO first. The default Morph varies cc ∈ [−0.9, 0.9].
This function maps the interior of the unit disk bijectively
to itself, for every choice of cc with |cc| < 1. The behaviour
outside of the unit disk is obtained by reflection in the unit
circle, i.e., z → 1/z̄.
These maps have an interesting geometric interpretation:
they are isometries for the “hyperbolic metric” on the unit
disk. To understand this further, imagine that the unit
disk is a map of this two-dimensional hyperbolic world
and that the scale of this map is not a constant but equals
1/(1 − z z̄). This means that we do not obtain the lengthof a Rcurve t → z(t) as in the Euclidean plane by the integral |z 0 (t)|dt—we have to takeRthe scale into account anddefine its hyperbolic length by |z 0 (t)|/(1 − |z(t)|2 )dt. It
is this hyperbolic length of curves that is left invariant by
the “hyperbolic translations” z → (z + cc)/(1 + cc · z).
Locally the Pseudosphere (see: Surfaces of Revolution)
has the same hyperbolic geometry.
H.K.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Conformal TOC

Nonconformal Complex Map z 7→ conj(z) + aa · z 2 *Look first at other functions and their ATOs, for examplez → z 2 and exp. The default Morph varies aa ∈ [0, 1].
The map z 7→ conj(z) + aa · z 2 is a map from the complex
plane to itself. The harmless looking “conj” is responsible
for the fact that this map is not complex differentiable andtherefore not a “conformal” map, that is, a map for which
the angles between any two curves and their images are
the same. It is clearly visible in the image that the squaresof the domain grid are mapped to rectangles and even toparallelograms in the range.
The image also shows two“fold lines”. We observethat interior points of thedomain are mapped so thatthey lie on the boundary of
the image. For a complex
differentiable function thiscan never happen as isasserted by theOpen Mapping Theorem.
See the default morph.
H. K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Conformal TOC

The Complex Map z 7→

√

z *

2
See the functionsz→7z, z 7→ 1/z and their ATOs first.
√
The map z 7→ z should be looked at both in Cartesianand Polar Grids and in the default morph z ee , ee ∈ [ 21 , 1].

Note that since this function is the inverse of z → z 2 , we
expect to see related phenomena: circles around 0 go tocircles around 0, radial lines from 0 go to radial lines from0, but now with half the angle between them (since welook at the inverse map). A neigbourhood of 0 was verymuch contracted by z → z 2 , now we see the opposite, thedistance of points from zero is increased very much (beyond
any Lipschitz bound).
A more√ complicated aspect is the fact that the function
z 7→ z is not really a well defined map until we makesome choices. Namely observe: all z =/ 0 have two distinct
square roots, differing by a factor of −1.
√
The function z used by 3D-XplorMath maps the upper
half plane to the first quadrant, the (strict) lower halfplane to the fourth quadrant, and the negative real axis
to the positive imaginary axis—so there is no continuity
from above to below the negative real axis (which is therefore called a “branch cut”).
The Cartesian grid lines are mapped to two families of hyperbolae which intersect each other orthogonally.
H.K.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Conformal TOC

The Complex Logarithm *z 7→ log z
Look at the function z → ez and its ATO first. The default
Morph shows a · log(z) + (1 − a) · z for a ∈ [0, 1].
The complex Logarithm tries to be the inverse function of
the complex Exponential. However, exp is 2πi-periodic, sosuch an inverse can only exist as a multivalued function.
From the differential equation exp0 = exp follows that thederivative of the inverse is not multivalued and in fact very
simple:log0 (z) = 1/z.
Integration of the geometric seriesP
1/z = 1/(1 − (1 − z)) = k (1 − z)k
0
P
k+1
=
/(k + 1)k −(1 − z)gives the Taylor expansion around 1 of log. The so called“principal value” of the complex Logarithm is defined in
the whole plane, but slit along the negative real axis, forexample by integrating the derivative log0 (z) = 1/z in that
simply connected domain along any path starting at 1.
Different values of log z differ by integer multiples of 2πi,e.g. i = exp(πi/2) implies log i = πi/2 + 2πi · Z.
H.K.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Conformal TOC

The Complex Sine z → sin(z) *Look at the functions z → z 2 , z → 1/z, z → z 2 + 2z,
z → ez and their ATOs first. The default Morph varies thefamily fa (z) = a · sin(z) + (1 − a) · z for a ∈ [0, 1].
While the behaviour of the one-dimensional real functions
x 7→ exp(x) and x 7→ sin(x) are quite dissimilar (exp isconvex and positive, while sin is periodic and bounded),as complex functions they are very closely related:
exp(iz) − exp(−iz)
sin(z) =,2i
an identity that explains why the image grid under sin of
the default Cartesian grid looks exactly like the image gridunder z → z + 1/z applied to a Conformal Polar Grid outside the unit circle. For if we put w(z) := exp(iz)/i, thensin(z) = (w(z) + 1/w(z)))/2, and: recall that exp maps
the standard Cartesian Grid to the Conformal Polar Gridaround 0. The parameter curves in the image grid of sine
are therefore the same orthogonal and confocal ellipses andhyperbolas as in the image of z 7→ z + 1/z.
H.K.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Conformal TOC

Hyperbolic Tangent*
Definition:exp(z) − exp(−z)
exp(z) + exp(−z)
w−1=with w = exp(z)2
w+1
sin(z)tan(z) := −i · tanh(i z) =cos(z)In 3DXM:
tanh(aa · z/2) + bb · z/2, bb = 1 − aatanh(z) :=default:

aa = 1, bb = 0, morph: 0 ≤ aa ≤ 1.

The default visualization of this function is on the standard
Cartesian Grid. The exponential map transforms this grid
into a polar grid. The rational map w 7→ (w − 1)/(w + 1)
sends the polar centers at 0, ∞ to polar centers at −1, 1.
The default morph connects z/2 7→ tanh(z/2) to the identity.
Recall that all real functions that have power series representations can be extended to be functions over part of thecomplex plane. Of course this includes all functions thathave simple definitions in terms of the exponential map.
H.K.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Conformal TOC

Stereographic Projection and its Importance *Stereographic projection was invented for making maps ofthe earth and the celestial sphere. It is an angle preserving map of the sphere minus one point onto the Euclidean
plane.
Complex differentiability (of maps from the plane to theplane) is the same as real differentiability plus preservation of oriented angles (except were the derivative vanishes). Angle preserving differentiable maps are therefore
important for the theory of complex functions. The stereographic projection can be interpreted as mapping the 2sphere to the Euclidean plane, compactified by a point at
infinity. All the complex rational functions can be extended to infinity, i.e. they can be considered as differentiable, angle preserving maps from the 2-sphere to the2-sphere. The stereographic projection turns this “canbe considered as” into a simple explicit relation. The 2sphere, in this context, is called the Riemann Sphere andthe image plane is called the Gaussian Plane.
In 3D-XplorMath a 3D image of the stereographic projection can be reached via the last entry of the Action Menu
in the Conformal Map Category. Stereographic projection
is defined as the central projection from one point on thesphere onto the opposite tangent plane.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Conformal TOC

Formulas for the Stereographic Projection
Algebraically it is slightly more convenient to map thesphere not to the opposite tangent plane but to the parallel plane through the midpoint. More explicitly, the unit
sphere S2 ⊂ R3 is projected from (0, 0, −1) to the plane{z = 0} (and similar formulas work in all dimensions):
1
St(x, y, z) := 1+z· (x, y, 0), where x2 + y 2 + z 2 = 1.

because the three points involved liexon aline:

 

 
x
0
ξ
 (1 + z) 


 
 

 
y
z
1

y
+
0
=
·
·
=:
 
η.
1+z  1+z 
 (1 + z) z
−1
00
The inverse map is given
 by:

2ξ


2η
St−1 (ξ, η) := 1+ξ21+η2 ·  ∈ S2 ,1 − ξ2 − η2because, again,  the three points
 lie on
 a line:
ξ
0
2
2
 


−1+ξ +η
2
−1
·
η
·
0
+
=
St
(ξ, η) =




2
2
2
2
1+ξ +η
+1+ξ +η
0
−1


2ξ


1
2η
·

.
2
2
1+ξ +η
1 − ξ2 − η2Claim: Stereographic Projection
maps circles on S2 to circles and lines in the plane.
The case of lines is easier: The line in the image plane andthe projection center define a plane. This plane intersects
the sphere in the preimage circle of the line. – In otherwords: all circles on the sphere which pass through the
projection center are mapped to lines.
For every circle {(ξ, η); (ξ − m)2 + (η − n)2 = r2 } in theimage plane one can easily compute the plane of the preimage circle, while the other direction needs a case distinction
because of the lines.
We put ξ 2 + η 2 := 2mξ + 2nη + r2 − m2 − n2 into the third
component of the preimage formula. Then clearly(1 + ξ 2 + η 2 ) · (z + mx + ny + r2 − m2 − n2 − 1) = 0,
which shows the equation for the plane of the preimage.
Claim: Stereographic Projection preserves angles.
Through every point p ∈ S2 and tangential direction v
there exists a circle on the sphere which passes throughthe projection center and through p and is tangential tov. Every such circle is mapped to a line which is parallelto the tangent of that circle at the projection center. The
image lines of two such circles therefore intersect with thesame angle as the two preimage circles, and the two preimage circles were chosen to represent an arbitrary angle on
the sphere. – One can also prove this by computing withthe derivative of the stereographic projection.

The image in 3D-XplorMath shows a nice proof of why
circles are mapped to circles. A circle on the sphere andthe projection center define a quadratic cone (unless thecircle passes through the center and the image is a line).
In general, therefore, the image of the circle is an ellipse.
And all planes parallel to the image plane cut this cone insimilar ellipses.

To prove that they are circles we show that their axes areequal. We add the line which joins the projection centerand the midpoint of the image ellipse and we intersect
it, at m, with the diameter of the circle (on the sphere)that is in the symmetry plane of the figure. Now consider
the ellipse which intersects the cone in the parallel planethrough m. Its axis orthogonal to the symmetry plane andthe mentioned diameter of the circle are two intersecting
secants of the circle. For their two subsegments holds:

axisA2 = D1 · D2. The other, axisB, has its two halfs asedges of similar triangles which also have the edges D1, D2.
So we conclude D1 · D2 = axisB2 , hence axisA = axisB.
– Note that the three shaded triangles are similar: two ofthem are bounded by parallel segments and the bottom
triangle has the angle at the cone vertex equal to the angleof the top triangle opposite axisB.
A remarkle property of the formulas for stereographic projection and its inverse is: They map points with rational
coordinates to points with rational coordinates! For example, if we stereographically project a tangent line to theunit circle, then the rational points on the line give us all
rational points on the circle! Multiply (p/q)2 + (s/t)2 = 1
by the denominators to get all Pythagorean triples.
A differential geometric result in Dimension 3 and higher
says: Any angle preserving map between spaces of constantcurvature maps planes and spheres to planes and spheres.
For people with that background it is therefore obviousthat stereographic projection maps circles to circles.
In 3D-XplorMath one has stereo vision available so that
one can see visualizations of images on the Riemann spherein 3D. This shows the symmetries, for example, of the
doubly periodic functions much more clearly than theirimages in the Gaussian plane do.
H.K.

Conformal TOC

Elliptic Functions of Jacobi Type *What to do in 3D-XplorMath at the end of this text.

Elliptic functions are doubly periodic functions in the complex plane. A period of a function f is a number ω ∈ Csuch that f (z) = f (z + ω) for all z ∈ C. Doubly periodic means that the function has two periods ω1 , ω2 with
ω1 /ω2 ∈/ R. The set of all period translations is a lattice
Γ, and Γ has some parallelogram as fundamental domain.
Period translations identify parallel edges of this parallelogram to a torus and elliptic functions can therefore beviewed as functions on such a torus or equivalently as conformal maps from the torus to the Riemann sphere.
The simplest such functions are two-to-one maps from thetorus to the sphere. They have either one double pole
and are not very different from the Weierstrass ℘-function:f (z) = a·℘(z)+b, or they have two simple poles; the oldestof these are Jacobi’s functions sn, cn, dn, more below.
These doubly periodic functions have various properties
in common with the singly periodic trigonometric functions. They are inverse functions of certain integrals andtherefore are solutionsof first order ODEs, just like the inRz pverse function of 0 1/ 1 − ζ 2 dζ is a solution of the nonLipschitz ODE f 0 (z)2 = 1 − f (z)2 . And, as in this trigonometric case, differentiation of the first order ODE gives asecond order ODE which is Lipschitz, here f 00 (z) = −f (z).
The trigonometric functions have more symmetries than* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Conformal TOC

their translations. These symmetries give all values fromthe values on [0, π/2] × i · R. For the two-to-one elliptic
functions all values occur already on half a torus and thefurther symmetries compute all these values from the values on one eighth of the torus. If the torus happens to berectangular, it has further symmetries: reflections which
are anticonformal. In this case, from the values on one
sixteenth of the torus, one can obtain all other values via
Möbius transformations. And, one can define interesting
elliptic functions with the Riemann mapping theorem:
The rectangle in the first quadrant of the picture below is
one sixteenth of a rectangular torus. The Riemann mapping theorem is used to map this rectangle to the quarter
unit circle in the first quadrant of the second picture. Riemann’s theorem allows to specify that three corners of therectangle go to 1, 0, i and at the 4th corner the derivative
vanishes and two edges are mapped to the quarter circle:

Torus 41 -fundamental domain

Image of JD -function

The extension of this definition to a two-to-one map fromthe torus to the (Riemann) sphere is made possible by
Schwarz Reflection.
version, for complex power
P∞ Its simplest
series f (z) = k=0 ak z k with ak ∈ R, says f (z̄) = f (z).
We use the next step: Instead of reflection in R, reflection
in any straight or circular boundary arc (in domain andrange) extends the definition of the function.
While all rational functions can be obtained by rational operations from the single polynomial P (z) = z, one needs
two elliptic functions on a given torus to obtain all othersby rational operations from these. Two choices for such asecond function are obtained if one maps three other vertices of the rectangle in the first quadrant to 0, 1, i on thequarter circle in the first quadrant (but always origin to
origin). In each case the derivative of the map vanishes atthe last corner and the 90◦ angle between adjacent edges
is opened to the 180◦ angle between the image arcs:

Image of JE -function

Image of JF - or sn-function

The three functions JD , JE , JF were developed in work on

minimal surfaces. In this context they have two advantages over Jacobi’s sn,cn,dn:a) JD , JE , JF are defined on the same torus while sn,cn,dn
are defined on three different tori, which are closely related, namely doubly covered by a common rectangulartorus. On this larger rectangular torus Jacobi’s functions
are of degree 4.
b) At points z1 , z2 of the torus which are related by a symmetry of the four branch points of JD , JE or JF , the values J(z1 ), J(z2 ) are related by isometries of the Riemannsphere, while for sn, cn, dn the relations between the values
are by more general Möbius transformations. In applications to minimal surfaces such isometric relations translate
into symmetries of the minimal surface, while Möbius relations do not.
Example: On each rectangular torus we have Riemann’s
embedded minimal surface and its conjugate; the Gaussmap of these surfaces is the geometrically normalized Weierstrass ℘-function (denoted ℘g ), not the original ℘-function.
We have ℘g = JE · JF . If σ is a 180◦ rotation around a
midpoint between the double zero and double pole of ℘g ,then ℘g (σ(z)) = −1/℘g (z).
Jacobi’s sn-function and our JF -function (on rectangulartori) are very closely related:
2
The branch values of sn are {±1, ±k}, modul m =k.
√
The branch values of JF are {±F, ±F −1 }, F := k.
The fundamental domain of sn is such that sn0 (0) = 1.
2
The fundamental domain of JF is such that JF0 (0) = F +1/F.

The function dn is also defined√on a rectangular torus andhas real branch values {±1, ± 1 − k 2 }.
The function cn is not defined on a rectangular torus.
cn(0), dn(0) =/ 0, cn0 (0) = 0, dn0 (0) = 0.
On non-rectangular tori we cannot define Jacobi type elliptic functions by the Riemann mapping theorem.
In Symmetries of Elliptic Functions we construct themwith a more abstract tool: One can rotate any parallelogram torus by 180◦ around any of its points. This symmetry has four fixed points which are the vertices of aparallelogram with half the edgelength as the fundamental
domain of the torus.
The quotient by such a symmetry is a conformal sphere!
The uniformization theorem of complex analysis states thatevery conformal sphere is biholomorphic to the Riemannsphere. Therefore we can make the quotient map into afunction by specifying three points on the torus and calltheir images on the Riemann sphere 0, 1, ∞. Example:

Torus 14 -fundamental domain

Image of JD -function

180◦ rotation of the torus around the edge midpoints of thequarter domain and the corresponding 180◦ Möbius rotations extend the shown portion to a two-to-one conformal
map from the torus to the sphere.
Our computation of these images uses the ODE. We scaled
the function so that its branch values are {±B, ±B −1 }.
Our three Jacobi type functions JD , JE , JF satisfy the ODE:

0 2
0
2
4
2
−2
2
(J ) = J (0) · J − (B + B ) · J + 1 .
The two functions on both sides of the equality sign agree
because they have the same zeros and poles, hence areproportional, and J 0 (0)2 is the correct proportionality factor. Differentiation of this ODE and cancellation of 2J 0
give the more harmless 2nd order nonlinear ODE (which
is needed because Runge-Kutta cannot integrate the 1storder ODE in the vicinity of the zeros of the right side,
called the branch values of J):

00
0
2
2
2
−2
J = J (0) · 2J − (B + B ) · J.
These equations can be used for JD , JE , JF . To be on thesame torus one has to transform the branch values as:
E=

D−1
D−i
E−i
, F =i·, F =−
,
D+1
D+i
E+i

and to use the same scale for the fundamental domain:0JD(0) = 1, JE0 (0) =D − 1/D
D + 1/D, JF0 (0) =.
2i
2

In 3D-XplorMath the morphing parameter for Jacobi’s sn
is the modul m and for JD , JE , JF it is the branch value
D of JD in the 1st quadrant. Note that |D| = 1 for rectangular tori and Re(D) = Im (D) > 0 for rhombic tori.
Users cannot change the size of the domain of elliptic functions, it is always one half of the torus, chosen so that thevalues cover the Riemann Sphere once. In the domain we
use a grid made up of eight copies of one sixteenth of thetorus. The number of grid lines is the same in both directions so that the grid meshes are proportional to thefundamental domain of the torus. The default picture is,
as always for our conformal maps, the image grid and thegrid meshes show approximately the conformal type of thetorus.
If one selects in the Action Menu Show Image on Riemann
Sphere one can see the symmetries of these elliptic functions more clearly.
The entry Show Inverse Function in the Action Menu
offers a second visualization. It assumes that standard polar coordinates on the Riemann Sphere are well known.
The preimage of this polar grid is shown. Note that thepreimages of all latitudes – except the equator – are pairsof congruent smooth closed curves. The preimage of theequator consists of four squares. The preimages of thenorthern and southern hemissphere are therefore easily
recognized parts of the torus. – This visualization is completed only for rectangular tori.
H.K.
Conformal TOC

Symmetries Of Elliptic Functions *[The approach below to elliptic functions follows thatgiven in section 3 of ”The Genus One Helicoid and the
Minimal Surfaces that led to its Discovery”, by David Hoffman, Hermann Karcher, and Fusheng Wei,
published in Global Analysis and Modern Mathematics, Publish or Perish Press, 1993. For convenience, the full text of section 3 (without diagrams)
has been made an appendix to the chapter on the
Conformal Map Category in the documentation of
3D-XplorMath.]
An elliptic function is a doubly periodic meromorphic function, F (z), on the complex plane C. The subgroup L of C
consisting of the periods of F (the period lattice) is isomorphic to the direct sum of two copies of Z, so that the quotient, T = C/L, is a torus with a conformal structure, i.e.,a Riemann surface of genus one. Since F is well-definedon C/L, we may equally well consider it as a meromorphic
function on the Riemann surface T .
It is well-known that the conformal equivalence class ofsuch a complex torus can be described by a single complex
number. If we choose two generators for L then, withoutchanging the conformal class of C/L, we can rotate and* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Conformal TOC

scale the lattice so that one generator is the complex number 1, and the other, τ , then determines the conformal classof T. Moreover, τ1 and τ2 determine the same conformalclass if and only if they are conjugate under SL(2, Z).
The simplest elliptic functions are those defining a degree
two map of T to the Riemann sphere. We will be concerned with four such functions, that we call JD, JE, JF,
and WP. The first three are closely related to the classical
Jacobi elliptic functions, but have normalizations that are
better adapted to certain geometric purposes, and similarly WP is a version of the Weierstrass ℘-function, witha geometric normalization. Any of these four functions
can be considered as the projection of a branched covering over the Riemann sphere with total space T , and assuch it has four branch values, i.e., points of the Riemannsphere where the ramification index is two. For JD there
is a complex number D such that these four branch values
are {D, −D, 1/D, −1/D}. Similarly for JE and JF there
are complex numbers E and F so that the branch values
are {E, −E, 1/E, −1/E} and {F, −F, 1/F, −1/F } respectively, while for WP there is a complex number P such thatthe branch values are {P, −1/P, 0, ∞}. The cross-ratio, λ,of these branch values (in proper order) determines τ andlikewise is determined by τ .
The branch values E, F, and P of JE, JF, and WP can beeasily computed from the branch value D of JD (and hence

from dd) using the following formulas:
E = (D − 1)/(D + 1), F = −i(D − i)/(D + i),
P = i(D2 + 1)/(D2 − 1),and we will use D as our preferred parameter for describing
the conformal class of T . In 3D-XplorMath, D is related
to the parameter dd (of the Set Parameter... dialog) by
D = exp(dd), i.e., if dd = a + ib, then D = exp(a) exp(ib).
This is convenient, since if D lies on the unit circle (i.e., if
dd is imaginary) then the torus is rectilinear, while if D has
equal real and imaginary parts (i.e., if b = π/4) then thetorus is rhombic. (The square torus being both rectilinear
and rhombic, corresponds to dd = i · π/4).
To specify an elliptic function in 3D-XplorMath, choose
one of JD, JE, JF, or WP from the Conformal Map menu,
and specify dd in the Set Parameter... dialog. (Choosing
Elliptic Function from the Conformal map menu will givethe default choices of JD on square torus.)
When elliptic functions where first constructed by Jacobi
and by Weierstrass these authors assumed that the latticeof the torus was given. On the other hand, in Algebraic
Geometry, tori appeared as elliptic curves. In this representation the branch values of functions on the torus are
given with the equation, while an integration of a holomorphic form (unique up to a multiplicative constant) is
required to find the lattice. Therefore the relation between
the period quotient τ (or rather its SL(2, Z)-orbit) and thecross ratio lambda of the four branch values has been wellstudied. More recently, in Minimal Surface Theory, it wasalso more convenient to assume that the branch values ofa degree two elliptic function were given and that the periods had to be computed. Moreover, symmetries became
more important than in the earlier studies.
Note that the four branch points of a degree two elliptic function (also called ”two-division points”, or Zweiteilungspunkte) form a half-period lattice. There are three
involutions of the torus which permute these branch points;
each of these involutions has again four fixed-points andthese are all midpoints between the four branch points.
Since each of the involutions permutes the branch points, ittransforms the elliptic function by a Moebius transformation. In Minimal Surface Theory, period conditions could
be solved without computations if those Moebius transformations were not arbitrary, but rather were isometricrotations of the Riemann sphere—see in the Surface Category the minimal surfaces by Riemann and those named
Jd and Je. This suggested the following construction: As
degree two MAPS from a torus (T = C/L) to a sphere, wehave the natural quotient maps T /−id; these maps havefour branch points, since the 180 degree rotations havefour fixed points. To get well defined FUNCTIONS wehave to choose three points and send them to {0, 1, ∞}.
We choose these points from the midpoints between the
branch points, and the different choices lead to different
functions. The symmetries also determine the points thatare sent to {−1, +i, −i}. In this way we get the most symmetric elliptic functions, and they are denoted JD, JE, JF.
The program allows one to compare them with Jacobi’s
elliptic functions. The function W P = JE · JF has a double zero, a double pole and the values {+i, −i} on certain
midpoints (diagonal ones in the case of rectangular tori).
Up to an additive and a multiplicative constant it agrees
with the Weierstrass ℘-function, but in our normalization
it is the Gauss map of Riemann’s minimal surface on eachrectangular torus.
We compute the J-functions as follows. If one branchvalue is called +B, then the others are {−B, +1/B, −1/B}.
Therefore the function satisfies the differential equations(J 0 )2 = (J 0 (0))2 (J 4 + 1 − (B 2 + 1/B 2 )J 2 ) = F (J),J 00 = (J 0 (0))2 (2J 3 − (B 2 + 1/B 2 )J = F 0 (J)/2).
Numerically we solve this with a fourth order scheme√thathas the analytic continuation of the square root J 0 = J 0 2
built into it:
Let J(0), J 0 (0) be given. Compute J 00 (0) := F 0 (J(0))/2
and, for small z,00
Jm := J(0) + J 0 (0) · z/2 + J 00 (0) · z 2 /8, Jm:= F 0 (Jm )/2,00
J(z) := J(0) + J 0 (0) · z + (J 00 (0) + 2 · Jm
) · z 2 /6.

Finally let J 0 (z) be that square root of F (J(z)) that is00
closer to J 0 (0) + Jm
· z (analytic continuation!). Repeat.
H.K.

Conformal TOC

Fractal Curves And Dynamical Systems

Go To Page 1
Fractal Curves(Click the Names)

1.) Dragon Curve
2.) Koch Snowflake3.) Hilbert Square Filling Curve
4.) Sierpinski Curve5.) Hilbert Cube Filling Curve
Dynamical Systems
6.) Henon Attractor
7.) Feigenbaum Tree
8.) Userdefined Feigenbaum Iteration
9.) Userdefined x y Iteration
10.) Mandelbrot Set And Its Julia Sets

About The Dragon Curve *see also: Koch Snowflake, Hilbert SquareFillCurveTo speed up demos, press DELETE
The Dragon is constructed as a limit of polygonal approximations Dn . These are emphasized in the 3DXM default
demo and can be described as follows:
1) D1 is just a horizontal line segment.
2) Dn+1 is obtained from Dn as follows:a) Translate Dn , moving its end point to the origin.
pb) Multiply the translated copy by 1/2.
c) Rotate the result of b) by −45◦ degrees and call the
result Cn .
d) Rotate Cn by −90◦ degrees and join this rotated copy
to the end of Cn to get Dn+1 .
The fact that the limit points of a sequence of longer
and longer polygons can form a two-dimensional set is not
surprising. What makes the Dragon spectacular is that it
is a continuous curve whose image has positive area—
properties that it shares with Hilbert’s square filling curve.
There is a second construction of the Dragon that makes
it easier to view the limit as a continuous curve and is alsosimilar to the constructions of the following curves. Selectin the Action Menu: Show With Previous Iteration.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Fractal TOC

This demo shows a local construction of the Dragon: We
obtain the next iteration Dn+1 if we modify each segmentof Dn by replacing it by an isocele 90◦ triangle, alternatingly one to the left of the segment, and the next to theright of the next segment. This description has two advantages:(i) Every vertex of Dn is already a point on the limit curve.
Therefore one gets a dense set of points, c(j/2n ), on thelimit curve c.
(ii) One can modify the construction by decreasing theheight of the modifying triangles from aa = 0.5 to aa = 0.
The polygonal curves are, for aa < 0.5, polygons without
self-intersections. This makes it easier to imagine the limit
as a curve. In fact, the Default Morph shows a deformationfrom a segment through continuous curves to the Dragon—
more precisely, it shows the results of the (ee = 11)th iterations towards those continuous limit curves.
The Dragon is a fractal tile for the plane., see several versions at http://en.wikipedia.org/wiki/Dragon− curve.
For two beautiful possibilities select from the Action Menu
Tile Plane With Dragon Pairs, or: Tile Plane With
Dragon Quartetts.
Finally, one can choose in the Action Menu to map any
selected Fractal curve by either the complex map z → z 2
or by the complex exponential. The program waits for amouse click and then chooses the mouse point as origin.
R.S.P., H.K.

Fractal TOC

About the Koch Snowflake (or Island) *The Koch Snowflake Curve (aka the Koch Island) is a fractal planar curve of infinite length and dimension approximately 1.262. It is defined as the limit of a sequence ofpolygonal curves defined recursively as follows:
1) The first polygon is an equilateral triangle.
2) The (n+1)-st polygon is created from the n-th polygon
by applying the following rule to each edge: construct
an equilateral triangle with base the middle third of
the edge and pointing towards the outside of the polygon, then remove the base of this new triangle.
Note that at each step the number of segments increases
by a factor 4 with the new segments being one third thelength of the old ones. Since all end points of segments
are already points on the limit curve we see that no partof the limit curve has finite length.
Actually this is true for a 1-parameter family of similar
constructions: Vary the parameter aa (Set Parametersin the Settings Menu) in the interval [0.25, 0.5] and watch
how the iterations evolve or choose Morph in the Animation
Menu and observe the deformation of the limit curves.
Hausdorff Dimension: Consider the union of those disks
which have a segment of one polygonal approximation asa diameter, then this union covers all the further approximations. From one step to the next the diameter of the* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Fractal TOC

disks shrinks to one third while the number of disks is multiplied by 4—so that the area of these covering disk unions
converges to zero. The fractal Hausdorff d-measure is defined as the infimum (as the diameter goes to zero) of thequantity (diameter)d × (number-of-disks), and the fractal
Hausdorff dimension is the infimum of those d for which thed-measure is 0. This shows that the Hausdorff dimensionof the Koch curve is less than or equal to log(4)/ log(3),and since the union of the disks of every second segment
does not cover the limit curve one can conclude that theHausdorff dimension is precisely log(4)/ log(3).
The artist Escher has made rather complicated fundamental domains for tilings of the plane by modifying the boundary between neighboring tiles. This idea can be used to illustrate the flexibility of fractal constructions: Select fromthe Action Menu of the Koch Snowflake Choose Escher
Version and observe:
The new polygonal curves remain boundaries of tiles of theplane under the iteration steps that make them more andmore complicated.
Finally, one can choose in the Action Menu to map any
selected Fractal curve by either the complex map z → z 2
or by the complex exponential. The program waits for amouse click and then chooses the mouse point as origin.
Note that one gets the graph of a continuous function if
one plots the x-coordinate of a continuous curve against
the curve parameter. This can be viewed with the last
Action Menu entry. – H.K.
Fractal TOC

About Hilbert’s Square Filling Curve *See also: Koch Snowflake, Dragon Curve
Speed up demos by pressing DELETE
In 1890—the year the German Mathematical Society was
founded, David Hilbert published a construction of a continuous curve whose image completely fills a square. At
the time, this was a contribution to the understanding ofcontinuity, a notion that had become important for Analysis in the second half of the 19th century. Today, Hilbert’s
curve has become well-known for a very different reason—
every computer science student learns about it because thealgorithm has proved useful in image compression. In thisapplication one has to enumerate a first square, its four
half size subsquares, their sixteen quarter-size subsquares
and so on, in such a way that squares whose numbers areclose are also close to each other geometrically. In otherwords, the continuity of this space filling curve is now important, in contrast to the fact that the curve was considered a pathological example of continuity for many years
after Hilbert’s discovery.
It was known in 1890 that such a curve, i.e., a continuous
map c of [0, 1] onto [0, 1] × [0, 1], could not be one-to-one,i.e., Certain pairs of points t1 , t2 of the interval [0, 1] musthave the same image c(t1 ) = c(t2 ) ∈ [0, 1] × [0, 1]. This
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Fractal TOC

led Hilbert to give a special twist to his construction: Hegave a sequence of polygon approximations of the strange
limit curve that, surprisingly, were all one-to-one! In retrospect it seems almost as if Hilbert foresaw what would beneeded a century later in image compression; when people
say that they are using Hilbert’s square-filling curve, theymean more precisely that they are using Hilbert’s approximations to that curve!
The basis of Hilbert’s construction is a single step that isrepeated over and over again. We first explain a simplified version, although this does not exactly give Hilbert’s
one-to-one approximations that made the construction sofamous. Assume that we already have a curve inside thesquare and joining the left bottom corner to the right bottom corner. 3DXM offers four different initial such curves,
leading to quite different pictures. The basic construction
step is to scale the square and its curve by 12 and put four
copies of this smaller square side by side in the originalsquare, in such a way that these four smaller copies of
the curve fit together to form a new curve from the left
bottom corner to the right bottom corner of the originalsquare. But instead of reading more words, we suggestthat you view the default approximations of the Hilbert
curve in 3DXM. We use a rainbow coloration to emphasize
the continuous parametrization, and we repeat the colors
four times to emphasize that four copies of the previous
approximation make up the new one.
The two end points of the curve (to which this basic iter-

ation step is applied) play a special role, on the one hand
they lead at each iteration step to more points that are already points on the limit curve, on the other hand exactly
these easy points lead to double points on the approximations! Hilbert therefore removed small portions of thecurve near its two end points before he applied the above
iteration step. One can see how these Hilbert approximations manage to stay one-to-one and how they wander
through all the little squares of the current subdivision of
the original square—and these are just the properties usedin image compression.
In 3DXM one can choose with the parameter cc between
several initial curves. An even number and the following odd number choose the same curve, but for even cc theHilbert iteration is done without the endpoints and for odd
cc including the endpoints. In the Action Menu one canswitch between Hilbert’s approximation (cc=0) and onethat emphasizes the iteration of the endpoints (cc=5).
Finally we add to the above descriptive part some more
technical explanations, namely how to understand the limit
as a continuous curve. Select the Action Menu entry “Emphasize Limit Points”. The first shown step (for our default value cc = 5) is a curve that is mostly a straightsegment, but has also two little wiggles, that emphasize
the initial point c(0) and the end point c(1). The second
step is a curve with four straight segments that join five
wiggles, the points c(j/4), j := 0, . . . , 4. These points arereally points on the limit curve because they remain fixedunder all further applications of Hilbert’s basic construction step. In the third step we get 17 wiggles, the pointsc(j/16), j := 0, . . . , 16 of the limit curve, and so on. The
3DXM demo shows six such iterations. One can deducefrom this the continuity of the limit curve if one proves forthese approximations:|t2 − t1 | ≤

1
1
⇒
|c(t
)
−
c(t
)|
≤
.
2
1
n
n
4
2

Early iterations for bb = 0.4 (left), for bb = 0.5 (right, theHilbert case).
H.K.

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The Sierpinski Triangle, The Sierpinski Curve *The Sierpinski Triangle is a well known example of a “large”
compact set without interior points. It is defined by thefollowing construction:
Start with an equilateral triangle and subdivide it into
four congruent equilateral triangles. Remove the middle
one. Subdivide the remaining triangles again and removein each the middle one. Repeat this procedure. Each step
reduces the area by a factor 3/4. – But more is true:
Sierpinski’s Triangle is the image of a continuous curve.
As in the other fractal curves in 3DXM we have to define
an iteratively defined and uniformly convergent sequenceof polygonal curves. As in the case of the Hilbert squarefilling curve there is an easier construction by non-injective
curves which, however, can be modified to give better looking injective approximations. In the following illustration
we have chosen the 3DXM parameter bb = 0.49, because
for bb < 0.5 the easier construction also gives injective approximations. (bb = 0.5 gives Sierpinski’s curve.)* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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The starting polygonal curve has the vertices and the edge
midpoints of an equilateral triangle as its vertices. The initial point is the midpoint of the bottom edge. The curvethat joins every second vertex of the starting curve is thetriangle in the middle. We view the starting curve as passing through two edges of each of the three outer triangles.
We only have to describe for one of these triangles how thenext iteration is obtained. We will obtain curves that always run through two edges of each triangle, and the basic
iteration can always be applied. If we join every second
vertex of the resulting curves then we obtain the injective
approximations of the Sierpinski Curve.
The basic iteration step, for one triangle:
First add the two midpoints of the traversed edges of thetriangle. Two more points are added, one over the firstand one over the last of the four subsegments. The points
lie in the inside of the traversed triangle and they are thetips of isocele triangles whose base is the first, resp. thelast, of the four subsegments. In the case of the Sierpinski Curve these isocele triangles are in fact equilateral. If
the parameter bb is smaller than 0.5 then the height of
the isocele triangle is by the factor bb/0.5 smaller than theheight of the equilateral triangle – thus avoiding the creation of double points of the approximation.
The iterated polygonal curve joins the initial point of thefirst edge to the first tip, continues to the first edge-midpoint,
passes through the vertex of the original triangle to thesecond edge-midpoint, continues through the second tip
and ends at the final point of the last segment. The iterated polygonal curve traverses three triangles, two edgesin each. Therefore the iteration step can be repeated.
The default Morph from the Animation Menu of 3DXM
varies bb from 1/3 to 1/2 thus joining the first triangle
contour by a family of continuous (and injective) curves to
the Sierpinski Curve.
Finally, one can choose in the Action Menu to map any
selected Fractal curve by either the complex map z → z 2
or by the complex exponential. The program waits for amouse click and then chooses the mouse point as origin.
H.K.

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About Hilbert’s Cube Filling Curve *See also: The more famous Hilbert SquareFillCurve.
Hilbert’s cube filling curve is a continuous curve whose image fills a cube. It is a straight forward generalization of
the continuous square filling curve. It is shown in anaglyphstereo via a sequence of polygonal approximations. Eachapproximation is a polygon that joins two neighboring vertices of the cube.
The iteration step goes as follows:
The cube with the given (initial or a later) approximation is scaled with the factor 1/2. Eight of these smaller
copies are put together so that they again make up theoriginal cube, and this is done in such a way that the endpoint of the curve in the first cube and the initial point of
the curve in the second cube fit together, and so on withall eight cubes. The result of one iteration therefore is a
curve that is four times as long as the previous curve andthat runs more densely through the cube. In 3DXM, if
one rotates the cube with the mouse then the cube and its
first subdividing eight cubes are shown together with one
iteration of the initial curve.
To achieve a better feeling for the iteration step, one can* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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set the parameter cc to integer values between 0 and 5.
This will select different initial curves. An even value ofcc and the following odd value give the same initial curve,
but for even cc the Hilbert iteration is done without theendpoints, while for odd cc the endpoints are included in
the iteration. (Using the Action Menu, one can switch between Hilbert’s default (cc=0) and a case that emphasizes
the iteration of the endpoints, cc=5.)
We have the same situation as in the two-dimensional case:
The endpoints and their iterates are points that already lie
on the limit curve because they are not changed under further iterations. One can say that the endpoints and their
iterates are related to the limit curve in a very simple way.
On the other hand, the approximating polygons develop
double points at these iterates and the result is that theapproximations look much more confusing if the endpoints
and their iterates are included in the iteration. This is why
we offer the choice between iterating with and without theendpoints.
H.K.

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The Quadratic Henon Map and its Attractor *The Henon Map visualization gives the orbit under iteration of the map (x, y) → (y + 1 − aa x2 , bb x).
The default values are aa = 1.4 and bb = 0.3. The initialpoint is (x, y) = (cc, dd), defaults cc = 1.0, dd = 1.0. The
number of iterations plotted is ee, but the first ff iterates
are omitted. The defaults are ee = 3000 and ff = 20. The
map has two fixed points, they are obtained by solving aquadratic equation and they are marked by small circles.
Also from a quadratic equation one obtains an orbit ofperiod 2, marked by small spades. Numerically we foundno indication of an orbit of period 3. We found one orbitof period 4 numerically and marked it by small squares.
— The derivative of the Henon map has determinant −bb,i.e. the Henon map reduces area with a uniform rate. One
can view the eigen directions of the derivative via an Action Menu entry. — The inverse of the Henon map is theaa 2
1
Y, X − 1 + bb
quadratic map (X, Y ) → ( bb
2 Y ).
In 3DXM, to move the finished image, drag the image withthe mouse. To zoom in our out, drag vertically with theShift key pressed. (If you zoom in, you might want toincrease parameter ee using Settings > Set Parameters.)
To zoom into a particular region, hold down Command andthen drag a rectangle in the window, then the program will
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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zoom into that region of the Henon attractor, allowing you
to see it in greater detail.
Are there more periodic points on the Henon attractor?
An Action Menu entry allows to search for orbits of periodii. Since numerical inaccuracies, e.g. of the period 2 orbit,
build up after about 60 iterations to visibly leave the orbit,
we restricted to: ii < 80. Each numerically found orbit is
iterated until the user stops the iteration. If this stop is
by mouse button, the orbit is saved and the last orbit ofthese is added to the visually indicated periodic points.
The Henon map depends strongly on the parameters. For
example, we can obtain an attracting orbit of period 6 withaa = 1.45. To see it, change aa, stop the iteration and clickin the Action Menu: Do 500 Iterations.
(Morphing aa and bb works, but there is no default morph,so first select Set Morphing... from the Settings menu to
set up the morph—be sure to click the Init To Current
Values button, then change aa0 aa1, bb0 and bb1.)
Finally, we added to the Action Menu: Use Hit Count
Coloration. This changes the representation by adding
vertical bars over the points of the Henon attractor. The
length of these bars shows how often that - pixelsized point of the attractor is visited during the iteration. The
graphic therefore illustrates the iteration-invariant measure on the attractor. For a 1-dimensional such hit countsee the Feigenbaum Tree.
H.K.

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About The Feigenbaum Tree *See also: Julia Set of z → (z 2 − c)
The Feigenbaum Tree is one of the earliest examples ofparameter dependent behavior of a dynamical system. The
dynamical system in question is called the Logistic Map:fµ (y) := 4µ · y(1 − y), y ∈ [0, 1], µ ∈ [1/4, 1].
Since both the parameter space, [1/4, 1], and the dynamical space, [0, 1], are 1-dimensional, one can illustrate in a(µ, y)-plane how the dynamical behavior changes as the parameter µ varies. The usual experiment (and the one usedin 3DXM) goes as follows: Starting with a set of initialvalues {yk ; yk ∈ [0, 1], k = 1, . . . , K} (and with as many
parameter values µ as one wants to handle) one computes
many iterations fµ◦n (yk ), n = 1, . . . , N with N large.
If one plots only the iterations with say n ≥ 1000, then one
observes in the (µ, y)-plane the Feigenbaum Tree: for smallµ the iterated points fµ◦n (yk ) converge to a stable fixed
point of the map fµ , yf = fµ (yf ), yf := 1−1/4µ. Observethat the derivative f 0 at the fixed point is 2 − 4µ ≤ 0. Atµ = 3/4 the derivative at the fixed point is −1, so thatthe fixed point stops being attractive. It turns out that
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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for larger µ the orbit of period 2 is attractive for a while– until µ reaches another bifurcation point after which anorbit of period 4 becomes attractive.
This period doubling “cascade” continues up to a certainµ-value, past which there is for a while no longer an attractive orbit. All this is clearly visible in the 3DXM demo.
One should use the Action Menu entry: Iterate Mouse
Point Forward to watch how arbitrary initial points areiterated and how these iterations converge to the attracting orbits of period 2d in the left, period doubling, part of
the Feigenbaum Tree. —Speed-Up Note: If one presses
DELETE either during the default iterations or during theiteration of a point chosen by mouse, then all delays are
skipped and the result of the iteration is reached quickly.
After the period doubling in the left part has been observed one wants to look at the right part of the Feigenbaum Tree more closely. The µ-interval which the illustration uses is the interval [bb, cc]. It can be changed in
the Parameter entry of the Settings Menu. Since the attractive orbit of period 2 appears after µ = 0.75, one loses
only the simple attractors if one increases bb from 0.25 to
0.75, and one gains that the remaining part of the Treeis stretched by a factor of 3. In the same way one canmagnify any part of the parameter space. Of course thedynamical space is always fully shown—unless one decides
to use SHIFT+MOUSE to scale the image to see part ofthe dynamical space magnified. In this case translation
using CONTROL+MOUSE-DRAG may be useful.

The most obvious feature in the right part of the Feigenbaum Tree are gaps, three fairly large ones and any numberof thinner ones. The three large ones belong to parameterintervals where the map fµ has attractive orbits of period
6, period 5, resp. period 3. If one magnifies a gap enough,one can experimentally check that the gaps belong to attractive orbits (use in the Action Menu Iterate Mouse
Point Forward). One also observes that at the right endof these intervals the periods double again, and again. In
other words, the Feigenbaum Tree illuminates, almost at
the first glimpse, many properties of this 1-parameter family of iterated maps.
The Action Menu has been expanded by four entries Iteration Invariant Density (either with mouse choice of
aa = µ or previous aa) and Density Function (again withmouse choice of aa or previous value). Before one chooses
any of these one should look at Iterate Mouse Point
Forward, where one sees how the iterated point, given by
the vertical coordinate y, jumps around with fixed µ. The
Iteration Invariant Density expands this: 1000 different y-values are chosen and represented in the left-mostcolumn on the screen. These points are iterated and shownin the second column, iterated again and shown in the thirdcolumn, and so on, 400 times. Except for the first few
columns one clearly sees a density pattern develop: all thevertical columns look essentially alike. This can be studied
further with the entry Density Function: Here we count
how often each pixel-sized interval of the dynamical (=ver-tical) interval is visited during the iterations and we plot
the counting result (normalized to fit on the screen). We
observe a function that describes the probability densitywith which each pixel interval is visited. – These demos
explain why the curves that represent attractors do extend
into the chaotic regions.
Finally we remark that the Feigenbaum Tree is related to
the real part of the Mandelbrot set because the Mandelbrot
set also parametrizes quadratic maps z → fc (z) := (z 2 − c)according to their dynamical properties. If c is chosenfrom the big bottom apple then fc has an attractive fixed
point. As one passes on the real axis from the apple tothe disk above it, the fixed point changes from attractive
through indifferent to unstable and the orbit of period 2
becomes attractive. As one moves (always along the realaxis) towards the top of the Mandelbrot set one continues
to meet exactly the same kind of dynamical behavior as
one sees in the Feigenbaum Tree. For more details see thedocumentation for Julia Set of z → (z 2 − c).
H.K.

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User Defined Feigenbaum Iteration *Please see first: About The Feigenbaum Tree
The question “How can one find periodic attractors of, say,a family of Newton Iterations?” led to the developmentof this exhibit. The user can input an iteration functionthat depends on one parameter aa (which is, as in the
Feigenbaum case, represented horizontally). The dynamical space consists of some interval of arguments y of
the function. We can view them as starting values of theiteration. The default iteration in 3DXM is the Newton iteration for the zeros of the polynomial y → (y 2 −3)2 +3aa.
The Feigenbaum picture is ideally suited to watch how
the aa-family of iterations behaves: One quickly spots attractive fixed points or attractive orbits with small period;
but one also observes the density curves in a seemingly
chaotic region. If one expands the scale, i.e. stretches avery small aa-interval over the whole screen, then one sees
easily whether there are in this interval periodic attractors,
or whether still only chaos is visible (then choose a different
aa-interval or expand the current interval further).
The remaining details are the same as for the classicalFeigenbaum Tree and are explained in detail in the documentation for that exhibit.
H.K.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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The Area Preserving Henon Twist Map *User Defined Example
The User Defined entry is designed to study the behaviourof 2-dimensional maps under forward iteration near an isolated, neutral fixed point. (We want a fixed point inside
the window since otherwise most of the iterated points will
move out of sight.) Our example is Henon’s quadratic, area
preserving twist map F :cos(aa) · x − sin(aa) · (y − ebb · x|x|)F (x, y) :=.
sin(aa) · x + cos(aa) · (y − ebb · x|x|)
Henon used x2 instead of x|x| for the perturbation term.
See below.
The main parameter aa controls the derivative of F at the
fixed point (0, 0); dF |(0,0) is the rotation matrix with angle
aa. The behaviour of the iterations changes strongly withaa. Try also −aa. F is area preserving since the Jacobian
determinant det(dF ) = 1 everywhere.
By default ebb = 1. This parameter serves to choose thesize of the neighborhood of the fixed point, because of thescaling property
F (~x; ebb ) = e−bb · F (ebb · ~x; 1).
We use exp(bb) instead of bb, because the scaling parameteris a multiplicative rather than an additive parameter.
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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The iteration is applied to the segment [0, 1] · (cc, dd). The
number of points on this segment is tResolution. The default number of iterations is ee = 2000. The next 2000
iterations are obtained from the Action Menu Entry: Continue Curve Iteration.
Since the graphic rendering is much slower than the thecomputation of iterations one can increase the parameterhh from its default value hh = 1 and then only one outof hh iterations is shown on the screen. This is useful if
one needs to see the result of a large number of iterations.
(For example hh = 4 · n in the case aa = π/2.)
The Action Menu Entry Iterate Mouse Point Forward
allows to iterate a single point. During the selection the
point coordinates appear on the screen. If DELETE ispressed during the iteration then the waiting time at eachstep is cancelled so that the point races through its orbit.
The Action Menu Entry Choose Iteration Segment By
Mouse allows to Mouse-select initial and final point of asegment on which ff points will be distributed and iterated(by default ff = 16). The parameter hh speeds up theiteration as above. After the first ee iterations an Action
Menu Entry is activated and allows to iterate further.
As ususal one can translate the image by dragging or one
can scale it by depressing SHIFT and dragging vertically.

One can also morph the images. They change rather drasti-

cally with aa. As default morph bb is decreased so that theneighborhood of the fixed point gets expanded. One observes that most of the iterated points travel on invariant
curves around the fixed point. Occasional periodic points
clearly show up in the image. If aa is an irrational multipleof π then the visible periods do increase as the neighborhood of the fixed point expands with decreasing bb. (Forthe default morph the number ee of iterations is restricted
to 500 to reduce waiting times.)
The Henon twist map can be written as a rotation plus a
quadratic perturbation:  
cos(aa) − sin(aa)xF (x, y) :=·+ perturb,sin(aa) + cos(aa)y+ sin(aa)perturb := ebb · x|x| ·.
− cos(aa)The scalar product between the perturbation and the tangent to the rotation circles is the
Forward Perturbation =−ebb |x|3 · (sin2 (aa) + cos2 (aa)).
This explains why we changed the Henon map. Our negative forward perturbation means that the images underF stay behind the rotation image, and more so the larger|x|. This is the usual behavior of a monotone twist map.
Henon’s perturbation has the factor x3 instead of |x|3 , sothat the twist in the left half plane partially cancels thetwist in the right half plane. In our definition do the elliptical islands around periodic points appear more easily,
while with Henon’s definition the behaviour near the fixed
point, in the case when aa is a rational multiple of π (e.g.
aa = π/2), is much more complicated.
We recommend that users try out also Henon’s definitionand definitions of their own.

H.K.

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The Mandelbrot Set And Its Julia Sets *If one wants to study iterations of functions or mappings,
f ◦n = f ◦ . . . ◦ f , as n becomes arbitrarily large then Julia
sets are an important tool. They show up as the boundaries of those sets of points p whose iteration sequencesf ◦n (p) converge to a selected fixed point pf = f (pf ). Oneof the best studied cases is the study of iterations in thecomplex plane given by the family of quadratic maps
z → fc (z) := z 2 − c.
The Mandelbrot set will be defined as a set of parametervalues c. It provides us with some classification of thedifferent ‘dynamical’ behaviour of the functions fc in thefollowing sense: If one chooses a c-value from some specificpart of the Mandelbrot set then one can predict rather well
how the iteration sequences zn+1 := fc (zn ) behave.
1) Infinity is always an attractor.
Or, more precisely, for each parameter value c we can define a Radius
Rc ≥ 1 such that for |z| > Rc the iteration sequencesf ◦n (z) converge to infinity. Proof: The triangle inequality
shows that |fc (z)| ≥ |z|2 − |c| and then |fc (z)| > |z| is certainly true if |z 2 | − |c| > |z|. Therefore it is sufficient to* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

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pdefine Rc := 1/2 + 1/4 + |c|, which is the positive solution of R2 − R − |c| = 0.
This implies: if we start the iteration with z1 > Rc thenthe absolute values |zn | increase monotonically—and indeed faster and faster to infinity. Moreover, any starting
value z1 whose iteration sequence converges to infinity will
end up after finitely many iterations in this neighborhoodof infinity, U∞ := {z ∈ C | |z| > Rc }. The set of all points
whose iteration sequence converges to infinity is therefore
an open set, called the attractor basin A∞ (c) of infinity.
2) Definition of the Julia set Jc . On the other hand,the attractor basin of infinity
p is never all of C, since fc
has fixed points zf = 1/2 ± 1/4 + c (and also points ofperiod n, that satisfy a polynomial equation of degree 2n ,namely f ◦n (z) = z).
Definition. The nonempty, compact boundary of the attractor basin of infinity is called the Julia set of fc ,Jc := ∂A∞ (c).
Example. If c = 0 then the exterior of the unit circle is theattractor basin of infinity, its boundary, the unit circle, isthe Julia set J0 . The open unit disk is the attractor basinof the fixed point 0 of fc . The other fixed point 1 lies on
the Julia set; 1 is an expanding fixed point since fc0 (1) = 2;
its iterated preimages −1, ±i, . . . all lie on the Julia set.
Qualitatively this picture persists for parameter values cnear 0 because the smaller fixed point remains attractive.

However, the Julia set immediately stops being a smooth
curve—it becomes a continuous curve that oscillates sowildly that no segment of it has finite length. Its image is one of those sets called a fractal for which a fractional dimension between 1 and 2 can be defined. Our
rainbow coloration is intended to show Jc as a continuously parametrized curve. We next take a more carefull
look at attractive fixed points.
3) c-values for which one fixed point of fc is attractive.
There is a simple criterion for this: if the derivative at
the fixed point satisfies |fc0 (zf )| = |2zf | < 1 then zf isa linearly attractive fixed point; if |2zf | > 1 then zf isan expanding fixed point; if the derivative has absolutevalue 1 then no general statement is true (but interesting
phenomena occur for special values of the derivative).
Since the sum of the two fixed points is 1, the derivative
fc0 can have absolute value < 1 at most at one of them.
Let wc be that square root of 1 + 4c having a positive
real part. Then |1 − wc | is the smaller of the absolutevalues (of the derivatives of fc at the fixed points). The
set of parameter values c with a (linearly) attractive fixed
point of fc is therefore the set {c | |1 − wc | < 1}, or{c = (w2 − 1)/4 | |1 − w| < 1}. In other words, thenumbers 1 + 4c are the squares of numbers w that lie ina disk of radius one with 0 on its boundary. The apple
shaped boundary is therefore the square of a circle through
0. It is called a cardioid.

4) The definition of the Mandelbrot set in the parameter plane.
The behavior of the iteration sequence zn+1 := fc (zn ) in
the z-plane depends strongly on the value of the parameterc. It turns out that for those c satisfying |c| > Rc , theset of points z whose iteration sequences do not converge
to infinity has area = 0. Such points are too rare to befound by trial and error, but one can still compute manyas iterated preimages of an unstable fixed point. It follows
from |c| > Rc that only the points of the Julia set Jc do notconverge to infinity. Moreover, the Julia set is no longer acurve, but is a totally disconnected set: no two points of
the Julia set can be joined by a curve inside the Julia set.
(In this case our coloration of Jc has no significance.)
The Mandelbrot set is defined by the opposite behaviourof the Julia sets:
Mandelbrot Set : M := {c | Jc is a connected set}
There is an 80 year old theorem by Julia or Fatou thatsays:
M = {c ; fc◦n (0)stays bounded}= {c ; |fc◦n (0)| < Rc for all n}.
This provides us with an algorithm for determining thecomplement of M; namely c 6∈ M if and only if the iteration sequence {fc◦n (0)} reaches an absolute value > Rc forsome positive integer n. (But, the closer c is to M, thelarger this termination number n becomes).

On the other hand, if fc has an attractive fixed point, thenit is also known that {fc◦n (0)} converges towards that fixed
point. The interior of the cardioid described above is therefore part of the Mandelbrot set, and in fact it is a largepart of it.
As experiments we suggest to choose c-values from theapple-shaped belly of the Mandelbrot set and observe how
the Julia sets deform as c varies from 0 to the cardioid
boundary. For an actual animation, choose the deformation interval with the mouse (Action Menu) and then select‘Morph’ in the Animation Menu. To see how the derivative at the fixed point controls the iteration near the fixed
point, choose ‘Iterate Forward’ (Action Menu) and watch
how chosen points converge to the fixed point. This is very
different for c from different parts of the Mandelbrot belly.
5) Attractive periodic orbits. As introduction let us
determine the orbits of period 2, i.e., the fixed points offc ◦ fc that are not also fixed points of fc . Observe that:fc ◦ fc (z) − z = z 4 − 2cz 2 − z + c2 − c= (z 2 − z − c)(z 2 + z − c + 1).
The roots of the first quadratic factor are the fixed pointsof fc , the roots of the other quadratic factor are a pair ofpoints that are not fixed points of fc , but are fixed pointsof fc ◦ fc , which means, they are an orbit of period 2,
clearly the only one. Such an orbit is (linearly) attractive
if the product of the derivatives at the points of the orbit
has absolute value < 1. The constant coefficient in thequadratic equation is the product of its roots, i.e. theproduct of the points of period 2 is 1 − c. Therefore:
The set of c-values for which the orbit of period 2 is attractive is the disk {c ; |1 − c| < 1/4}.
Again, this disk is part of the Mandelbrot set since {fc◦n (0)}has the two points of period 2 as its only limit points.
The interior of the Mandelbrot set has only two components that are explicitly computable. These are the cvalues giving attractive fixed points or attractive orbits ofperiod 2. For example, the points of period 3 are the zerosof a polynomial of degree 6, namely:fc ◦ fc ◦ fc (z) − z /(z 2 − z − c)= z 6 + z 5 + (1 − 3c)z 4 + (1 − 2c)z 3 ++ (1 − 3c + 3c2 )z 2 + (c − 1)2 z + 1 − c(c − 1)2 .
But since this polynomial cannot be factored (with c a parameter) into two polynomials of degree 3 it does not provide us with a description of the attractive orbits of period
3. However, it does give those c-values for which the period
3 orbits are superattractive (i.e. (f ◦3 )0 (orbit point) = 0),
since in this case the constant term must vanish. Approximate solutions of 1 − c(c − 1)2 = 0 are c = 1.7549, c =0.12256 ± 0.74486i. One can navigate the Mandelbrot set
and observe that the complex solutions are between thetwo biggest blobs that touch the primary apple from either side.

Linearly attractive orbits always have c-values which belong to open subsets of the Mandelbrot set (in particular all the blobs touching the two explicit components),
but the closure of these open subsets does not exhaust theMandelbrot set. For example for c = i the orbit of 0 is
0 7→ −i 7→ −1 − i 7→ i 7→ −1 − i . . ., i.e., after two preliminary steps it reaches an orbit of period 2. Since thisorbit stays clearly bounded we have i ∈ M (by the criterium quoted before). On the other hand, if the iterationz 7→ z 2 − i had any attractor (besides ∞), then the orbitof 0 would have to converge to the attracting orbit. Therefore there is no attractor and no attractor basin. In fact,
the complement of the Julia set is the (simply connected)
attractor basin of ∞. Because of its appearance, this Julia
set is called a dendrite.
To generalize this observation, consider, for any c, theorbit of 0: 0 7→ −c 7→ c2 − c 7→ c4 − 2c3 + c2 − c 7→
(c4 − 2c3 + c2 − c)2 − c 7→ . . .. If 0 is on a periodic orbit
for some c, then this orbit is superattractive. If the periodicity starts later then this periodic orbit may not be an
attractor even though the orbit of 0 reaches it in finitely
many steps. For example c2 − c is periodic of period 3, if
c3 · (c − 2) · (c3 − 2c2 + c − 1)2 · (c6 − 2c5 + 2c4 − 2c3 +c2 + 1) = 0; c = 2 is the largest point on the Mandelbrot set, the third factor has as roots the three c-values(mentioned before) for which the iteration has superattractive orbits of period 3. The last factor has the rootc = 1.239225555 + 0.4126021816 · i, its Julia set is anotherdendrite. A third dendrite is obtained, for example, if the
4th point c4 − 2c3 + c2 − c in the orbit of 0 is a fixed point,
which is the case if c4 (c − 2)(c3 − 2c2 + 2c2 − 2) = 0; here
the last factor has the numerical solutions c = 1.543689
and c = 0.2281555 ± 1.1151425 · i.
6) Suggestions for experiments. The final entry in
the Action Menu for the Julia set fractal is a hierachical menu with five submenus, each of which lists a number of related c-values that you may select. The c-values
in these menus were selected because they typify eithersome special topological property of the associated Julia
set or some dynamical property of the iteration dynamics
of z 7→ z 2 − c, and these properties are referenced by special abbreviations added to the menu item. (In additionsome menu items also list a “name” that is in common use
to refer to the Julia set, usually deriving from its shape).
For convenience we will list in the next couple of pages all
the items from these five menus, but first we explain theabbreviations used to describe them.
Abbreviations used in the following lists of interesting Cvalues. ‘FP’ means ‘fixed point’, the corresponding cvalues are from the belly of the Mandelbrot set. ‘cyc k’means ‘cyclic of period k’, the corresponding c-values arefrom the blobs directly attached to the belly; its Julia sets
have a fixed point which is a common boundary point of kcomponents of the attractor basin and the attractive orbit
wanders cyclicly through these k components. ‘per 2 · 3’means: this c-value has an attractor of period 6 and thec-value is from a blob which is attached to the disk in M
(which gives the attractive orbits of period 2). By contrast,‘per 3 · 2’ means that the c-value is from the biggest blobwhich is attached to a period-3 blob (attached to the belly);
its attractor has also period 6, but the open sets throughwhich the attractive orbit travels are arranged quite differently in the two cases. One should compare both of themwith the cyclic attractors of period 2 resp. 3. The abbreviation ‘tch 1-2’ means that the c-value is in the Mandelbrot
set a common boundary point between the belly (i.e. thecomponent of attractive fixed points) and the componentof attractors of period 2. For the ‘Siegel disks’ see Nr. 8of this ATO first; the column entry in the list gives therotation number of the derivative (of the iteration map)at the fixed point. In the dendrite section of the list wemean by ‘ev per 2’ that the orbit of 0 is ‘eventually periodic
with period 2’, as explained in Nr5 of this ATO. Finally, if
c 6∈ M then the Julia set is a totally disconnected Cantor
set and there are no such easy distinctions between different kinds of behaviour of the iteration on the Julia set (allother points are iterated to ∞).

Interesting C-values From the Action Submenus
Attractors Menu
C - values
0.0
0.0
0.127
-0.3530.74551.0
1.0
0.12271.756-0.2818
1.3136-0.3795
0.5045-0.3909
0.11361.1409-0.3773-0.1205-0.36
0.3614-0.3273
1.0
1.37950.0318-0.0500-0.4068
0.53410.9205
1.21140.6977
0.48640.65842566307252

+
0.0
·i
+
0.1
·i
+
0.6435·i0.1025
·i
+
0.0
·i
+
0.0
·i
+
0.2
·i
+
0.7545
·i
+
0.0
·i
+
0.5341
·i
+
0.0
·i
+
0.3386·i+
0.5659·i+0.2159·i+
0.8636·i+0.2409·i+0.1455·i+
0.6114·i0.1·i+0.6182·i+
0.5659·i+
0.2659
·i
+
0.0
·i
+
0.7932·i+
0.6318·i+
0.3409·i+
0.6023·i+
0.2477·i+0.1545·i+
0.2818·i+
0.6023·i- 0.44980525145595 · iPopular Name Behaviour
CircleFP
Rough Circle
FP
Near-Rabbit
FP
Near-Dragon
FP
Near San Marco FP
cyc 2
cyc 2
Rabbitcyc 3
Airplane
cyc 3
cyc 4
per 2·2
cyc 5
cyc 5
cyc 6
per 3·2
Rabbit’s Shadow per 2·3
cyc 7
cyc 7
Dragoncyc 8
cyc 8
per 4·2
per 2·4
per 2·2·2
Rabbit Triplets
per 3·3
cyc 10per 5·2
per 5·2
per 2·5
per 2·5
cyc 11
Quintuple Rabbits per 5·3
Super Attractor
per 21Interesting C-values From the Action Submenus.
C - values
Between Attractors Menu:
0.75

+ 0.0

1.25

+ 0.0

0.125

+ 0.64952-0.35676+ 0.32858

Popular Name

Behaviour· i San Marcotch 1-2· i S.Marco’s Shadow tch 2-2·2· i Balloon Rabbit
tch 1-3·i
tch 1-5

Siegel Disks Menu:
0.390540870218+ 0.586787907347-0.08142637539+ 0.61027336571

0.66973645476- 0.316746426417

·i
·i
·i

2π · i· gold√
2π · i/ 2√2π · i/ 5

One Simply Connected Open Component Menu:
0.0
+ 1.0
· i Dendrite
ev per 2
0.2281554936539+ 1.1151425080399 · i Dendrite
FP after 3
1.2392255553895- 0.4126021816020 · i Dendrite
ev per 3-0.4245127190500 - 0.2075302281667 · i
FP after 7
1.1623415998840 + 0.2923689338965 · i
per 2 after 7
Outside Mandelbrot set Menu:
0.765+ 0.12-0.4- 0.25-0.4253- 0.2078· i Cantor set· i Cantor set· i Cantor set

An experiment which one should always make after one hascomputed a Julia set for some c from the Mandelbrot set:
Remember from which part of M c came and then ‘Iterate
Forward’ (Action Menu) mouse selected points until theyvisually converge to a periodic attractor. Observe how
the shape of the Julia set lets one guess the period of its
attractor and how this relates to the position of c in M.
7) Computation of the Julia set. In addition to theattractor at infinity there is at most one further attractor in
the z → (z 2 − c) systems. All preimages of non-attractive
fixed points or non-attractive periodic orbits are points
on the Julia set. Since |fc0 | > 1 along the Julia set (withsome exceptions), the preimage computation is numericallystable. This is a common method for computing Juliasets.
In our program we compute preimages starting from thecircle {z; |z| = Rc } around the wanted Julia set. Under
inverse images these curves converge from outside to theJulia set. Such an approximation by curves allows us tocolor the Julia set in a continuous way and thus emphasize
that, despite its wild looks it is the image of a continuous
curve—at least for c ∈ M , otherwise we recall that theJulia set is totally disconnected, so in particular is not theimage of a curve. Our computation works also for c 6∈ M,
since our ‘curves’ of course consist of only finitely many
points, and the inverse images of each of these points havetheir limit points on the Julia set.

8) Self-similarity of a Julia set. A well advertised property of these Julia sets is their so called ‘self-similarity’.
By this one means: Take a small piece of the Julia set
and enlarge it; the result looks very much like a larger
piece of that same Julia set. For the Julia sets of thepresent quadratic iterations, this self-similarity is easily
understood from the definitions: The iteration map fc is a
conformal map that stretches its Julia set 1:2 onto itsself.
In other words, the iteration map itself maps any small
piece of its Julia set to roughly twice as large a piece, andit does so in an angle preserving way. From this point ofview self-similarity should come as no surprise.
9) Siegel Disks. We next would like to explain an experimentally observable phenomenon that mathematicians
find truly surprising, but this needs a little preparation.
Simplifying Mappings. Imagine that we want to describe
something on the surface of the earth, for example a walk.
For a long time, people have been more comfortable giving the description on a map of the earth rather than on
the earth itself. Mathematicians view a map of the earth
more precisely as a mapping F from the earth to a pieceof paper and they describe (or even prove) properties ofthe map by properties of the mapping F . An example ofa useful property is ‘conformality’: angles between curves
on the earth are the same as the angles between the corresponding curves on the map.
Conjugation by simplifying mappings. Let us consider oneof the above iteration maps fc and assume that it has an at-tractive fixed point zf with derivative q := fc0 (zf ), |q| < 1.
The simplest map with the same derivative is the linear
map L(z) := q · z. It is the definition of derivative thatthe behaviour of fc near the fixed point looks ‘almost’
like the behaviour of L near its fixed point 0, and ‘almost’ means: the smaller the neighborhoods of the fixed
points (on which the maps are compared) the more themaps look alike. But more is true for fc because of theassumption |q| < 1, we have the theorem: There exists
on a fixed(!) neighborhood of the fixed point zf a simplifying map F to a neighborhood of 0 ∈ C that makes fc
look exactly like its linear approximation L, by which wemean: fc = F −1 ◦ L ◦ F . In particular, this tells us everything about the iterations of fc in terms of the iterationsof L because they also look the same when compared using(‘conjugation’ by) F :fc◦n = F −1 ◦ L◦n ◦ F .
Siegel’s Theorem. The previous result cannot be true ingeneral if |q| = 1. For example if q = exp(2πi/k), thenL◦k = id, but fc◦k =/ id. Therefore they cannot look alikeunder a simplifying (i.e., ‘conjugating’) mapping F . But if
z → q · z is an irrational rotation and if some further
con√
dition is satisfied, for example if q := exp(2πi/ 2), thenthere is again such a simplifying mapping F such that fc
looks near that fixed point exactly like its linearization,
namely: fc = F −1 ◦ L ◦ F .
Experiment. While Siegel’s proof insures only very smallneighborhoods on which the simplifying mapping F exists,these neighborhoods are surprisingly large in the present

case. One can ‘observe’ Siegel’s theorem by first choosing c = ((1 − q)2 − 1)/4 such that fc0 (zf ) = q with q =√exp(2πi · k/ p), p prime (or square free), then one chooses
points on a fairly straight radial curve from the fixed point
almost out to the Julia set. Under repeated iterations thesepoints travel on closed curves around the fixed point (’circles’ when viewed with F ) and all of them travel with thesame angular velocity, i.e., one observes that they remain
on non-intersecting radial curves.
H.K.

Fractal TOC

Ordinary Differential Equations, ODEs
Go To Page 1(Click the Names)
1.) About ODEs
2.) ODE Equations in 1D, 2D3.) ODE Equations in 3D
4.) ODEs with Central Forces
More Details for Selected Equations
5.) The Duffing Oscillator
6.) Charged Particles in a Magnetic FieldOrdinary Differential Equations *3D-Filmstrip knows how to calculate and display solutionsof the initial value problem for first and second order systems of ordinary differential equations (ODE) in one, two,or three dependent variables.
Let us recall briefly what this means. We will be dealing
with vector-valued functions x of a single real variable t
called the “time”. Here x can take values in R, R2 , orR3 . The problem is to find x from a knowledge of how x0
depends on x and t (in the first order case) or a knowledgeof how x00 depends on x, x0 , and t (in the second ordercase). Thus in the first order case the ODE we are trying
to solve has the form x0 = f (x, t) and in the second ordercase it is x00 = f (x, x0 , t).
In the first order case, the so-called Local Existence Theorem for First Order ODEs tells us that, provided the function f is continuously differentiable, given an “initial time”
t0 , and an “initial position” x0 , then in some sufficiently
small interval around t0 , there will be a unique solution
x(t) to the ODE with x(t0 ) = x0 .
There is a similar local existence theorem for second order
ODEs (which in fact is an easy consequence of the firstorder theorem). It says that given an initial time t0 , an
* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

ODEs

initial position x0 , and an initial velocity v0 then, in somesufficiently small interval around t0 , there will be a uniquesolution x(t) of the ODE with x(t0 ) = x0 and x0 (t0 ) = v0 .
Theory provides not only an abstract existence theorem,
but also many explicit numerical algorithms for finding approximating solutions, in terms of the function f and theinitial data. One of the all-time favorites for general purposes is the so-called Fourth Order Runge-Kutta Method,
and this is the one that 3D-Filmstrip uses.
Although the overall approach to solving such ODEs isquite similar for first and second order ODEs and for thevarious dimensions, the details for giving initial conditions
and for displaying solutions are different for each case, andfor that reason instead of having a single ODE category,it turns out to be convenient to have six. The namingof these categories is fairly self-evident. For example, theODE(1D) 1stOrder Category deals with the case that x
take values in R, and the equation is first order, while theODE(3D) 2ndOrder Category deals with the case that x
take values in R3 , and the equation is second order.
For the ODE(1D) 2ndOrder category, the usual reduction
to a first order system in two variables (x, u), where u
represents x0 , is made via (x, u)0 = (u, f (x, u, t)).
In the case of the ODE(2D) 1stOrder category, the orbit isdrawn dotted, with a constant time interval between dots.
This gives a valuable visual clue concerning the velocity at
which the orbit is traced out, but if you wish to turn thisfeature off, just set Dot Spacing to zero using the ODE
Settings... dialog (see below).
In all the ODE categories, when you have chosen either aparticular pre-programmed example (or set up your own,
using the User Defined. . . feature) then as with the othercategories you will first see a visualization of a default solution. This display will usually stop quickly on its own, but
you can also click the mouse button (or type Commandperiod) to stop it. You may then choose the item “ODE
Settings. . .” in the Settings menu and this will allow you
to set the various data the program needs to compute anddisplay an orbit, namely:a) the initial time,b) the time-span,c) the step-size (used in the Runge-Kutta method),d) the initial value of x, and (in a second order case)e) the initial value of x0 .
Choosing Create from the Action Menu will then display
the solution for these newly selected settings.
There is an ODE control panel that opens by default justbelow the main display window. This has buttons to do
more easily things you can also do with the menus (Create,
Erase, Continue, double or half the scale, and bring up
the dialog to set initial conditions, step-size, time-span,and dottedness). In addition there are buttons for singlestepping the ODE forward or backward, and there is aread-out of the current time, position, and velocity. This
control panel can be hidden using the Hide ODE Controls
command of the Action menu (and then may be re-opened
with the Show ODE Controls command).
The main display shows the evolution of an orbit in thephase space. By default, the program also shows projections of the orbit on the coordinate axes (using different
colors to distinguish the projections). This display occurs
in a second pane of the graphics window that opens automatically below the main pane. This pane can be hidden by choosing ”Hide Direction Fields” from the Action
menu. There is a rectangular button at the right edge ofthe screen where the two panes meet. If you press on thisbutton, the button itself will disappear and be replaced bya horizontal line. Drag the horizontal line to where you
would like the new boundary between panes and release
the mouse. (At least twenty percent of the total screen
height must be devoted to each pane.)
For first order ODE in one and two dimensions, the program by default displays the direction field defined by thecurrent ODE. (Since second order ODEs in one variable
are reduced to first order ODEs in two variables, the direction field is also shown in this case.) The direction fieldof a time dependent ODE is updated every few integration steps. (As far as I know, 3D-Filmstrip is the only
publicly available program that shows direction fields fortime-dependent ODE.) For the ODE(3D) 2ndOrder category there is also a direction field shown (when the display

is in stereo) for the special case of a charged particle in amagnetic field—but be careful, the field shown is the magnetic field, not the direction of the Lorentz force acting on
the particle.
It is fairly easy to do a rough “phase space analysis” bykeeping the other data fixed and varying the initial values.
To make this easier, in certain categories it is possible tochoose the initial conditions using the mouse. For example,
in the ODE(1D) 1stOrder category, each time you click themouse in the window, the forward and backward orbits are
drawn through that initial value. Similarly, clicking themouse in the ODE(1D) 2ndOrder category and ODE(2D)
1st Order category also creates an orbit with the mousepoint as initial condition. Surprisingly, something similareven works for the ODE(2D) 2ndOrder category. Here tochoose an initial condition AND velocity, either select IC
By Mouse [Drag] from the Action menu or type Control
I. You may then click and drag the mouse to choose theinitial position (click) and velocity (drag).
Numerical Methods for 1st Order ODEs
Numerical solutions are computed at a discrete set of time
points t0 , t1 , . . . , tn and usually one such time step of amethod is described, i.e. how to get from x(t0 ) to x(t1 ).
One then has to repeat this step as long as one wants to.
The simplest of all methods is the Euler method: Fromthe initial data t0 , x0 = x(t0 ) one computes first theinitial derivative: x0 (t0 ) := f (x0 , t0 ), then theEuler step: x(t1 ) := x0 + x0 (t0 ) · (t1 − t0 ).

The following picture shows five such (large) Euler steps
and the corresponding exact solutions for the same timeinterval. Clearly, if the exact solution curves happen to
be convex, the Euler Step solutions stay outside and movefarther away from the exact solution with each step.

Five Euler Steps
Recall that the tangent of a parabola wich is parallel tosome secant touches the parabola in the middle of the secant interval. This suggests a substantial improvement
over the Euler method, it is called the Halfstep Method:
Stepsize: ∆t := t1 − t0
Initial Derivative: x00 := f (x0 , t0 )Half Step towards Middle: xm := x0 + x00 · ∆t/2
Middle Derivative: x0m := f (xm , t0 + ∆t/2)
Final Step: x(t1 ) := x0 + x0m · ∆t.
Again we illustrate this method with a picture, but using a larger stepsize than in the Euler case. The magenta
vector is the derivative at the approximate midpoint xm
multiplied by the stepsize, i.e. x0m · ∆t. This vector is thedifference between x(t1 ) and x(t0 ). One can see that thismethod follows the solution curve rather well, but traverses

it too fast.

Three Halfstep Steps

Some first order ODEs have the property that the second
initial derivative can be computed rather easily, more eas∂∂f · x0 + ∂t
f,
ily than differentiation of the ODE, x00 = ∂xsuggests. In those cases one can use a second order Taylor
step instead of the first order step used in Euler’s method.
We call this the Taylor-2 Method:
Initial 1st and 2nd Derivative: x00 , x00002Taylor Step: x(t1 ) := x0 + x0 · ∆t + x00
0 · (∆t) /2.
The endpoints of the magenta segments lie on the quadratic
parabola p(s) := x0 + x00 · s + x000 · s2 /2, s ∈ [0, ∆t] (withthe same initial tangent (green) as the exact solution). The
method has the same order of precision as the previous one,so that one will often prefer the Halfstep method. But
in the vicinity of a point where f (x, t) = 0, the Taylor-2
method is usually better.

Three Taylor-2 Steps
Next we try to explain the famous Runge-Kutta Method.
First we need to understand what numerical analysts callan order k method. If we compute one Euler step from x0 ,but with various stepsizes s, then all the computed points
lie on the tangent x0 + x00 · s. The difference to the exact
solution is controlled by a bound B2 on the second derivative of solutions near x0 as: error(s) ≤ B2 · s2 . This iscalled a first order method.
For the Taylor-2 method these stepsize dependent numerical values lie on a parabola which was already mentioned:p(s) := x0 + x00 · s + x000 · s2 /2. The difference to the exact
solution x(s) is controlled in terms of a bound B3 for third
derivatives as: error(s) ≤ B3 · s3 . This is called a second
order method.
How can one make such considerations work for the Halfstep method? Clearly the approximate midpoint is a function of the stepsize: xm (s) = x0 +x00 ·s/2 and thereforethe
final point also: xf (s) := x0 + f xm (s), t0 + s/2 · s. We
may call this curve the method-curve. Computation showsthat the exact solution at t0 and the method-curve at s = 0
have the same first and second derivative: x00 (t0 ) = x0f (0),x000 (t0 ) = x00f (0). The error, therefore, is again ≤ B3 · s3 ,and the method is also called a second order method.
What we saw in these examples is true in general: Consider the initial data x0 , t0 as fixed. The numerical value ofthe computation of one step will then only depend on thestepsize s, giving us the method-curve xf (s) for the considered numerical solution. If the first k initial derivatives ofthe exact solution agree with the first k initial derivativesof the method-curve, then we speak of an order k method.
The Runge-Kutta Method can be seen as a generalization of the halfstep method: Instead of computing two
derivatives x00 , x0m , Runge-Kutta computes four and averages them for the final step.
initial derivative: x00 := f (x0 , t0 )1st intermediate point: xa (s) := x0 + x00 · s/2
1st intermediate derivative: x0a (s) := f (xa (s), t0 +s/2)
2nd intermediate point: xb (s) := x0 + x0a · s/2
2nd intermediate derivative: x0b (s) := f (xb (s), t0 +s/2)
3rd intermediate point: xc (s) := x0 + x0b · s
3rd intermediate derivative: x0c (s) := f (xc (s), t0 + s)derivative average: x0RK (s) := (x00 + 2x0a + 2x0b + x0c )/6
final step: xf (s) := x0 + x0RK (s) · s.
We can compute the first four derivatives of this methodcurve and check that the above defines a 4th order method.
It is one of the most celebrated ODE solving numericalmethods. A visualisation of one Runge-Kutta step is:

The three straight segments from the initial point end at
the three intermediate points xa , xb , xc . Note how far they
are apart from each other compared to the small error of
the composite step. (The dotted green curve is the exact
solution, the black dots lie on the method-curve.)
In 3D-XplorMath ODEs are not only objects of visualisations, they are an essential tool for many computations in
the program. The ODE for elliptic functions f , namely:
f 0 (z)2 = P (f (z)), P a polynomial of degree 3 or 4
is an example where the second derivative can be moreeasily computed than the first: f 00 (z) = P 0 (f (z))/2. We
therefore use a different 4th order method:
Initial Derivatives: x00 , x00
02Intermediate Point: xm (s) := x0 + x00 · s/2 + x00
0 · s /80Intermediate 2nd Derivative: x00
m (s) := P (xm (s))/2

00
2Final Step: xf (s) := x0 + x00 · s + (x00
0 + 2xm (s)) · s /6

Although zeros of the polynomial P are constant solutionsof the first order ODE, the just described method can start
to compute the non-constant function f at points z wheref 0 (z) = 0 as initial value. A Runge-Kutta solution would
stay constant.
R.S.P., H.K.
ODEs

1D-, 2D- ODE examples in 3D-XplorMath *See First: About This Category (DocumentationMenu)
1D 1st Order:
logistic:ẋ = aa · x(1 − x/bb)
mass action: ẋ = aa · x + bb · x2
1D 2nd Order:
harmonic Osc: ẍ = −aa2 x − bb u
forced Osc:ẍ = −aa2 x − bb u + cc cos(dd(t − ee))forced Duffing: ẍ = −hh x − ii x3 − aa u + bb cos(cc t)
van der Pol:ẍ = hh x − aa(x2 − 1)u + bb cos(cc t)
Pendulum:ẍ = −aa sin(x) − bb u
2D 1st Order:
harmonic Osc:



Pendulum:

x
y ˙=

x
y ˙=

0
1
x
·
−aa −bbyy−aa sin(x)−bb ylinear 2D:

x
y ˙=



aa bb
cc dd

VolterraLotka:xy ˙=(aa−bb y)·x
(cc x−dd)·y







·



x
y





2D 2nd Order:
coupled Osc:(ẋ = u, ẏ = v)−aa2 x+dd(y−x)−gg uxy ¨= −bb2 y+dd(x−y)−hh v−aa2 x+bb cos(cc t)
x
forced Osc:y ¨= −dd2 y+ee cos(ee t)−aa2 x+2 bb sin(π cc/180)v
x
Foucault Pend: y ¨= −aa2 y−2 bb sin(π cc/180)ubb := earth’ angular velocity, cc := pendulum latitude* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

ODEs

3D-ODE examples in 3D-XplorMath *See First: About This Category (DocumentationMenu)

3D 1st Order: 0   xaa bb ccx y  =  dd ee ff  ·  y 
linear:z
gg hh iiz
 0 
x−10x + 10y
Lorenz:  y  =  aa x − y − xz z−(8/3)z + xy 0 
x−bb x + yz
Rikitake:  y  =  −aa x − bb y + xz z
1 − xy 0 
x−y − zRössler:  y  =  x + aa y zbb − cc z + xz

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

ODEs

(x0 = u, y 0 = v, z 0 = w)

3D 2nd Order:

coupled Oscillator: 00 2x
−aa x + dd(y − x) + ee(z − x) − gg u y  =  −bb2 y + dd(x − y) + ff (z − y) − hh v z−cc2 z + ee(x − z) + ff (y − z) − ii w
forced Oscillator: 00 2x
−aa x + bb cos(cc · t)
 y  =  −dd2 y + ee cos(ff · t) z
−gg 2 z + hh cos(ii · t)
The ODE for a charged particle in a magnetic field M :~x 0 = ~u, ~x 00 = ~u 0 = ~u × M (x, u), charge and mass are 1.
const Magnetic Field:

M (~x) := (aa, bb, cc)t

Magnetic Field of straight wire:√~ := (aa, bb, cc)t / aa2 + bb2 + cc2
wire~ · wire,~px~ := ~x − h~x, wirei
rr := |px|
~ 2
~ × px)M (~x) := dd/rr · (wire~Toroidal Magnetic Field:
M (~x) := (2y, −2x, 0)/(x2 + y 2 )Magnetic Dipole Field:
rr := x2 + y 2 + z 2
eDOT p := aa x + bb y + cc z
M (~x) := (

3x
3y
3z
eDOT p,,)rr − aa rr − bb rr − cc rr3/2

ODEs with Central Forces*
This category models the motion of a particle moving according to Newton’s Law “F = ma”, where the force F
is a “central” force, i.e., one directed towards (or awayfrom) the origin and whose magnitude depends only on
the distance of the point from the origin. (As we will recall below, this also models the motion of two particles inspace when the force between them is directed along theline joining them and has a magnitude that depends only
on their separation.)By “conservation of angular momentum”, the particle moves
in a fixed plane, and this is taken as the plane of the screen,with the origin at the screen center. By default, the x andy axes are visible but, as usual, you can reverse this using
the View Menu.
There are several pre-programmed central forces:
Coulomb: F (r) = −aa/r2(default: aa = 2.5)

Power Law: F (r) = −aa rbb(defaults: aa = 2.5,
bb = −2.01)

Yukawa: F (r) = −aa exp(−bb r)(bb/r + 1/r2 )(defaults: aa = 2.5, bb = 0.25)
Hooke’s Law : F (r) = −aa r(default: aa = 2.5)

Higgs: F (r) = aa r(1 − (r/cc))(aa = 3.0, cc = 2).

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

ODEs

(And there is also a provision for a “User Defined” central
force law.)After choosing one of the pre-programmed examples, anorbit will be displayed for a default initial position (xinitial,yinitial) and initial velocity (xdotinitial,ydotinitial).
The time evolution of the orbit will continue until themouse is clicked. The same initial position (0.5, 0.5) andsame initial velocity (−1.25, 2.0) is used for the Coulomb’slaw, the Power Law, and Yukawa Law. (This, in particular, demonstrates the fact that having all bounded orbits
closed is rather special to the case of a power law withexponent −2.)
To display an orbit with different initial conditions, choose“IC by mouse (Drag)” from the Central Force menu andthen click to choose an initial value, and drag to choose an
initial velocity—the velocity vector will be proportional
to the vector difference of the MouseUp and MouseDown
points. (If instead you choose “IC by mouse (Throw)”from the menu, then the initial point is again the MouseDown point, but the velocity is computed using the position of the mouse one-half second after MouseDown rather
than the MouseUp position. (In either case, the new orbit will start being displayed immediately, and continue
to evolve until the mouse is again clicked). There is stilla third way to choose initial conditions; namely, choose“ODE Settings. . . ” from the Settings menu. This will display a Dialog in which you can choose the initial valuesof x and y. In addition x- and y- components of velocity.

(In this Dialog, you can also set the value of the mass,
the step-size used in the Runge-Kutta integration, and the“dotted-ness” of the orbit. You can also change the values of any parameters on which the chosen central force
depends by choosing Set Parameters. . . from the Settings
menu. (To check dependence of the current central force on
the parameters aa, . . . , ii, choose either “About this Object” from the Central Force menu.) After making changes
to the parameters, the mass, initial conditions, stepsize,and dotted-ness of the orbit (see below for more about thelatter two) choose Create from the Central Force menu tostart the display of the new orbit. (If you want the orbit
drawn onto a clean screen, choose Erase before choosing
Create.)Instead of using one of the pre-programmed central forces,
the user can define a central force law by choosing User
Defined. . . in the Central Force menu and entering a function of the radius, r, and the usual usual nine parameters aa, . . . , ii. (After clicking the OK button, either
choose an initial position and velocity with the mouse as
described above, or use the Set Parameters. . . and ODE
Settings. . . and then choose Create.)
If the Time step-size is set equal to zero in the ODE Settings Dialog, then the stepsize will be set by an adaptivealgorithm that makes the stepsize (roughly) inversely proportional to the magnitude of the velocity. This has theadvantage of making the stepsize very small where the velocity is large, avoiding numerical errors that are otherwiseinherent in such situations. The downside is that all timing
information is lost—the orbit is traced out equal lengths
in equal times rather than by Kepler’s Law of equal areas
in equal times.
The Dot Spacing in the ODE Settings. . . Dialog sets thetime between which successive dots are drawn along theorbit. If Dot Spacing is set equal to zero, then the orbit isdrawn as a solid curve.
Finally, we recall that the real significance of a central
force is that it models the Newtonian motion of a twoparticle system (say with masses m1 and m2 ) in which theforce F (r) on each particle is equal and opposite to thaton the other, is along the line joining the two particles
and has a magnitude depending only on their distance ofseparation, r. First, by conservation of linear momentum,
the center of mass of the two particles moves with constantvelocity in a straight line, so making a Gallilean coordinatetransformation we can assume the center of mass is fixed at
the origin. The problem then is to find the time evolutionof the vector ~r joining particle one to particle two, and
it is easily seen that ~r satisfies Newton’s equations for aparticle moving in response to the central force F (r) andwith mass the harmonic mean of m1 and m2 (the so-called“reduced mass”). For details see any good text on classicalmechanics, for example, Chapter 3 of Goldstein.
R.S.P.

ODEs

Forced Duffing Oscillator *What is it?
What we shall call the Forced Duffing Oscillator Equationis the second order ODE for a single variable x,
d2 x
dx
3
=
−hh
x
−
ii
x
−
aa
(1)
2
dt
dt + bb cos(cc t)
whose solutions we display via the equivalent (non-autonomous) first order system in two variables, x and y:
dy
dx
3
=
y,
(2)
dt
dt = −hh x − ii x − aa y + bb cos(cc t)
which in turn can be made into an autonomous first ordersystem in three variables, T , x and y:
dx
dT
= 1,
= y,
dt
dt
dy
= −hh x − ii x3 − aa y + bb cos(cc T ).
(3)
dt
We discuss the interpretation and significance of the fiveparameters, aa, bb, cc, hh, ii below. Their default values
are: aa = 0.25, bb = 0.3, cc = 1.0, hh = −1.0, and ii = 1.0.
If bb·cc =/ 0 then the forcing period 2π/cc is shown by yellow
dots on the orbit.
Why is it interesting?
Here are two of the considerations that make the oscillator equation (1) worth studying. First, with appropriate choices of parameter values it reduces to a variety of* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

ODEs

mathematically and physically interesting oscillator models; some classical such as the harmonic oscillator (withand without damping and forcing) and others that are
more exotic, such as the classic Duffing oscillator introduced by Duffing in 1918. By putting these together in aparametric family, we can investigate how various featuresof these systems behave as we move around in the parameter space. Secondly—and more importantly—it was in in
the study of the Duffing Oscillator that symptoms of thephenomena we now call “chaos” and “strange attractor”
were first glimpsed (although their significance was only
appreciated later). By the Poincaré-Bendixson Theorem,
three is the smallest dimension in which an autonomous
system can exhibit chaotic behavior, and the Duffing system is so simple that it lends itself very easily to the study
and visualization of the phenomena related to chaos.
The Newtonian Particle Interpretation.
Note that (1) becomes Newton’s equation of motion for aparticle of unit mass moving on the x-axis if we define the“force”, F (x, dx
dt , t), acting on the particle to be the righthand side of (1). Let’s interpret the various terms of F
from this point of view.
If hh is positive then the term −hh x by itself gives Hooke’s
Law for a spring, that “stress is proportional to strain”
and the parameter hh has the interpretation of Hooke’s
proportionality factor between the extension of the spring,x, and the restoring force. If also ii = 0 then we have apure Hooke’s Law force that gives the Harmonic Oscillator,
d2 x
dt2 = −hh x. But a real spring only satisfies Hooke’s Law
approximately, and the term −ii x3 represents the nextterm in the Taylor expansion of the restoring force under
the reasonable assumption that this force is an odd function of the spring extension, x. (If ii is positive it is called
a “hardening” spring and if negative a “softening” spring.)
For the classic Duffing Oscillator, hh is negative and ii is
positive and there is not a good interpretation of the force
in terms of a spring. Rather, the sum of the two terms−hh x − ii x3 should be interpreted as the force on a particle that is moving in a double-well potential as we will
discuss in more detail below.
The term −aa dx
dt represents a “friction” force of the sortthat would be experienced by a particle like a bullet traveling through air or a bead sliding on a wire; that is, assuming that the “damping” or “friction” coefficient aa ispositive, it describes a force acting on the particle in the
direction opposite to the velocity and with a magnitude
that is proportional to the magnitude of the velocity.
Under the sum of the above terms of the force law F , theparticle will (in general) oscillate back and forth—which ofcourse is why it is called an oscillator—however if aa > 0
these oscillations will gradually die down as the kinetic energy is absorbed by friction. The final term in the force
law, bb cos(cc t) is a periodic forcing term that will act on
and perturb the motion of this oscillating particle, and wenote that it is solely a function of the time and is independent of both the position and velocity of the particle.
We will discuss a possible physical interpretation of thisterm later. The parameter bb is clearly the amplitude ofthis forcing term, i.e., its maximum magnitude, and theparameter cc is the angular velocity of its phase in radians
per unit time, so that the period of the forcing term is 2πcccc
and its frequency is 2π . As we shall see, it is the energythat is fed into the system by this forcing term that is essential for the interesting chaos related effects to occur. In
fact the most interesting behaviors of solutions of (1) arepresent when all the above terms are present in F , thatis when the oscillator is both forced and damped, and infact the way damping and forcing can balance each other iscrucial to understanding the general behavior of solutions.
However we will begin by analyzing the simpler situation
when both the damping and forcing terms are missing.
The Undamped, Unforced Case.
We now assume that aa and bb are both zero, so theforce F (x) = −hh x − ii x3 is a function of x alone. Now
in one-dimension, whenever this is the case the force isconservative, that is, it is minus the derivative of a “potential”
function, U (x). Indeed, if we define U (x) :=Rx− 0 F (ξ) dξ, then clearly F (x) = −U 0 (x). If as above we
1 2,
definethekinetic
energybyK(y):=write y := dx
dt2y
and define the Hamiltonian or total energy function bydydx
0H(x, y) := K(y) + U (x), then dH
=
y
+
U
(x)
dt
dt
dt =

dy
0
+
U
(x)).
So,
if
Newton’s
Equationissatisfied,
y( dy
dt
dt =
d2 x
dH
0
dt2 = F (x) = −U (x), so dt = 0. This of course is thelaw of conservation of energy: the total energy function
H(x, y) is constant along any solution of Newton’s Equations. In one-dimension this provides at least in principle a way to solve Newton’s Equation for any initial conditions x = x0 and y = y0 at time t = t0 . Namely,
the path or orbit of the solution is a curve in the x-yplane, and by conservation of energy this curve is givenby the implicit equation H(x, y) = H(x0 , y0 ). And sincedx 2= y 2 = 2K(y) = 2(H(x0 , y0 ) − U (x)), we find:
dt

1
dt
p
=
,
dx
2(H(x0 , y0 ) − U (x))
so we can find t as a function of x by a quadrature, andthen invert this relation to find x as a function of t.
In the Harmonic Oscillator case, with hh = 1 andii = 0, U (x) = 12 x2 so H(x, y) = 12 (x2 + y 2 ), so the orbits
are circles, and it is easy to carry out the above quadratureand inversion explicitly, to obtain x(t) = x0 cos(t − t0 ) +y0 sin(t − t0 ).
The Universal “Sliding Bead on a Wire” Model.
In one-dimension there is a highly intuitive physical modelthat makes it easy to visualize the motion of a particle
under a given force. Moreover this model is “universal” inthe sense that it works for all forces that are function of
position only and hence, as we noted above, are of the formF (x) = −U 0 (x) for some potential function U . Namely,
imagine that we string a bead on a frictionless piece ofwire that lies along the graph of the equation y = U (x).
If the bead has mass m = 1 and if we choose units sothat g, the acceleration of gravity, equals one, then thegravitational potential of the bead is mgh = h where h is
its height. So if as usual we interpret the ordinate of apoint as its height, then the gravitational potential of thebead when it is at the point (x, y) = (x, U (x)) is just U (x),and the sliding motion of the bead along the wire under
the attraction of gravity will exactly model whatever othersystem we started from!
In the case of the Harmonic Oscillator, where F (x) = −x
and U (x) = 21 x2 , the graph is the parabola, y = 12 x2 and
it is easy to imagine the bead oscillating back and forthalong this parabola.
For the unforced and undamped Duffing Oscillator theforce is F (x) = −hh x − ii x3 , where for simplicity in what
follows we will assume that ii > 0 and hh < 0. The poten2 ii 4
tial is U (x) = hh
2 x + 4 x , which we note can be consideredas the first two terms in the Taylor expansion for an arbitrary symmetric potential with local maximum at 0. It iseasily checked that limx→±∞ U (x) = +∞ and in addition
to the local maximumqat 0, there are two other criticalpoints of U , at x = ± −hh
ii , where U has local minima.
For the default values, hh = −1 and ii = 1, the force is
F (x) = x(1 − x2 ), and the potential is U (x) = 14 x2 (x2 − 2),so the local minima are at ±1. We graph this force F (x)and potential U (x) below, and show a selection of the resulting orbits. It should be clear why U is called a doublewell potential.

F (x)

U (x)

Some orbits of the Unforced, Undamped Duffing Oscillator

The Unforced, Damped Duffing Oscillator.
We now still assume bb = 0 (so there is no forcing) but weassume that aa > 0, so there is damping. In the bead ona wire picture, aa dx
dt = aa y is the friction from the beadrubbing against the wire, and the force is now given by
F (x) = −U 0 (x) − aa y. If we again calculate dH
dt as we diddH
2
above, we now find not dH
dt = 0 but instead dt = −aa y .
The result is that instead of the orbits of the bead in the
x-y-plane being closed curves of constant total energy H,
the energy decrease along the obits, and they cut acrossthe H = constant curves and spiral in towards the two
minima of H at the bottom of the two potential wells. We
show a selection of the resulting orbits below.

Some orbits of the Unforced, Damped Duffing Oscillator

The Forced Duffing Oscillator.
We now add back the forcing term bb cos(cc t). First aword about how to interpret this force in the sliding bead
picture. If we assume that there is an alternating electricfield parallel to the x direction and with strength cos(cc t)
at time t, then bb cos(cc t) will be the electric force felt by
the bead if we give the bead an electric charge of magnitudebb.

Some orbits of the Forced, Damped Duffing Oscillator

Chaos, Strange Attractors, and Poincaré Sections.

Two Time Slices of the Duffing Attractor
R.S.P.

ODEs

3-dim ODE of 2nd Order: Charged Particles *The Motion of Charged Particles
in Magnetic Fields
The path p(t) of a particle with electric charge e and mass
m in a magnetic field B is given bym · p00 (t) = e · p0 (t) × B(p).
(The right hand side is called the Lorentz Force.)
This implies that, for an arbitrary magnetic fields, B, the
0
0
hp
,
p
i(t) is constant in time.
kinetic energy E(t) = m
2
One should first convince oneself in the case of aConstant Magnetic Field
that a particle can move tangentially to the field lines, incircles around the field lines and in helices around the fieldlines, i.e., in any linear combination of the first two special
cases.
Put in Settings, ODE Settings:
vx = 0.003, vy = 0.003, vz = 0.5, to obtain almost circles
around the field lines.
And put vx = 0.2, vy = 0.2, vz = 0.001 to obtain approximate straight lines parallel to the field lines. The Default* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

ODEs

Settings give a general helix. We will also try to understandcharged particle motions in nonlinear fields by looking at
such special cases.
We consider next motion of a charged particle in the:Field of an Electric Current
along the x-axis. The field lines are circles parallel to the yz-plane with centers on the x-axis. In this case, the Default
Settings give initial conditions in the x-y-plane (a symmetry plane of the field) that are orthogonal to the fieldlines. The solution curves therefore remain in this plane,
and are, for small velocities, almost circles around the fieldlines. But, because the absolute value of the field is decreasing with r, these solution curves are more strongly
curved the nearer they are to the wire. They are therefore
rolling curves with a translational period in the directionof the wire. (See Plane Curves, Cycloid and put aa = 1
and bb = 6.5 in the Set Parameters dialog.) If in ODE
Settings one increases the velocity to vy = 0.5, then thetranslational part is so large that the consecutive loops do
not intersect.
We obtain solution curves which almost follow the fieldlines if we make the initial velocity tangential to the fieldlines and fairly small:vx = 0, vy = 0, vz = 0.02,
Time span = 450, Step-size = 0.2.
Now slowly increase vz , e.g., to vz = 0.2, to obtain an-other family of solutions follows the field lines, but windingaround them in small loops.
Next in Settings, ODE Settings, put:vx = 0.02, vy = 0.02, vz = 0.01
leaving Time span = 450, Step-size = 0.2, as before.
Finally we increase the initial velocity to obtain solution
curves that look fairly wild at first but can be seen to follow the pattern which we recognized for more special initialconditions, namely put
vx = 0.2, vy = 0.2, vz = 0.1
to see solutions that follow helices with wide loops aroundthem. Try by all means to view this in stereo!
Finally we consider the so-called Störmer Problem, namely
the motion of charged particles in a Magnetic Dipole Field.
Since the magnetic field of our Earth is a dipole field, suchmotions occur in the van Allan Belt when charged particles
from the Sun’s plasma meet the Earth. A dipole field B(p)with a magnetic moment mm is given by:mmp.
B(p) = 3hmm, pi 5 −|p||p|3
The Default ODE Settings give a fairly general but somewhat complicated solution curve. To see solutions that almost follow the field lines use ODE Settings to set a small
initial velocity tangential to the field lines, say vx = 0, vy =0, vz = 0.05. To see solutions that almost circle the fieldlines in the equator plane of the dipole, in the ODE Settings dialog, choose small initial conditions in the equatorplane, e.g., vx = 0.1, vy = 0.1, vz = 0. The resulting curves
are close to rolling curves. (Compare Plane Curves, Circle using, Parameter Settings: hh = −0.125, ii = 4, andincrease t-Resolution to 200, then choose Generalized Cycloids from the Action Menu.) Since the absolute valueof the field increases along the field lines from the equator
towards the poles, one cannot have solutions that almost
follow the field lines while circling them in narrow loops,
however one can approximate such behavior with the initial condition vx = 0.035, vy = 0.035, vz = 0.05.
In a Plenary address on Dynamical Systems he gave at the1998 International Congress of Mathematicains in Berlin,Jürgen Moser had an interesting discussion of the Störmer
Problem that we reproduce below from Documenta Mathematica, Extra Volume, ICM 1998, pp. 381–402. (After thelecture, one of us approached Moser and showed him the visualization of the Störmer Problem in 3D-XplorMath. He
appeared to be delighted by it, but said something looked
not quite right to him, a remark that helped us eliminate
a small bug!)
Here is the extract from Moser’s lecture.
R.S.P. & H.K.

ODEs
The Störmer Problem.

Another large scale confinement region is known in themagnetic field of the earth. With the advent in 1957 ofsatellites it was soon discovered that the earth was sur-rounded by (two) belts of charged particles caused by its
magnetic field. Since the beginning of the century it was
known that such charged particles were present above theatmosphere and were responsible for the aurora borealis(and australius). It was Störmer (incidentally president of
the ICM 1936 in Oslo) who made calculations of the orbitsof these charged particles moving in the magnetic field of
the earth, which he modelled as a magnetic dipole field.
This is an interesting nonlinear Hamiltonian system.
The satellite measurements led to the discovery of two
regions surrounding the earth, the so-called van Allan belts,
in which charged particles were trapped. It turns out that
it is an example of a magnetic bottle to which the stability
theory is applicable (M. Braun 1970).
It is interesting to realize the dimensions involved:
For electrons, the “cyclotron radius” is of the order of afew kilometers and the corresponding periods of oscillation
about one millionth of a second! The “bounce period” oftravel from north pole to south pole and back is a fractionof a second.
In addition to these natural van Allan belts several
artificial radiation belts have been made by the explosionof high-altitude nuclear bombs since 1958. Some of thoseso created belts had a lifetime up to several years—whichshows the long stability of these experiments as well as theirresponsibility for carrying them out! Some 30 years ago
these tests have been stopped.

Störmer Problem

Van Allen Belt
ODEs

Lattice Models
Go To Page 1( Click the Names )1.) Summary on Lattice Models2.) FERMI-PASTA-ULAM Lattice3.) Toda Lattice Model
4.) Discrete Sine-Gordon
5.) Discrete Klein-Gordon

Lattice Models *The Lattice Model category integrates and displays theequations of motion for a one dimensional lattice of “oscillators” with nearest neighbor interactions. This is interesting for itself of course, but in addition it provides animportant general method for making discrete approximations to many continuous wave equations.
An excellent introduction to lattice models can be foundin Michel Remoissenet’s “Waves Called Solitons: Concepts
and Experiments” (Springer-Verlag, 1994, ISBN 3–540–
57000–4). Indeed, the implementation of the Lattice Model
category in 3D-Filmstrip, and the choice of particular models was very strongly influenced by Remoissenet’s treatment of the subject. For this reason we will only cover
below what a user of 3D-Filmstrip needs to know to use
the Lattice Model category, and refer to the above text for
the theoretical background.
For various reasons the number N of oscillators in the lattice is restricted to be a power of 2. Currently N can be
any one of16, 32, 64, 128, 256, or 512. (One not so obvious
reason is that choice greatly speeds up the computationof the amplitudes of the normal modes for the linearized
lattice.) The characteristics that determine the equationsof motion of the lattice are:* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Lattice TOC

1) The mass of an oscillator,
2) The “internal” force an oscillator exerts on a neighboring oscillator,
3) The external force on an oscillator.
The mass of an oscillator is not set directly, but rather isderived from a parameter called the density, which has adefault value of 1, but can be set to another value by theuser. The mass satisfies the relation that the number N ofoscillators times the mass of an oscillator is equal to thelength of the lattice times its density.
The force between two neighboring oscillators is always assumed to be a function F (d) of the distance d separatingthem. If xi is the displacement of the i-th oscillator from its
equilibrium position, then the total force on the i-th oscillator due to its two neighbors is F (xi −xi−1 )−F (xi+1 −xi ).
The external force on the i-th oscillator is also assumed tobe some function G(xi ) of the displacement xi from equilibrium.
There are four preprogrammed Lattices: Fermi-Pasta-Ulam,
Toda, Discrete Sine-Gordon, and Discrete Klein-Gordon.
The internal forces between oscillators for these four are
as follows:
1) Fermi-Pasta-Ulam: F (d) = −aa d−bb d2 −cc d3 −dd d42) Toda: F (d) = aa(exp(−bb d) − 1)3) Discrete Sine-Gordon: F (d) = (cc/aa)d4) Discrete Klein-Gordon: F (d) = (cc/aa)dand the external forces are:1) Fermi-Pasta-Ulam: G(d) = 0
2) Toda: G(d) = 0
3) Discrete Sine-Gordon: G(d) = bb2 sin(d)4) Discrete Klein-Gordon: G(d) = bb2 d.
In addition their is a User Defined... item in the Latticemenu that brings up a dialog that lets the user define internal and external forces.
The equations of motion can be integrated either with zeroboundary conditions (i.e., x1 = xN = 0) or with periodic
boundary conditions (x1 = xN ).
Various choices including initial conditions can be changed
from their default values using the dialog brought up by
the Set Lattice Parameters... menu item of the Lattice
Model menu.
R.S.P.

Lattice TOC

The FERMI-PASTA-ULAM Lattice *Background
The Fermi-Pasta-Ulam lattice is named after the experiments performed by Enrico Fermi, John Pasta, and Stanislaw Ulam in 1954-5 on the Los Alamos MANIAC computer, one of the first electronic computers. As reported
in Ulams autobiography [Ul], Fermi immediately suggested
using the new machine for theoretical work, and it was decided to start by studying the vibrations of a string under the influence of nonlinear internal forces. Nonlinearity makes the computations very difficult and the problem
could not be attacked by standard mathematical methods.
However, physical intuition suggested that the motion ofsuch a string would eventually “thermalize”. The purpose of the experiment was to investigate the rate of thermalization. For the computer calculations, the string wasapproximated by a finite sequence of point particles withnearest-neighbor interactions—a so-called “lattice model”.
The motion of such a lattice is governed by a system of ordinary differential equations, and it is these equations thatthe computer solves, numerically.
The results more than justified the trouble of carefully formulating the experiment and programming it on a very
primitive machine, for they went completely against all* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Lattice TOC

expectations. In fact the motion did not seem to to thermalize at all! The previously accepted beliefs thus had
to be re-examined, and this re-examination led eventuallyto the discovery of soliton theory, which became a major
theme of twentieth Century mathematics.
How To View the Demonstration
The Fermi-Pasta-Ulam string is the black curve. Although
it appears continuous, it is in fact made up of the motion ofindividual points (the blue curve represents the velocitiesof these points). The number of points (i.e., the number ofparticles in the lattice) can be changed by choosing LatticeParameters in the Action menu, and various other adjustments are possible. Fermi- Pasta-Ulam actually used alattice with at most 64 particles.
The energies of the so-called normal modes are the red/green
bars in the left hand corner. Thermalization means, roughly
speaking, that the heights of the red/green bars should
equalize after a sufficiently long period of time. More precisely, it is the time averages of their heights that should
equalize, and these are represented by the blue bars.
The fact that this does not happen is the surprising featureof the Fermi-Pasta-Ulam experiment. What happens, andwhat Fermi-Pasta-Ulam observed, is first of all that only
the first few normal modes are excited, and in addition
that the motion is almost periodic. This can be detectedby watching the motion of the red/green bars—after sometime they return (apparently) to the original configuration.

Various initial shapes can be chosen from the Lattice Parameters menu, but the almost periodicity seems to occur
regardless of the initial shape of the string (or the initialconfiguration of the red/green bars).
Further Aspects
To put this observation into context, it is necessary to consider the effects of different internal forces in the string. As
explained in the Lattice Models ATC the internal force is
specified by a function T (y). By choosing the User Lattice
Model from the Lattice Models menu, the function T (y)can be specified (as well as the usual Lattice Parameters).
To switch on the energies of the normal modes, select Show
Normal Mode Display from the Action menu. For example, T (y) = ky (where k is a positive number) gives the
motion of a “standard” string, which is exactly periodic.
The red/green bars behave even more simply—they remain
constant. The Fermi-Pasta-Ulam case is T (y) = y + αy 2 ;the default value of α in 3D-XplorMath is 0.3. (FermiPasta-Ulam also considered y + αy 3 and a piecewise linear function, with similar results.) The effect of the small
nonlinear term αy 2 is to “perturb” the periodic behaviourof the linear case, and the red/green bars no longer remain constant. But, as we have observed, some vestiges of
periodicity remain.
This is surprising because a randomly chosen nonlinear
internal force T (y) generally leads to the thermalizationthat Fermi-Pasta-Ulam expected. For physical reasons,T (y) should be of the form ky + N (y) where N (y) is anonlinear term which remains small during the motion,for example, T (y) = y + 100y 3 + 5y 4 + 5y 5 .
Why is the Fermi-Pasta-Ulam lattice so special? This
question took some time to answer, because the necessary
mathematical tools were insufficiently developed at thattime. In addition there were various sources of distraction:
the mathematical string is only an idealized model of areal physical situation, the discrete lattice is in turn onlya model of that string, and furthermore any computer calculations are subject to numerical and rounding errors. In
fact the philosophical questions surrounding such “simulation” experiments, and the subject of “experimental mathematics” in general, are still being debated (see [We]).
However, it is by now generally agreed that the FermiPasta-Ulam lattice is a genuine phenomenon and that it
can be explained by a combination of powerful theories:
the Kolmogorov-Arnold-Moser (or KAM) theory, the theory of completely integrable Hamiltonian systems, and soliton theory. The KAM theory explains, very roughly speaking, that a system which is close enough to a completely
integrable Hamiltonian system retains some of the predictable behavior of such a system. The Fermi-Pasta-Ulamlattice happens to be close to the Toda lattice, which (some20 years after the FPU experiments) was discovered to be
a completely integrable Hamiltonian system. The Toda
lattice is discussed further in the ATO for the Toda lattice(select Toda from the Lattice Models menu).

Soliton theory refers to the study of special solutions (solitons) of certain nonlinear wave equations such as the Kortewegde Vries (KdV) equation. These solutions have unexpectedly persistent behavior (in contrast to randomly chosennonlinear waves, which either disperse or break after a sufficiently long time). It turns out that the differential equations of the FPU lattice can be considered as a discrete
approximation to the KdV equation; hence it is plausiblethat the FPU lattice admits soliton-like solutions as well.
References[Pa] R.S. Palais, The Symmetries of Solitons, Bull.A.M.S.
34, 1997 pp. 339-403.(See pages 355-359 for a discussionof the FPU experiments).
[To1] M. Toda, Nonlinear Waves and Solitons., Mathematics and its Applications, 5, Kluwer, 1989.
[To2] M. Toda, Theory of Nonlinear Lattices, Springer Series in Solid-State Sciences, 20, Springer-Verlag, 1989.
[Ul] S.M. Ulam, Adventures of a Mathematician, University of California Press, 1991. (See chapter 12.)[We] T.P. Weissert, The Genesis of Simulation in Dynamics: Pursuing the Fermi-Pasta-Ulam Problem, Springer,
1997. (This recounts the history of the FPU lattice andthe ensuing mathematical and philosophical fallout).
R.S.P.

Lattice TOC

The TODA Lattice *Background
The Toda lattice is named after Morikazu Toda, who discovered in the 1960s that the differential equations for alattice with internal force T (y) = α(eβy − 1) (with α, βconstant) admit solutions which can be written in termsof elliptic functions. This extends the fact that the standard lattice with linear internal force T (y) = ky can besolved using trigonometric functions.
Since nonlinear ODE are usually much more complicatedthan linear ODE, the very fact that it admits explicit solutions at all means that the Toda lattice is already a remarkable example. (The Fermi-Pasta-Ulam lattice, in contrast,
has no analogous explicit solutions.) A few years later, apossible explanation of this property appeared, when it was
discovered that the Toda lattice is an example of a “completely integrable Hamiltonian system”. This, in turn, led
to an explanation of the unexpectedly simple behaviour of
the Fermi-Pasta-Ulam lattice (see the ATO for the FermiPasta-Ulam lattice): on the one hand, for a completely integrable Hamiltonian system, (almost) periodic behaviour
can be predicted, and on the other hand, the Fermi-PastaUlam lattice can be regarded as an approximation to the
Toda lattice when the vibrations are small. We shall say* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Lattice TOC

more about completely integrable Hamiltonian systems below.
How To View the Demonstration
It can be seen that the Toda lattice behaves in a very
similar way to the Fermi-Pasta-Ulam lattice. Instead ofthermalizing, the lattice motion appears to be periodic.
Further Aspects
Mathematically, however, the Toda lattice is much easier
to analyze than the Fermi-Pasta-Ulam lattice, because it is
a completely integrable Hamiltonian system. This means,practically speaking, that it has the maximum possible
number of conserved quantities. For a system of n ordinary differential equations of second order, this maximum
number is n. For example, for the equation y 00 = −ky 2 ,2the total energy 21 y 0 + 12 ky 2 (kinetic energy plus potential
energy) is a conserved quantity, i.e., it is constant when yis a solution of the differential equation, and there are noothers. The standard linear lattice is of this type; each of
the normal mode energies is a conserved quantity, because
each represents the total energy of an uncoupled (independent) oscillator.
The fact that the Toda lattice has the maximum possible
number of conserved quantities is not obvious, and certainly not on physical grounds. A mathematical explanation comes from the fact that the equations of motion may
be written in the form L0 = [L, M ], where L and M arematrix functions. This type of equation is called a Lax
equation (after Peter Lax), and the Lax equation for the
Toda lattice was discovered by Hermann Flaschka in the
1970s. For any Lax equation, the eigenvalues of the matrix function L turn out to be conserved quantities, andthis gives the required number of conserved quantities for
the Toda lattice—though their physical meaning remains
unclear.
The conserved quantities greatly constrain the motion of
the system. In fact, the Arnold-Liouville Theorem saysthat the motion of an n-dimensional completely integrable
Hamiltonian system must, if it is bounded (as in our case),be equivalent to linear motion on an n-dimensional torus.
When n = 1 this means that the motion must be periodic.
When n = 2 (and similarly if n > 2) the motion must
be expressable as (e2πiat , e2πibt ) for some real numbers a,b. If a/b is rational the motion is periodic, otherwise it isalmost-periodic.
Although the Fermi-Pasta-Ulam lattice and the Toda lattice are approximately the same when the vibrations are
small, it can be shown that the Fermi-Pasta-Ulam lattice
is not a completely integrable Hamiltonian system. Nevertheless, the almost periodic motion of the Toda lattice
is inherited by the Fermi-Pasta-Ulam lattice, at least forsmall vibrations, and this explains the absence of thermalization in both cases.
Completely integrable Hamiltonian systems are important,but quite special, and it is an important area of current

research to identify and study more general types of “integrable systems”.
References[Ar] V. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Math. 60, Springer, 1978.
[To1] M. Toda, Nonlinear Waves and Solitons., Mathematics and its Applications, 5, Kluwer, 1989.
[To2] M. Toda, Theory of Nonlinear Lattices, Springer Series in Solid-State Sciences, 20, Springer-Verlag, 1989 .
[We] T.P. Weissert, The Genesis of Simulation in Dynamics. Pursuing the Fermi-Pasta-Ulam Problem, Springer,
1997.
R.S.P.

Lattice TOC

Soliton Traveling Waves
Go To Page 1( Click the Names )1.) The Waves Category
2.) The Korteweg-de Vries Equation
3.) The Sine-Gordon Equation
4.) The Nonlinear Schrödinger Equation

The Waves Category *The subject of wave phenomena is an exceedingly rich andvaried one, playing an important rôle in everyday life, inthe pure and applied sciences, and in many areas of mathematics. Water waves can be small ripples generated bya passing breeze or enormous, destructive tsunamis generated by earthquakes, which themselves are waves thattravel in the Earth’s crust. Maxwell’s Equations of electromagnetism are wave equations that describe the behavior of light, and of the signals responsible for radio, TV,
and the World Wide Web. The probability waves of quantum theory govern the behavior of elementary particles atthe smallest spatial scales, and the search for gravitational
waves at the largest scales is a major concern of modern
cosmology.
Our goal in the 3D-XplorMath Wave Category is to show
examples of waves from a fairly limited region of this vastlandscape. Namely, the program displays the evolution intime of a one-dimensional “wave-form”, by which we meana real or complex valued function u(x, T ) of the “time”, T ,and a single “space” variable x. At a time T , the wave has
a given “shape”, the graph of the function x 7→ u(x, T ),and the program displays this graph with x as abscissa andu(x, T ) as ordinate.
[NOTE: The resolution of this graph is tResolution, and its* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Waves TOCdomain runs from tMin to tMax. That is to say, the interval [tMin,tMax] is divided in tResolution points xi , i =(tMax−tMin)
1, . . . , tResolution with spacing xStep = (tResolution−1),
and the graph is plotted at the points (xi , yi ) where xi =tMin+(i − 1) xStep and yi = u(xi , T ), and then these plot
points are joined into a polygonal graph. Note that the tof tMax tMin and tResolution have nothing to do with thetime T ! Thus to make the graph smoother, you must increase tResolution in the Set Resolution & Scale... dialog,
and to change the domain of the graph, you must changetMin and tMax in the Set t,u,v Ranges... dialog.]
The time evolution of the wave-form is shown by the standard flip-book animation technique; i.e., we plot the abovegraph forT = InitialTime + k·StepSize, k = 0, 1, . . .,until the user clicks the mouse. The variables InitialTime
and StepSize are set in the dialog brought up by choosing
ODE Settings... from the Settings menu. To make theanimation slower (but smoother) decrease StepSize, andconversely increase StepSize to make the wave-form evolution proceed more rapidly, but more jerkily.
As usual, formulas defining a wave form u(x, T ) can dependon the parameters aa,bb, ..., ii as well as on x and T , andfor each of the canned wave-forms these formulas can bechecked by choosing About This Object... from the Waves
main menu.
In trying to understand features of a wave-form, it helps

to see not only the animated evolution as above, but also
to display the graph of the function u(x, T ) as a surface,over the (x, T )-plane or to show “time-slices” of this graph.
Therefore the Waves menu has choices to permit the user
to switch between these display methods. (These two alternate formats, being three dimensional objects, can beviewed in stereo.)
There is also a User Wave Form... item in the Wave menu,
which brings up a dialog permitting the user to enter aformula for u(x, t) as a function of x, t,aa, . . . ,ii.
Interesting wave-forms generally arise as solutions of socalled “wave equations”. These are partial differential equations of evolution type for a function u(x, t). That is,
the function u(x, t) is determined as the unique solutionof a PDE satisfying some initial conditions. Perhaps thesimplest example is the so-called “linear advection equation” ut + vux = 0 (where v is some constant “velocity”). This has the general solution u(x, t) = f (x − vt).
But there are also many interesting non-linear wave equations, in particular the so-called soliton equations, including the Korteweg-DeVries (KDV), Sine-Gordon, and Cubic
Schroedinger equations. Many of our examples are puresoliton solutions of these latter examples.
For a fairly detailed account of the elementary theory ofwave equations, select Introduction to Wave Equations fromthe Topics submenu of the Documentation menu.
R.S.P.

Waves TOCThe Korteweg-de Vries Equation *NOTE: For a fuller understanding of the following, you may find it helpful to first read“About This Category” from the Documentation Menu and “Introduction to Wave Equations”
from its Topics submenu.

The partial differential equation,ut (x, t) + u(x, t)ux (x, t) + uxxx (x, t) = 0
for a real-valued function, u, of two real variables x and t(space and time) is known as the Korteweg-de Vries Equation (or simply KdV). It was first derived in 1895 by D.J.
Korteweg and G. de Vries to model water waves in a shallow canal. Their goal was to settle a long-standing question; namely whether a solitary wave could persist underthose conditions. Based on personal observations of suchwaves in the 1830’s, the naturalist John Scott Russell insisted that such waves do occur, but several prominent
mathematicians, including Stokes, were convinced they were
impossible.
Korteweg and de Vries proved Russell was correct by finding explicit, closed-form, traveling-wave solutions to theirequation that moreover decay rapidly spatially, and so represent a highly localized moving lump. In fact, they founda one-parameter family of such solutions:u(x, t) = 2a2 sech 2 (a(x − 4a2 t),(in 3DXM: a = cc)* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/

Waves TOC

and these are the traveling wave solutions whose evolutions are displayed by 3D-XplorMath, with its parameter
cc playing the rôle of the constant a. Note that the wave’s
velocity, 4a2 , is proportional to its amplitude, 2a2 , meaning that the taller waves of this family move faster. Boththe fact that such a solution to a non-linear equation could
exist and that one could write it in explicit form werelater recognized to be highly important, although originally these facts were relatively unnoticed.
The KdV equation did not receive significant further attention until 1965, when N. Zabusky and M. Kruskal published results of their numerical experimentation with theequation. Their computer generated approximate solutions to the KdV equation indicated that any localized
initial profile, when allowed to develop according to KdV,
asymptotically in time evolved into a finite set of localized
traveling waves of the same shape as the original solitary
waves discovered in 1895. Furthermore, when two of the localized disturbances collided, they would emerge from thecollision as another pair of traveling waves with a shift inphase as the only consequence of their interaction *. Since
the “solitary waves” which made up these solutions seemed
to behave like particles, Zabusky and Kruskal coined thename “soliton” to describe them. The formula
12 gg (3 + 4 cosh(2x − 8t) + cosh(4x − 64t)pu(x, t) =(3 cosh(x − 28t) + cosh(3x − 36t))* To see this phase shift, choose Display as Graph from the Action Menu.

is one of these solutions, consisting of two lumps and therefor called two-soliton solution of KdV, and it is one of the
KdV solutions whose evolution can be displayed using 3DXplorMath. Shortly after that, another remarkable discovery was made concerning KdV; a paper by C. Gardner,
J. Greene, M. Kruskal, and R. Miura demonstrated that
it was possible to derive many exact solutions to the equation by using ideas from scattering theory. In particular,
the solutions whose evolution are shown by 3D-XplorMath
are exact solutions that can be found by this method.
Using modern terminology, we could say that these authors
showed that KdV was an integrable non-linear partial differential equation. It was the first to be discovered, butsince then, many other such equations have been foundto be integrable and admit soliton solutions, in particular the Sine-Gordon Equation (SGE) and the Nonlinear
Schrödinger Equation (NLS)—solutions to both of which
are also shown by 3D-XplorMath. However, KdV is usually considered the canonical example, in part because it
was the first equation known to have these properties.
Alex Kasman

Waves TOCThe Shepard Tones Demo
Go To Page 1( Click the Names )1.) About this Demo
2.) What Are The Shepard Tones?

The Sound Category *While all the other 3D-XplorMath categories are concernedwith the visualization of mathematics—and so make useof your eyes and sense of sight, in this one you will beusing your ears and your sense of hearing. For this reason
it is important that while using with this category yourcomputer should be attached to a reasonably high fidelity
sound system—either headphones or stereo loudspeakers(preferably with a subwoofer).
So far the only exhibit in the Sound Category is the Shepard Tones. For an explanation of this exhibit, select About
This Object after selecting Shepard Tones from the Soundmenu of 3D-XPlorMath or see the following text.

* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Shepard Tones

Shepard Tones *Ascending and Descending by M. Escher* This file is from the 3D-XplorMath project. Please see:
http://3D-XplorMath.org/
Shepard Tones

If you misread the name of this exhibit, you may have expected to hear some pretty music played on a set of shepherd’s pipes. Instead what you heard was a sequence ofnotes that probably sounded rather unmusical, and may
have at first appeared to keep on rising indefinitely, “onenote at a time”. But soon you must have noticed that it
was getting nowhere, and eventually you no-doubt realizedthat it was even “cyclic”, i.e., after a full scale of twelve
notes had been played, the sound was right back to where
it started from! In some ways this musical paradox is an
audible analog of Escher’s famous Ascending and Descending drawing.
While this strange “ever-rising note” had precursors, inthe form in which it occurs in 3D-XplorMath, it was firstdescribed by the psychologist Roger N. Shepard in a paper titled Circularity in Judgements of Relative Pitch published in 1964 in the Journal of the Acoustical Society ofAmerica.
To understand the basis for this auditory illusion, it helps
to look at the sonogram (next page) that is shown while
the Shepard Tones are playing. What you are seeing is agraph in which the abcsissa represents frequency (in Hertz)
and the ordinate represents the intensity at which a soundat a given frequency is played. Note that the Gaussian orbell curve in this diagram shows the intensity envelope atwhich all sounds of a Shepard tone are played.

A single Shepard tone consists of the same “note” played
in seven different octaves—namely the octave containing
A above middle C (the center of our bell curve) and threeoctaves up and three octaves down. Thus the first or “A”
Shepard tone consists of simultaneously playing the following frequencies (in Hertz): 55, 110, 220, 440, 880 1760,
3520, with each frequency being played at the intensity
given by the height of the Gaussian at that frequency. Of
course, what our ear hears it interprets as the note A of thescale—not the pure tone of A in some particular octave,
but rather a rich harmonic mixture of the A notes fromseven adjacent octaves of which the central one, A above
middle C, sounds most strongly, while the others get lessintense as they go up or down from there.
To get the next (or “B”) Shepard tone, each note of theabove mixture is increased by a “half-tone”, that is, itsfrequency is multiplied by the twelfth root of two. And ofcourse the same bell curve again gives the intensity of eachnote of the mixture. Now our ear hears the note B (again
as a rich harmonic mixture) and by our long experience
with the scale, it seems that we have now gone “up” onehalf-tone from the first note.
Well, you can take it from there. To get the next (third)in the sequence of Shepard Tones, we again multiply every
frequency of the B tone by twelfth root of two (i.e., “ascend
a half-tone”) and play each tone at the appropriate intensity for that frequency, as given by the bell curve. Again
it feels to us that this third Shepard tone is a half-tone“higher” than the second one. And so on for the next nine
Shepard tones.
But then as you will see we are right back to where we
started!
R.S.P.
Shepard Tones