aarjaneiro/planetmath huggingface dataset:
circle of curvature

circle of curvature

Let a given

plane curve

γ

have a definite

curvature

(

http://planetmath.org/CurvaturePlaneCurve

)

κ

in
a point

P

of

γ

.  The

circle of curvature

of

γ

in the
point

P

is the circle which has the radius

1

|

κ

|

and which has with the curve the common

tangent

in

P

and which
in a neighbourhood of

P

is on the same side as the curve.

The radius

ϱ

of the circle of curvature is the

radius of curvature

of

γ

in

P

.  The center of the
circle of curvature is the

center of curvature

of

γ

in

P

.

When the curve

γ

is given in the

parametric form

x

=

x

⁢

(

t

)

,

y

⁢

Ã

⁢



⁢

Â

⁢

´

=

y

⁢

(

t

)

,

the

coordinates

of the center of curvature belonging to the point

(

x

,

y

)

of the curve are

ξ

=

x

-

(

x

′

⁣

2

+

y

′

⁣

2

)

⁢

y

′

x

′

⁢

y

′′

-

x

′′

⁢

y

′

,

η

=

y

+

(

x

′

⁣

2

+

y

′

⁣

2

)

⁢

x

′

x

′

⁢

y

′′

-

x

′′

⁢

y

′

.

Example.

Since the curvature of the

parabola

y

=

x

2

in the origin is

-

2

,  the corresponding radius of curvature is

1

2

and the center of curvature

(

0

,

1

2

)

.

Furthermore, it is possible to define the circle of curvature without
first knowing about curvature of the curve. (In fact, using this

definition

,
one could reverse the procedure and define curvature as the radius of
the circle of curvature.) We may define the circle of curvature
a point

P

of

γ

as the unique circle

passing through

P

which
makes a second-order contact with

γ

at

P

.

Title

circle of curvature

Canonical

name

CircleOfCurvature

Date of creation

2013-03-22 16:59:46

Last modified on

2013-03-22 16:59:46

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53A04

Synonym

osculating circle

Related topic

CurvatureOfaCircle

Related topic

OsculatingCurve

Defines

radius of curvature

Defines

center of curvature

Defines

centre of curvature

Generated on Fri Feb 9 22:19:01 2018 by

LaTeXML
circular helix

circular helix

The

space curve

traced out by the parameterization

𝜸

⁢

(

t

)

=

[

a

⁢

cos

⁡

(

t

)

a

⁢

sin

⁡

(

t

)

b

⁢

t

]

,

t

∈

ℝ

,

a

,

b

∈

ℝ

is called a

circular helix

(plur.

helices

).

Its

Frenet frame

is:

𝐓

=

1

a

2

+

b

2

⁢

[

-

a

⁢

sin

⁡

t

a

⁢

cos

⁡

t

b

]

,

𝐍

=

[

-

cos

⁡

t

-

sin

⁡

t

0

]

,

𝐁

=

1

a

2

+

b

2

⁢

[

b

⁢

sin

⁡

t

-

b

⁢

cos

⁡

t

a

]

.

Its

curvature

and

torsion

are the following

constants

:

κ

=

a

a

2

+

b

2

,

τ

=

b

a

2

+

b

2

.

A circular helix can be conceived of as a space curve with constant,
non-zero curvature, and constant, non-zero torsion. Indeed, one can
show that if a space curve

satisfies

the above constraints, then there
exists a system of

Cartesian coordinates

in which the curve has a
parameterization of the form shown above.

Figure 1:

A plot of a circular helix with

a

=

b

=

1

, and

κ

=

τ

=

1

/

2

.

An important

property

of the circular helix is that for any point of it, the angle

φ

between its

tangent

and the helix axis is constant. Indeed, if we consider the position vector of that arbitrary point, we have (where

𝐤

is the unit vector

parallel

to helix axis)

d

⁢

𝜸

d

⁢

t

⋅

𝐤

=

[

-

a

⁢

sin

⁡

t

a

⁢

cos

⁡

t

b

]

⁢

[

0 0 1

]

=

b

≡

∥

d

⁢

𝜸

d

⁢

t

∥

⁢

cos

⁡

φ

=

a

2

+

b

2

⁢

cos

⁡

φ

.

Therefore,

cos

⁡

φ

=

b

a

2

+

b

2

⁢

constant

,

as was to be shown.

There is also another

parameter

, the so-called

pitch of the helix

P

which is the

separation

between two consecutive turns.
(It is mostly used in the manufacture of screws.)
Thus,

P

=

γ

3

⁢

(

t

+

2

⁢

π

)

-

γ

3

⁢

(

t

)

=

b

⁢

(

t

+

2

⁢

π

)

-

b

⁢

t

=

2

⁢

π

⁢

b

,

and

P

is also a constant.

Title

circular helix

Canonical

name

CircularHelix

Date of creation

2013-03-22 13:23:25

Last modified on

2013-03-22 13:23:25

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53A04

Related topic

SpaceCurve

Related topic

RightHandedSystemOfVectors

Defines

circular helices

Generated on Fri Feb 9 22:19:04 2018 by

LaTeXML
curvature of a circle

curvature of a circle

Let

C

r

be a circle of radius

r

centered at the origin.

A

canonical

parameterization of the curve is (counterclockwise)

g

⁢

(

s

)

=

r

⁢

(

cos

⁡

(

s

r

)

,

sin

⁡

(

s

r

)

)

for

s

∈

(

0

,

2

⁢

π

⁢

r

)

(actually this leaves out the point

(

r

,

0

)

but this could be treated via another parameterization taking

s

∈

(

-

π

⁢

r

,

π

⁢

r

)

)

Differentiating the parameterization we get

𝐓

=

g

′

⁢

(

s

)

=

(

-

sin

⁡

(

s

r

)

,

cos

⁡

(

s

r

)

)

and this results in the normal

𝐍

=

J

⋅

𝐓

=

-

(

cos

⁡

(

s

r

)

,

sin

⁡

(

s

r

)

)

=

-

g

⁢

(

s

)

r

Differentiating

g

a second time we can calculate the

curvature

𝐓

′

=

-

1

r

⁢

(

cos

⁡

(

s

r

)

,

sin

⁡

(

s

r

)

)

=

1

r

⁢

𝐍

and by

definition

𝐓

′

=

k

⁢

𝐍

∴

k

=

1

r

and thus the curvature of a circle of radius

r

is

1

r

provided that the

positive

direction on the circle is anticlockwise; otherwise it is

-

1

r

.

Title

curvature of a circle

Canonical name

CurvatureOfACircle

Date of creation

2013-03-22 15:50:30

Last modified on

2013-03-22 15:50:30

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

9

Author

cvalente (11260)

Entry type

Example

Classification

msc 53A04

Related topic

circle

Related topic

curvature

Related topic

Connection

Related topic

CircleOfCurvature

Generated on Fri Feb 9 22:19:09 2018 by

LaTeXML
curvature of Nielsen’s spiral

curvature of Nielsen’s spiral

Nielsen’s spiral

is the

plane curve

defined in the
parametric form

x

=

a

⁢

ci

⁢

t

,

y

=

a

⁢

si

⁢

t

(1)

where

a

is a non-zero constant, “

ci

” and
“

si

” are the

cosine integral

(

http://planetmath.org/sineintegral

)
and the

sine integral

(

http://planetmath.org/sineintegral

)
and

t

is the

parameter

(

http://planetmath.org/Parametre

) (

t

>

0

).

We determine the

curvature

(

http://planetmath.org/CurvaturePlaneCurve

)

κ

of this curve using the expression

κ

=

x

′

⁢

y

′′

-

y

′

⁢

x

′′

[

(

x

′

)

2

+

(

y

′

)

2

]

3

/

2

.

(2)

The

first derivatives

of (1) are

x

′

=

d

d

⁢

t

⁢

(

a

⁢

∫

∞

t

cos

⁡

u

u

⁢

𝑑

u

)

=

a

⁢

cos

⁡

t

t

,

(3)

y

′

=

d

d

⁢

t

⁢

(

a

⁢

∫

∞

t

sin

⁡

u

u

⁢

𝑑

u

)

=

a

⁢

sin

⁡

t

t

,

(4)

and hence the second derivatives

x

′′

=

-

a

⋅

t

⁢

sin

⁡

t

+

cos

⁡

t

t

2

,

y

′′

=

a

⋅

t

⁢

cos

⁡

t

-

sin

⁡

t

t

2

.

Substituting the

derivatives

in (2) yields

κ

=

a

2

⋅

(

cos

⁡

t

)

⁢

(

t

⁢

cos

⁡

t

-

sin

⁡

t

)

+

(

sin

⁡

t

)

⁢

(

t

⁢

sin

⁡

t

+

cos

⁡

t

)

t

⋅

t

2

:

(

a

2

⁢

cos

2

⁡

t

+

a

2

⁢

sin

2

⁡

t

t

2

)

3

2

,

which is easily simplified to

κ

=

t

a

.

(5)

The

arc length

(

http://planetmath.org/ArcLength

) of Nielsen’s spiral can also be obtained in a

closed form

(

http://planetmath.org/ClosedForm4

); using (3) and (4) we get:

s

=

∫

1

t

x

′

⁣

2

+

y

′

⁣

2

⁢

𝑑

t

=

∫

1

t

a

2

⁢

cos

2

⁡

t

t

2

+

a

2

⁢

sin

2

⁡

t

t

2

⁢

𝑑

t

=

∫

1

t

a

t

⁢

𝑑

t

,

i.e.

s

=

a

⁢

ln

⁡

t

.

(6)

Note.

The expressions for

x

′

and

y

′

allow us determine as well

d

⁢

y

d

⁢

x

=

y

′

x

′

=

sin

⁡

t

cos

⁡

t

=

tan

⁡

t

,

which says that the sense of the parameter

t

is the slope angle of the

tangent line

of the Nielsen’s spiral.

Figure 1:

Plot of Nielsen’s spiral for

2

≤

t

≤

50

.
Axis scaling is in units of

a

.

(Octave / MATLAB

http://aux.planetmath.org/files/objects/9350/nielsen.m

source program for plot;
in

http://aux.planetmath.org/files/objects/9350/nielsen.pdf

PDF format)

Title

curvature of Nielsen’s spiral

Canonical name

CurvatureOfNielsensSpiral

Date of creation

2015-02-06 12:53:54

Last modified on

2015-02-06 12:53:54

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

22

Author

pahio (2872)

Entry type

Example

Classification

msc 53A04

Synonym

arc length of Nielsen’s spiral

Related topic

CosineIntegral

Related topic

SineIntegral

Related topic

FamousCurvesInThePlane

Related topic

DerivativeForParametricForm

Defines

Nielsen’s spiral

Generated on Fri Feb 9 22:19:12 2018 by

LaTeXML
curvature (space curve)

curvature (space curve)

Let

I

⊂

ℝ

be an

interval

, and let

γ

:

I

→

ℝ

3

be
an

arclength parameterization

of an oriented space curve, assumed to
be

regular

, and free of

points of inflection

. We interpret

γ

⁢

(

t

)

as
the

trajectory

of a particle moving through 3-dimensional space. Let

T

⁢

(

t

)

,

N

⁢

(

t

)

,

B

⁢

(

t

)

denote the corresponding

moving trihedron

. The
speed of this particle is given by

v

⁢

(

t

)

=

∥

γ

′

⁢

(

t

)

∥

.

The quantity

κ

⁢

(

t

)

=

∥

T

′

⁢

(

t

)

∥

v

⁢

(

t

)

=

∥

γ

′

⁢

(

t

)

×

γ

′′

⁢

(

t

)

∥

∥

γ

′

⁢

(

t

)

∥

3

is called the

curvature

of the space curve. It is

invariant

with respect to
reparameterization, and is therefore a measure of an intrinsic

property

of the curve, a

real number

geometrically assigned to the point

γ

⁢

(

t

)

. If one parameterizes the curve with respect to the arclength

s

, one gets the more concise

relation

that

κ

⁢

(

s

)

=

1

⋅

∥

γ

′′

⁢

(

s

)

∥

⋅

sin

⁡

π

2

1

3

=

∥

γ

′′

⁢

(

s

)

∥

.

Physically, curvature may be conceived as the ratio of the normal

acceleration

of a particle to the particle’s speed. This ratio
measures the degree to which the curve deviates from the

straight line

at a particular point. Indeed, one can show
that of all the circles

passing through

γ

⁢

(

t

)

and lying on the

osculating plane

, the one of radius

1

/

κ

⁢

(

t

)

serves as the best
approximation to the space curve at the point

γ

⁢

(

t

)

.

To treat curvature analytically, we take the

derivative

of the relation

γ

′

⁢

(

t

)

=

v

⁢

(

t

)

⁢

T

⁢

(

t

)

.

This yields the following
decomposition of the acceleration vector:

γ

′′

⁢

(

t

)

=

v

′

⁢

(

t

)

⁢

T

⁢

(

t

)

+

v

⁢

(

t

)

⁢

T

′

⁢

(

t

)

=

v

⁢

(

t

)

⁢

{

(

log

⁡

v

)

′

⁢

(

t

)

⁢

T

⁢

(

t

)

+

κ

⁢

(

t

)

⁢

N

⁢

(

t

)

}

.

Thus, to change speed,
one needs to apply acceleration along the

tangent vector

; to change
heading the acceleration must be applied along the normal.

Title

curvature (space curve)

Canonical

name

CurvaturespaceCurve

Date of creation

2013-03-22 12:14:58

Last modified on

2013-03-22 12:14:58

Owner

slider142 (78)

Last modified by

slider142 (78)

Numerical id

13

Author

slider142 (78)

Entry type

Definition

Classification

msc 53A04

Synonym

curvature

Related topic

SpaceCurve

Related topic

Torsion

Related topic

ExpressionsForCurvatureAndTorsion

Related topic

SerretFrenetFormulas

Generated on Fri Feb 9 22:19:18 2018 by

LaTeXML
DNA inequality

DNA inequality

Given

Γ

, a convex

simple closed curve

(

http://planetmath.org/Curve

) in the plane, and

γ

a

closed curve

contained in

Γ

, then

M

⁢

(

Γ

)

≤

M

⁢

(

γ

)

where

M

is the

mean curvature

function.

This was a conjecture due to S. Tabachnikov and was proved by Lagarias and Richardson of Bell Labs. The idea of the proof was to show that there was a way you could reduce a curve to the boundary of its convex hull so that if it holds for the boundary of the convex hull, then it holds for the curve itself.

Conjecture

:

Equality holds iff

Γ

and

γ

coincide

.

It’s amazing how many questions are still open in the Elementary Differential Geometry of curves and

surfaces

. Questions like this often serve as a great research opportunity for undergraduates. It is also interesting to see if you could extend this result to curves on surfaces:

Theorem : If

Γ

is a circle on

S

2

, and

γ

is a closed curve contained in

Γ

then

M

⁢

(

Γ

)

≤

M

⁢

(

γ

)

.

It is not known whether this result holds for

Γ

a simple closed convex curve on

S

2

.

It is known also that this inequality does not hold in the

hyperbolic plane

.

Title

DNA inequality

Canonical name

DNAInequality

Date of creation

2013-03-22 15:31:14

Last modified on

2013-03-22 15:31:14

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

12

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 53A04

Generated on Fri Feb 9 22:19:30 2018 by

LaTeXML
easy calculation of the area of an ellipse

easy calculation of the area of an ellipse

Consider the unit circle

{

(

x

,

y

)

∈

ℝ

2

:

x

2

+

y

2

≤

1

}

. It’s a well known fact that the area of this set is

π

.

Now consider the following

linear transformation

(

x

,

y

)

→

(

u

,

v

)

=

(

a

⁢

x

,

b

⁢

y

)

.

The

determinant

of the

transformation

is

a

⁢

b

and the transformed circle is:

{

(

u

,

v

)

∈

ℝ

2

:

(

u

a

)

2

+

(

v

b

)

2

≤

1

}

an ellipse of axis

(

a

,

b

)

.

Now since the

Jacobian

of the transformation is

constant

, the change of variables in integral

theorem

(

http://planetmath.org/ChangeOfVariablesInIntegralOnMathbbRn

) allows us to say the area of the transformed set is

a

⁢

b

times the area of the original set.

Thus, the area of an ellipse is

π

⁢

a

⁢

b

.

Title

easy calculation of the area of an ellipse

Canonical name

EasyCalculationOfTheAreaOfAnEllipse

Date of creation

2013-03-22 15:44:18

Last modified on

2013-03-22 15:44:18

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

7

Author

cvalente (11260)

Entry type

Definition

Classification

msc 53A04

Generated on Fri Feb 9 22:19:46 2018 by

LaTeXML
equal arc length and area

equal arc length and area

We want to determine the nonnegative

differentiable

real functions

x

↦

y

whose graph has the property that the

arc length

between any two points of it is the same as the area (

http://planetmath.org/AreaOfPlaneRegion

) by the curve, the

x

-axis and the

ordinate

lines of those points.

The requirement leads to the equation

∫

a

x

1

+

(

d

⁢

y

d

⁢

x

)

2

⁢

𝑑

x

=

∫

a

x

y

⁢

𝑑

x

.

(1)

By the

fundamental theorem of calculus

, we infer from (1) the

differential equation

1

+

(

d

⁢

y

d

⁢

x

)

2

=

y

,

(2)

whence

d

⁢

y

d

⁢

x

=

y

2

-

1

.  In the case

y

≢

1

,  the

separation of variables

yields

∫

𝑑

x

=

∫

d

⁢

y

y

2

-

1

,

i.e.

x

+

C

=

arcosh

⁡

y

.

Consequently, the equation (2) has the

general solution

y

=

cosh

⁡

(

x

+

C

)

(3)

and the

singular solution

y

≡

1

.

(4)

The functions defined by (3) and (4) are the only satisfying the given requirement.  The graphs are a

chain curve

(which may be translated in the horizontal direction) and a line

parallel

to the

x

-axis.  Evidently, the line is the

envelope

of the

integral curves

given be the general solution.

Title

equal arc length and area

Canonical name

EqualArcLengthAndArea

Date of creation

2013-03-22 19:13:36

Last modified on

2013-03-22 19:13:36

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Example

Classification

msc 53A04

Classification

msc 34A34

Classification

msc 34A05

Classification

msc 26A09

Synonym

equal area and arc length

Related topic

Arcosh

Related topic

HyperbolicFunctions

Related topic

ChainCurve

Generated on Fri Feb 9 22:20:08 2018 by

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example of curvature (space curve)

example of curvature (space curve)

Example

space curves

and calculating their

curvatures

using the formula

κ

⁢

(

t

)

=

∥

𝐫

′

⁢

(

t

)

×

𝐫

′′

⁢

(

t

)

∥

∥

𝐫

′

⁢

(

t

)

∥

3

1.

𝐫

⁢

(

t

)

=

3

⁢

t

⁢

i

^

+

t

2

⁢

j

^

-

4

⁢

t

2

⁢

k

^

the

first derivative

𝐫

′

⁢

(

t

)

=

3

⁢

i

^

+

2

⁢

t

⁢

j

^

-

8

⁢

t

⁢

k

^

vector magnitude

of the

derivative

∥

𝐫

′

⁢

(

t

)

∥

=

3

2

+

(

2

⁢

t

)

2

+

(

-

8

⁢

t

)

2

∥

𝐫

′

⁢

(

t

)

∥

=

9

+

4

⁢

t

2

+

16

⁢

t

2

=

9

+

20

⁢

t

2

the second derivative

𝐫

′′

⁢

(

t

)

=

2

⁢

j

^

-

8

⁢

k

^

the

cross product

𝐫

′

⁢

(

t

)

×

𝐫

′′

⁢

(

t

)

=

|

i

^

j

^

k

^

3

2

⁢

t

-

8

⁢

t

0

2

-

8

|

=

(

-

16

⁢

t

+

16

⁢

t

)

⁢

i

^

-

(

-

24

)

⁢

j

^

+

6

⁢

k

^

𝐫

′

⁢

(

t

)

×

𝐫

′′

⁢

(

t

)

=

24

⁢

j

^

+

6

⁢

k

^

∥

𝐫

′

⁢

(

t

)

×

𝐫

′′

⁢

(

t

)

∥

=

576

+

36

=

612

=

2

⁢

153

∥

𝐫

′

⁢

(

t

)

∥

3

=

(

9

+

20

⁢

t

2

)

3

/

2

κ

⁢

(

t

)

=

2

⁢

153

(

9

+

20

⁢

t

2

)

3

/

2

2.

Calculate the curvature of the right

circular helix

as given in the plot below and defined as

𝐫

⁢

(

t

)

=

cos

⁡

t

⁢

i

^

+

sin

⁡

t

⁢

j

^

+

t

⁢

k

^

𝐫

′

⁢

(

t

)

=

-

sin

⁡

t

⁢

i

^

+

cos

⁡

t

⁢

j

^

+

k

^

∥

𝐫

′

⁢

(

t

)

∥

=

sin

2

⁡

t

+

cos

2

⁡

t

+

1

2

=

2

𝐫

′′

⁢

(

t

)

=

-

cos

⁡

t

⁢

i

^

-

sin

⁡

t

⁢

j

^

𝐫

′

⁢

(

t

)

×

𝐫

′′

⁢

(

t

)

=

|

i

^

j

^

k

^

-

sin

⁡

t

cos

⁡

t

1

-

cos

⁡

t

-

sin

⁡

t

0

|

=

sin

⁡

t

⁢

i

^

-

cos

⁡

t

⁢

j

^

+

(

sin

2

⁡

t

+

cos

2

⁡

t

)

⁢

k

^

𝐫

′

⁢

(

t

)

×

𝐫

′′

⁢

(

t

)

=

sin

⁡

t

⁢

i

^

-

cos

⁡

t

⁢

j

^

+

k

^

∥

𝐫

′

⁢

(

t

)

×

𝐫

′′

⁢

(

t

)

∥

=

sin

2

⁡

t

+

cos

2

⁡

t

+

1

2

=

2

∥

𝐫

′

⁢

(

t

)

∥

3

=

2

3

/

2

κ

⁢

(

t

)

=

2

2

3

/

2

=

1

2

Title

example of curvature (space curve)

Canonical name

ExampleOfCurvaturespaceCurve

Date of creation

2013-03-22 15:40:58

Last modified on

2013-03-22 15:40:58

Owner

bloftin (6104)

Last modified by

bloftin (6104)

Numerical id

8

Author

bloftin (6104)

Entry type

Example

Classification

msc 53A04

Related topic

PositionVector

Generated on Fri Feb 9 22:20:24 2018 by

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expressions for curvature and torsion

expressions for curvature and torsion

For a

regular

(

http://planetmath.org/Curve

),

parameterized curve

α

:

(

a

,

b

)

→

ℝ

3

, not necessarily unit speed, the

curvature

κ

⁢

(

t

)

and

torsion

τ

⁢

(

t

)

are given, respectively, by

κ

⁢

(

t

)

=

∥

α

′

⁢

(

t

)

×

α

′′

⁢

(

t

)

∥

∥

α

′

⁢

(

t

)

∥

3

;

τ

⁢

(

t

)

=

(

α

′

⁢

(

t

)

×

α

′′

⁢

(

t

)

)

⋅

α

′′′

⁢

(

t

)

∥

α

′

⁢

(

t

)

×

α

′′

⁢

(

t

)

∥

2

.

References

John McCleary,

Geometry

from a

Differentiable

Viewpoint

, Cambridge University Press, 1994.

Title

expressions for curvature and torsion

Canonical name

ExpressionsForCurvatureAndTorsion

Date of creation

2013-03-22 15:38:14

Last modified on

2013-03-22 15:38:14

Owner

Simone (5904)

Last modified by

Simone (5904)

Numerical id

7

Author

Simone (5904)

Entry type

Theorem

Classification

msc 53A04

Related topic

Torsion

Related topic

CurvatureOfACurve

Generated on Fri Feb 9 22:20:36 2018 by

LaTeXML
famous curves

famous curves

Closed curves

•

Circle (

http://planetmath.org/circle

)

•

Ellipse

•

Astroid

(

http://planetmath.org/clairautsequation

)

•

Cardioid

•

Cassini oval

(especially:

lemniscate

)

•

graphs of spherical functions

•

Lissajous curves

(

http://planetmath.org/lissajouscurves

)

Periodic curves

•

Sinusoid

•

Cycloid

•

Triangular-wave curve (

http://planetmath.org/triangularwavefunction

)

Unbounded

•

Line (

http://planetmath.org/LineInThePlane

)

•

Hyperbola

•

Parabola

•

Semicubical parabola

•

Cubic hyperbola (

http://planetmath.org/polytrope

)

•

Logarithmic spiral

•

Archimedean spiral

•

Nielsen’s spiral (

http://planetmath.org/curvatureofnielsensspiral

)

•

Gaussian

(

http://planetmath.org/areaundergaussiancurve

) and Lorentzian curves

•

graphs of

Bessel functions

•

graphs of

Legendre polynomials

•

Catenary

•

Tractrix

•

Clothoid

•

Lamé’s cubic (

http://planetmath.org/asymptoteoflamescubic

)

•

Witch of Agnesi

•

Cissoid of Diocles

•

Conchoid of Nicomedes

•

Quadratrix of Hippias

•

graphs of damped sine and

cosine functions

•

graphs of

sinc function

,

sine integral

,

gamma function

,

function

x

x

(

http://planetmath.org/FunctionXx

)

Space curves

•

Circular helix

•

Loxodrome

http://www.mathcurve.com/courbes2d/courbes2d.shtml

ENCYCLOPÉDIE DES FORMES MATHÉMATIQUES REMARQUABLES

Title

famous curves

Canonical name

FamousCurves

Date of creation

2015-02-06 13:05:15

Last modified on

2015-02-06 13:05:15

Owner

matte (1858)

Last modified by

pahio (2872)

Numerical id

51

Author

matte (2872)

Entry type

Topic

Classification

msc 53A04

Related topic

ClairautsEquation

Related topic

CurvatureOfNielsensSpiral

Related topic

Asymptote

Related topic

ArcLengthOfParabola

Generated on Fri Feb 9 22:20:38 2018 by

LaTeXML
Frenet frame

Frenet frame

Let

I

⊂

ℝ

be an

interval

and let

γ

:

I

→

ℝ

3

be a

parameterized space curve

, assumed to be

regular

(

http://planetmath.org/SpaceCurve

) and free of

points of inflection

. We
interpret

γ

⁢

(

t

)

as the

trajectory

of a particle moving through
3-dimensional space. The

moving trihedron

(also known as the Frenet
frame, the Frenet trihedron, the repère mobile, and the

moving
frame

) is an orthonormal basis of 3-vectors

T

⁢

(

t

)

,

N

⁢

(

t

)

,

B

⁢

(

t

)

,

defined
and named as follows:

T

⁢

(

t

)

=

γ

′

⁢

(

t

)

∥

γ

′

⁢

(

t

)

∥

,

the unit tangent;

N

⁢

(

t

)

=

T

′

⁢

(

t

)

∥

T

′

⁢

(

t

)

∥

,

the unit normal;

B

⁢

(

t

)

=

T

⁢

(

t

)

×

N

⁢

(

t

)

,

the unit binormal.

A straightforward application of the

chain rule

shows that these
definitions are covariant with respect to reparameterizations. Hence,
the above three vectors should be conceived as being attached to the
point

γ

⁢

(

t

)

of the oriented space curve, rather than being
functions of the parameter

t

.

Corresponding to the above vectors are 3 planes,

passing through

each
point

γ

⁢

(

t

)

of the space curve. The

osculating plane

at
the point

γ

⁢

(

t

)

is the plane

spanned by

T

⁢

(

t

)

and

N

⁢

(

t

)

; the

normal plane

at

γ

⁢

(

t

)

is the plane spanned by

N

⁢

(

t

)

and

B

⁢

(

t

)

; the

rectifying plane

at

γ

⁢

(

t

)

is the plane spanned by

T

⁢

(

t

)

and

B

⁢

(

t

)

.

Title

Frenet frame

Canonical name

FrenetFrame

Date of creation

2013-03-22 12:15:44

Last modified on

2013-03-22 12:15:44

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

16

Author

rmilson (146)

Entry type

Definition

Classification

msc 53A04

Synonym

moving trihedron

Synonym

moving frame

Synonym

repère mobile

Synonym

Frenet trihedron

Related topic

SpaceCurve

Defines

osculating plane

Defines

normal plane

Defines

rectifying plane

Defines

unit normal

Defines

unit tangent

Defines

binormal

Generated on Fri Feb 9 22:20:41 2018 by

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