包絡

喺幾何上，平面上一族曲線嘅包絡係咁樣一條曲線：喺佢上面每一點處，都有呢族曲線中嘅一條同佢相切。Classically, 包絡上嘅一個點可以認為係兩條「鄰近」曲線嘅交點，即係鄰近曲線嘅交點嘅極限。呢個諗法可以推廣到空間中曲面嘅包絡，同埋更高嘅維數。

一族曲線嘅包絡
設族入面嘅每條曲線Ct由ft(x, y)=0確定，其中t係參數。記 F(t, x, y)=ft(x, y)並且假設F可微。

於是曲線族Ct嘅包絡 is defined as the set of points for which 

for some value of t,
where  is the partial derivative of F with respect to t.

Note that if t and u, t≠u are two values of the parameter then the intersection of the curves Ct and Cu is given by 

or equivalently

Letting u→t gives the definition above.

An important special case is when F(t, x, y) is a polynomial in t. This includes, by clearing denominators, the case where F(t, x, y) is a rational function in t. In this case, the definition amounts to t being a double root of F(t, x, y), so the equation of the envelope can be found by setting the discriminant of F to 0.

For example, let Ct be the line whose x and y intercepts are t and 1−t, this is shown in the animation above. The equation of Ct is

or, clearing fractions,

The equation of the envelope is then

Often when F is not a rational function of the parameter it may be reduced to this case by an appropriate substitution. For example if the family is given by Cθ with an equation of the form u(x, y)cosθ+v(x, y)sinθ=w(x, y), then putting t=eiθ, cosθ=(t+1/t)/2, sinθ=(t-1/t)/2i changes the equation of the curve to

or

The equation of the envelope is then given by setting the discriminant to 0:

or

其他定義
 包絡E1係鄰近曲線Ct交點嘅極限。
 包絡E2係一條同所有Ct相切嘅曲線。
 包絡E3係由Ct所填充嘅區域嘅邊界。

於是乎，, 同埋，其中係由呢篇文開頭嘅定義所確定嘅曲線組成嘅集合。

例

例1

These four definitions of the discriminant, E1,E2, and E3, can lead to different sets. Consider for instance the curve  parametrised by  where . The one-parameter family of curves will be given by the tangent lines to γ.  

First we calculate the discriminant . The generating function is

Calculating the partial derivative . It follows that either  or . First assume that . Substituting into F: 
and so, assuming that t ≠ 0, it follows that  if and only if . Next, assuming that  and substituting into F gives .  So, assuming , it follows that  if and only if . Thus the discriminant is the original curve and its tangent line at γ(0):

Next we calculate E1. One curve is given by  and a nearby curve is given by  where ε is some very small number. The intersection point comes from looking at the limit of  as ε tends to zero. Notice that  if and only if

If  then L has only a single factor of ε. Assuming that  then the intersection is given by

Since  it follows that . The y value is calculated by knowing that this point must lie on a tangent line to the original curve γ: that . Substituting and solving gives y = t3. When , L is divisible by ε2. Assuming that  then the intersection is given by

It follows that , and knowing that  gives . It follows that

Next we calculate E2. The curve itself is the curve that is tangent to all of its own tangent lines. It follows that

Finally we calculate E3.  Every point in the plane has at least one tangent line to γ passing through it, and so region filled by the tangent lines is the whole plane. The boundary E3 is therefore the empty set. Indeed, consider a point in the plane, say (x0,y0). This point lies on a tangent line if and only if there exists a t such that

This is a cubic in t and as such has at least one real solution. It follows that at least one tangent line to γ must pass through any given point in the plane. If  and  then each point (x,y) has exactly one tangent line to γ passing through it. The same is true if  . If  and  then each point (x,y) has exactly three distinct tangent lines to γ passing through it. The same is true if  and . If  and  then each point (x,y) has exactly two tangent lines to γ passing through it (this corresponds to the cubic having one ordinary root and one repeated root). The same is true if  and . If  and , i.e., , then this point has a single tangent line to γ passing through it (this corresponds to the cubic having one real root of multiplicity 3). It follows that

例2
In string art it is common to cross-connect two lines of equally spaced pins. What curve is formed?

For simplicity, set the pins on the x- and y-axes; a non-orthogonal layout is a rotation and scaling away. A general straight-line thread connects the two points (0, k−t) and (t, 0), where k is an arbitrary scaling constant, and the family of lines is generated by varying the parameter t. From simple geometry, the equation of this straight line is y = −(k − t)x/t + k − t. Rearranging and casting in the form F(x,y,t) = 0 gives:

Now differentiate F(x,y,t) with respect to t and set the result equal to zero, to get

These two equations jointly define the equation of the envelope. From (2) we have t = (−y + x + k)/2. Substituting this value of t into (1) and simplifying gives an equation for the envelope in terms of x and y only:

This is the familiar implicit conic section form, in this case a parabola. Parabolae remain parabolae under rotation and scaling; thus the string art forms a parabolic arc ("arc" since only a portion of the full parabola is produced). In this case an anticlockwise rotation through 45° gives the orthogonal parabolic equation y = x2/(k√2) + k/(2√2). Note that the final step of eliminating t may not always be possible to do analytically, depending on the form of F(x,y,t).

例3

Let I ⊂ R be an open interval and let γ : I → R2 be a smooth plane curve parametrised by arc length. Consider the one-parameter family of normal lines to γ(I). A line is normal to γ at γ(t) if it passes through γ(t) and is perpendicular to the tangent vector to γ at γ(t). Let T denote the unit tangent vector to γ and let N denote the unit normal vector. Using a dot to denote the dot product, the generating family for the one-parameter family of normal lines is given by  where

Clearly (x − γ)·T = 0 if and only if x − γ is perpendicular to T, or equivalently, if and only if x − γ is parallel to N, or equivalently, if and only if x = γ + λN for some λ ∈ R. It follows that

is exactly the normal line to γ at γ(t0). To find the discriminant of F we need to compute its partial derivative with respect to t:

where κ is the plane curve curvature of γ. It has been seen that F = 0 if and only if x - γ = λN for some λ ∈ R. Assuming that F = 0 gives

Assuming that κ ≠ 0 it follows that λ = 1/κ and so

This is exactly the evolute of the curve γ.

例4
The following example generalizes the preceding one, and shows that in some cases the envelope of a family of curves may be seen as the topologic boundary of a union of sets, whose boundaries are the curves of the envelope. For  and  consider the (open) right triangle in a Cartesian plane with vertices ,  and 

Fix an exponent , and consider the union of all the triangles  subjected to the constraint , that is the open set

To write a Cartesian representation for , start with any ,  satisfying  and any . The Hölder inequality in  with respect to the conjugated exponents  and  gives:
,
with equality if and only if .
In terms of a union of sets the latter inequality reads: the point  belongs to the set , that is, it belongs to some  with , if and only if it satisfies

Moreover, the boundary in  of the set  is the envelope of the corresponding family of line segments

(that is, the hypotenuses of the triangles), and has Cartesian equation

Notice that, in particular, the value  gives the arc of parabola of the example 1, and the value  (meaning that all hypotenuses are unit length segments) gives the astroid.

Envelope of a family of surfaces
A one-parameter family of surfaces in three-dimensional Euclidean space is given by a set of equations

depending on a real parameter a. For example the tangent planes to a surface along a curve in the surface form such a family.

Two surfaces corresponding to different values a and a'  intersect in a common curve defined by

In the limit as a'  approaches a, this curve tends to a curve contained in the surface at a

This curve is called the characteristic of the family at a. As a varies the locus of these characteristic curves defines a surface called the envelope of the family of surfaces.

概括

The idea of an envelope of a family of smooth submanifolds follows naturally. In general, if we have a family of submanifolds with codimension c then we need to have at least a c-parameter family of such submanifolds. For example: a one-parameter family of curves in three-space (c = 2) does not, generically, have an envelope.

應用

常微分方程
Envelopes are connected to the study of ordinary differential equations (ODEs), and in particular singular solutions of ODEs. Consider, for example, the one-parameter family of tangent lines to the parabola y = x2. These are given by the generating family . The zero level set  gives the equation of the tangent line to the parabola at the point (t0,t02). The equation  can always be solved for y as a function of x and so, consider

Differentiating and eliminating t gives the ODE

Not surprisingly y = 2tx − t2 are all solutions to this ODE. However, the envelope of this one-parameter family of lines, which is the parabola y = x2, is also a solution to this ODE. Another famous example is Clairaut's equation.

偏微分方程
Envelopes can be used to construct more complicated solutions of first order partial differential equations (PDEs) from simpler ones.  Let F(x,u,Du) = 0 be a first order PDE, where x is a variable with values in an open set Ω ⊂ Rn, u is an unknown real-valued function, Du is the gradient of u, and F is a continuously differentiable function that is regular in Du.  Suppose that u(x;a) is an m-parameter family of solutions:  that is, for each fixed a ∈ A ⊂ Rm, u(x;a) is a solution of the differential equation.  A new solution of the differential equation can be constructed by first solving (if possible)

for a = φ(x) as a function of x.  The envelope of the family of functions {u(·,a)}a∈A is defined by

and also solves the differential equation (provided that it exists as a continuously differentiable function).

Geometrically, the graph of v(x) is everywhere tangent to the graph of some member of the family u(x;a). Since the differential equation is first order, it only puts a condition on the tangent plane to the graph, so that any function everywhere tangent to a solution must also be a solution. The same idea underlies the solution of a first order equation as an integral of the Monge cone.  The Monge cone is a cone field in the Rn+1 of the (x,u) variables cut out by the envelope of the tangent spaces to the first order PDE at each point. A solution of the PDE is then an envelope of the cone field.

In Riemannian geometry, if a smooth family of geodesics through a point P in a Riemannian manifold has an envelope, then P has a conjugate point where any geodesic of the family intersects the envelope. The same is true more generally in the calculus of variations: if a family of extremals to a functional through a given point P has an envelope, then a point where an extremal intersects the envelope is a conjugate point to P.

焦散曲線

In geometrical optics, a caustic is the envelope of a family of light rays. In this picture there is an arc of a circle. The light rays (shown in blue) are coming from a source at infinity, and so arrive parallel. When they hit the circular arc the light rays are scattered in different directions according to the law of reflection. When a light ray hits the arc at a point the light will be reflected as though it had been reflected by the arc's tangent line at that point. The reflected light rays give a one-parameter family of lines in the plane. The envelope of these lines is the reflective caustic. A reflective caustic will generically consist of smooth points and ordinary cusp points.

From the point of view of the calculus of variations, Fermat's principle (in its modern form) implies that light rays are the extremals for the length functional

among smooth curves γ on [a,b] with fixed endpoints γ(a) and γ(b).  The caustic determined by a given point P (in the image the point is at infinity) is the set of conjugate points to P.

惠更斯原理
Light may pass through anisotropic inhomogeneous media at different rates depending on the direction and starting position of a light ray.  The boundary of the set of points to which light can travel from a given point q after a time t is known as the wave front after time t, denoted here by Φq(t).  It consists of precisely the points that can be reached from q in time t by travelling at the speed of light.  Huygens's principle asserts that the wave front set  is the envelope of the family of wave fronts  for q ∈ Φq0(t). More generally, the point q0 could be replaced by any curve, surface or closed set in space.

Ruled surface

"Enveloppe d'une Famille de Courbes Planes" at Encyclopédie des Formes Mathématiques Remarquables

微分幾何
解析幾何