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Envelope (mathematics)

In mathematics, an envelope of a family of manifolds (especially a family of curves) is a manifold that is tangent to each member of the family at some point.

Envelope of a family of curves

The simplest formal expression for an envelope of curves in the $\left(x,y\right)$-plane is the pair of equations

$F\left(x,y,t\right)=0qquadqquad\left(1\right),$

$frac\left\{partial F\left(x,y,t\right)\right\}\left\{partial t\right\}=0qquadqquad\left(2\right),$

where the family is implicitly defined by (1). Obviously the family has to be "nicely" — differentiably — indexed by t.

The logic of this form may not be obvious, but in the vulgar: solutions of (2) are places where $F\left(x,y,t\right)$, and thus $\left(x,y\right)$, are "constant" in tie, where "adjacent" family members intersect, which is another feature of the envelope.

For a family of plane curves given by parametric equations $\left(x\left(t, p\right), y\left(t, p\right)\right),$, the envelope can be found using the equation

$\left\{partial xoverpartial t\right\}\left\{partial yoverpartial p\right\} = \left\{partial yoverpartial t\right\}\left\{partial xoverpartial p\right\}$

where variation of the parameter p gives the different curves of the family.

Examples

Example 1

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, kt) 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:

$F\left(x,y,t\right)=t^2 + t\left(y-x-k\right) + kx = 0,$ (1)

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

$frac\left\{partial F\left(x,y,t\right)\right\}\left\{partial t\right\}=2t+ y-x-k = 0,$ (2)

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:

$x^2 - 2xy + y^2 -2kx - 2ky + k^2 = 0,$

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).

Example 2

Another example: $\left(x-u\right)v\text{'}=\left(y-v\right)u\text{'}$ is a tangent of a parametrised curve $\left(u\left(t\right),v\left(t\right)\right)$. If we take $F\left(x,y,t\right)=\left(x-u\right)v\text{'}-\left(y-v\right)u\text{'}$ then $F_t\left(x,y,t\right)=xv$-yu-uv+vu and $F=F_t=0$ gives $\left(x,y\right)=\left(u,v\right)$ when $v$u'ne uv'. So a curve is the envelope of its own tangents except where its curvature is zero. (This could also be read as a validation of this analytical form.)

Envelope of a family of surfaces

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

$F\left(x,y,z,a\right)=0$

depending on a real parameter a (see ). 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

$F\left(x,y,z,a\right)=0,,,\left\{F\left(x,y,z,a^prime\right)-F\left(x,y,z,a\right)over a^prime -a\right\}=0.$

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

$F\left(x,y,z,a\right)=0,,,\left\{partial Fover partial a\right\}\left(x,y,z,a\right)=0.$

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.

References

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