Envelope (mathematics)

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This article is about envelopes in mathematics. For other uses, see Envelope (disambiguation).

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.

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

$F(x,y,t)=0\qquad\qquad(1)\,$

${\partial F(x,y,t)\over\partial t}=0\qquad\qquad(2)\,$

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 $F (x, y, t)$ , and thus $(x, y)$ , are "constant" in t — ie, where "adjacent" family members intersect, which is another feature of the envelope.

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

${\partial x\over\partial t}{\partial y\over\partial p} = {\partial y\over\partial t}{\partial x\over\partial p}$

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

[edit] Example

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 axes; a non-orthogonal layout is a rotation and scaling away. Then

$F(x,y,t)=(k-t)x+(k+t)y-(k-t)(k+t)\,$

for some fixed k, is suitable, and

$F_t(x,y,t)=2t-x+y.\,$

So $t = (x - y) / 2$ , giving

x 2 - 2 x y + y 2 - 4 k y - 4 k x + 4 k 2 = 0

which is the familiar implicit conic section form, in this case a parabola.

Parabolae remain parabolae under rotation and scaling; thus our answer is "parabolic arc" (since only a portion is produced).

See also ruled surface.

Another example: $(x - u) v' = (y - v) u'$ is a tangent of a parametrised curve $(u (t), v (t))$ . If we take $F (x, y, t) = (x - u) v' - (y - v) u'$ then $F t (x, y, t) = x v'' - y u'' - u v'' + v u''$ and $F = F t = 0$ gives $(x, y) = (u, v)$ when $v''u'\ne u''v'$ . 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.)