Envelope (mathematics)

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For other senses of this word, see envelope (disambiguation).

The form of envelope treated here is a manifold that manages to be tangent to some point of each member of a family of manifolds. Curves are the usual manifolds involved.

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)\,]

[=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

[ = ]

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

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^2-2xy+y^2-4ky-4kx+4k^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).

External links

[Mathworld]

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