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 <math>(x,y)</math>-plane is the pair of equations
- <math>F(x,y,t)=0\qquad\qquad(1)\,</math>
- <math>\frac{\partial F(x,y,t)}{\partial t}=0\qquad\qquad(2)\,</math>
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 <math>F(x,y,t)</math>, and thus <math>(x,y)</math>, 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 <math>(x(t, p), y(t, p))\,</math>, the envelope can be found using the equation
- <math>{\partial x\over\partial t}{\partial y\over\partial p} = {\partial y\over\partial t}{\partial x\over\partial p}</math>
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
- <math>F(x,y,t)=(k-t)x+(k+t)y-(k-t)(k+t)\,</math>
for some fixed k, is suitable, and
- <math>F_t(x,y,t)=2t-x+y.\,</math>
So <math>t=(x-y)/2</math>, giving
- <math>x^2-2xy+y^2-4ky-4kx+4k^2=0</math>
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: <math>(x-u)v'=(y-v)u'</math> is a tangent of a parametrised curve <math>(u(t),v(t))</math>. If we take <math>F(x,y,t)=(x-u)v'-(y-v)u'</math> then <math>F_t(x,y,t)=xv-yu-uv+vu</math> and <math>F=F_t=0</math> gives <math>(x,y)=(u,v)</math> when <math>vu'\ne uv'</math>. 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.)
