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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.
(For other forms of envelope, see Envelope (disambiguation); there are at least two others with the same frontier-of-figures flavour.)
The simplest formal expression for an envelope of curves in the <math>(x,y)<math>-plane is the pair of equations
- 1: <math>F(x,y,t)=0<math>
- 2: <math>{\partial F(x,y,t)\over\partial t}=0<math>
where the family is implicitly defined by (1). Obviously the family has to be "nicely" -- differentiably -- indexed.
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.
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).
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>v''u'\ne u''v'<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.)
External links
- Mathworld (http://mathworld.wolfram.com/Envelope.html)
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