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 (x,y)-plane is the pair of equations

1: F(x,y,t) = 0

2:

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 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.

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 x² - 2xy + y² - 4ky - 4kx + 4k² = 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).

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) = xv'' - yu'' - uv'' + vu'' and F = F_t = 0 gives (x,y) = (u,v) when . 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