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In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two "infinitesimally adjacent" curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.

To have an envelope, it is necessary that the individual members of the family of curves are differentiable curves as the concept of tangency does otherwise not apply, and there has to be a smooth transition proceeding through the members. But these conditions are not sufficient – a given family may fail to have an envelope. A simple example of this is given by a family of concentric circles of expanding radius.

Generalisations

The idea of an envelope of a family of smooth submanifolds follows naturally. In general, if we have a family of submanifolds with codimension c then we need to have at least a c-parameter family of such submanifolds. For example: a one-parameter family of curves in three-space (c = 2) does not, generically, have an envelope.

Applications

Ordinary differential equations

ordinary differential equations (ODEs), and in particular singular solutions of ODEs.[3] Consider, for example, the one-parameter family of tangent lines to the parabola y = x2. These are given by the generating family F(t,(x,y)) = t2 – 2tx + y. The zero level set F(t0,(x,y)) = 0 gives the equation of the tangent line to the parabola at the point (t0,t02). The equation t2 – 2tx + y = 0 can always be solved for y as a function of x and so, consider