Envelope (mathematics)

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 not apply otherwise, 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. Let each curve Ct in the family be given as the solution of an equation ft(x, y)=0 (see implicit curve), where t is a parameter. Write F(t, x, y)=ft(x, y) and assume F is differentiable. The envelope of the family Ct is then defined as the set of points (x,y) for which, simultaneously, for some value of t, where is the partial derivative of F with respect to t. If t and u, t≠u are two values of the parameter then the intersection of the curves Ct and Cu is given by or, equivalently, Letting u → t gives the definition above. An important special case is when F(t, x, y) is a polynomial in t. This includes, by clearing denominators, the case where F(t, x, y) is a rational function in t. In this case, the definition amounts to t being a double root of F(t, x, y), so the equation of the envelope can be found by setting the discriminant of F to 0 (because the definition demands F=0 at some t and first derivative =0 i.e. its value 0 and it is min/max at that t). For example, let Ct be the line whose x and y intercepts are t and 11−t, this is shown in the animation above.