Envelope (mathematics)

In geometry, an envelope of a family of curves in the plane 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.The envelope of a family of surfaces is tangent to each surface in the family along the characteristic curve in that surface. In geometry, an envelope of a family of curves in the plane 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. 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 D {displaystyle {mathcal {D}}} of points (x,y) for which, simultaneously, for some value of t,where ∂ F / ∂ t {displaystyle partial F/partial t} 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