ABSTRACT
We have seen in Chapter 11 that the normal lines to a plane curve are tangents to its evolute. If we draw all these normal lines they will 'envelop' the evolute. We consider now the more general case where a 'family' of general curves envelops a curve. As well as families of lines we will give examples of families of other curves, including families of circles, parabolae, or ellipses. A geometrical envelope of a family of curves is a curve which at each of its points is tangent to a curve of the family. We consider three types of envelopes, namely singular-set envelopes, discriminant envelopes, and limiting positions envelopes. In individual cases these may give rise to different sets and may differ from the geometrical envelope depending, for example, on the existence of singularities. As examples of envelopes, we consider e volutes and parallel curves as envelopes, and orthotomics and caustics as envelopes, including, especially, the orthotomics and caustics of the circle. We also prove that a curve is the envelope of its circles of curvature. The approach we take in this chapter is that of problem solving, rather than of proving the technical theorems which occur in the general theory of envelopes.
