Envelope curves and equidistant sets

Abstract

Given two sets of points A and B in the plane (called the focal sets), the equidistant set (or midset) of A and B is the locus of points equidistant from A and B. This article studies envelope curves as realizations of focal sets. We prove two results: First, given a closed convex focal set A that lies within the convex region bounded by the graph of a concave-up function h, there is a second focal set B (an envelope curve for a suitable family of circles) such that the graph of h lies in the midset of A and B. Second, given any function yD h.t/ with a continuous third derivative and bounded curvature, the envelope curves A and B associated to any family of circles of sufficiently small constant radius centered on the graph of h will define a midset containing this graph.