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 $y=h\left(t\right)$ 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.