Vol. 9, No. 5, 2016

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Envelope curves and equidistant sets

Mark Huibregtse and Adam Winchell

Vol. 9 (2016), No. 5, 839–856

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(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.

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equidistant set, envelope curve, midset
Mathematical Subject Classification 2010
Primary: 51M04
Received: 9 July 2015
Accepted: 20 October 2015
Published: 25 August 2016

Communicated by Kenneth S. Berenhaut
Mark Huibregtse
Department of Mathematics and Computer Science
Skidmore College
815 North Broadway
Saratoga Springs, NY 12866
United States
Adam Winchell
Skidmore College
815 North Broadway
Saratoga Springs, NY 12866
United States