Envelope curves and equidistant sets

Huibregtse, Mark; Winchell, Adam

doi:10.2140/involve.2016.9.839

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Envelope curves and equidistant sets

Mark Huibregtse and Adam Winchell

Vol. 9 (2016), No. 5, 839–856

DOI: 10.2140/involve.2016.9.839

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 $y = h (t)$ 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.

Keywords

equidistant set, envelope curve, midset

Mathematical Subject Classification 2010

Primary: 51M04

Milestones

Received: 9 July 2015

Accepted: 20 October 2015

Published: 25 August 2016

Communicated by Kenneth S. Berenhaut

Authors

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

Vol. 9, No. 5, 2016