Vol. 8, No. 2, 2015

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Bisection envelopes

Noah Fechtor-Pradines

Vol. 8 (2015), No. 2, 307–328

We study the envelope of the family of lines which bisect the interior region of a simple, closed curve in the plane. We determine this ‘bisection envelope’for polygons and show that polygons with no parallel pairs of sides are characterized by their bisection envelope. We show that the bisection envelope always has at least three and an odd number of cusps. We investigate the winding numbers of bisection envelopes, and use this to show that there are an infinite number of curves with any given bisection envelope and show how to generate them. We obtain results on the intersections of bisecting lines. Finally, we give a relationship between the ‘internal area’of a curve and that of its bisection envelope.

bisection envelope, area, winding number, envelope, geometry, bisection
Mathematical Subject Classification 2010
Primary: 26B15, 51M25
Received: 30 July 2013
Revised: 23 October 2013
Accepted: 12 November 2013
Published: 3 March 2015

Communicated by Frank Morgan
Noah Fechtor-Pradines
Harvard University
1405 Harvard Yard Mail Center
Cambridge, MA 02138
United States