Envelopes of Bisection Lines of Polygons

TL;DR

This paper explores different notions of discrete envelopes of bisection lines in polygons, analyzing their properties and relationships, which advances understanding of geometric partitions and their singularities.

Contribution

It introduces and compares three distinct definitions of discrete envelopes of bisection lines in polygons, highlighting their differences and connections.

Findings

Identified multiple notions of discrete envelopes for polygon bisection lines

Analyzed the singularities present in these envelopes

Established relationships between different envelope notions

Abstract

A bisection line divides a convex planar curve into two parts with equal areas. It is natural to study the envelope of these lines, which in general present singularities. The polygonal case is particularly inte\-resting, since there are several different notions of a discrete envelope. In this paper, we study three different notions of discrete envelopes of bisection lines and the connections between them.