Polygons with Parallel Opposite Sides

TL;DR

This paper studies polygons with parallel opposite sides, defining discrete geometric constructs that mimic smooth curve properties, revealing their similar behaviors and potential applications in geometric analysis.

Contribution

It introduces discrete analogs of classical smooth curve concepts for polygons with parallel opposite sides, demonstrating their properties and behaviors.

Findings

Discrete area evolute and symmetry set behave like their smooth counterparts.

Discrete equidistants and area parallels exhibit similar properties to smooth curves.

The study provides a framework for analyzing polygonal approximations of convex curves.

Abstract

In this paper we consider planar polygons with parallel opposite sides. This type of polygons can be regarded as discretizations of closed convex planar curves by taking tangent lines at samples with pairwise parallel tangents. For this class of polygons, we define discrete versions of the area evolute, central symmetry set, equidistants and area parallels and show that they behave quite similarly to their smooth counterparts.