Visibility-Monotonic Polygon Deflation

TL;DR

This paper investigates the deformation properties of polygons with no visibility crossings, establishing the minimal polygon size for non-deflatability and introducing a new combinatorial structure to analyze visibility.

Contribution

It introduces the directed dual, a novel combinatorial structure, and proves the existence of polygons that cannot be deformed into deflated polygons, answering a longstanding open question.

Findings

The smallest polygon that cannot be deformed into a deflated polygon has seven sides.

Any two deflated polygons with the same directed dual can be transformed into each other via visibility-preserving deformation.

The directed dual encodes the visibility properties of deflated polygons.

Abstract

A deflated polygon is a polygon with no visibility crossings. We answer a question posed by Devadoss et al. (2012) by presenting a polygon that cannot be deformed via continuous visibility-decreasing motion into a deflated polygon. We show that the least n for which there exists such an n-gon is seven. In order to demonstrate non-deflatability, we use a new combinatorial structure for polygons, the directed dual, which encodes the visibility properties of deflated polygons. We also show that any two deflated polygons with the same directed dual can be deformed, one into the other, through a visibility-preserving deformation.