Zero-Parity Stabbing Information

TL;DR

This paper introduces zero-parity stabbing information for polygons, analyzing its ability to distinguish visibility edges and its limitations, with specific results for certain polygon classes and smooth curves.

Contribution

It defines zero-parity stabbing information and completely characterizes its effectiveness in distinguishing visibility edges in polygons and smooth curves.

Findings

ZP information cannot distinguish internal from external visibility edges in general

ZP information suffices for polygons avoiding certain complex substructures

ZP information distinguishes internal from external visibility in smooth curves

Abstract

Everett et al. introduced several varieties of stabbing information for the lines determined by pairs of vertices of a simple polygon P, and established their relationships to vertex visibility and other combinatorial data. In the same spirit, we define the ``zero-parity (ZP) stabbing information'' to be a natural weakening of their ``weak stabbing information,'' retaining only the distinction among {zero, odd, even>0} in the number of polygon edges stabbed. Whereas the weak stabbing information's relation to visibility remains an open problem, we completely settle the analogous questions for zero-parity information, with three results: (1) ZP information is insufficient to distinguish internal from external visibility graph edges; (2) but it does suffice for all polygons that avoid a certain complex substructure; and (3) the natural generalization of ZP information to the continuous case of smooth curves does distinguish internal from external visibility.