Zero-Parity Stabbing Information

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.