Angle counts for isothetic polygons and polyhedra

Abstract

In the case of isothetic simple polyhedra there are only six different types of 3D angles. This article states and proofs a formula about counts of these angles. This complements formulas in combinatorial topology such as Euler’s polyhedron formula, or the previously known formula on angle counts for isothetic polygons. The latter formula and the shown equality for angle counts of isothetic simple polyhedra are useful formulas for analyzing isothetic boundaries in 2D digital images (e.g. classification into inner (boundary of a hole) or outer boundaries, see Voss [Discrete Images, Objects, and Functions in Zn, Springer, Berlin, 1993]) and isothetic surfaces in 3D digital images (e.g. necessary condition for a complete surface scan).