A Novel Point Inclusion Test for Convex Polygons Based on Voronoi Tessellations

Abstract: The point inclusion tests for polygons, in other words the point-in-polygon (PIP) algorithms, are fundamental tools for many scientific fields related to computational geometry, and they have been studied for a long time. The PIP algorithms get direct or indirect geometric definition of a polygonal entity, and validate its containment of a given point. The PIP algorithms, which are working directly on the geometric entities, derive linear boundary definitions for the edges of the polygons. Moreover, almost all direct test methods rely on the two-point form of the line equation to partition the space into half-spaces. Voronoi tessellations use an alternate approach for half-space partitioning. Instead of line equation, distance comparison between generator points is used to accomplish the same task. Voronoi tessellations consist of convex polygons, which are defined between generator points. Therefore, Voronoi tessellations have become an inspiration for us to develop a new approach of the PIP testing, specialized for convex polygons. The equations, essential to the conversion of a convex polygon to a Voronoi polygon, are derived. As a reference, a very standard convex PIP testing algorithm, \textit{the sign of offset}, is selected for comparison. For generalization of the comparisons, \textit{the ray crossing} algorithm is used as another reference. All algorithms are implemented as vector and matrix operations without any branching. This enabled us to benefit from the CPU optimizations of the underlying linear algebra libraries. Experimentation showed that, our proposed algorithm can have comparable performance characteristics with the reference algorithms. Moreover, it has simplicity, both from a geometric representation and the mental model.