Ray-casting point-in-polyhedron test

This paper considers a ray-casting point-in polyhedron test. Although it is conceptually the simple extension of a well-known point-in-polygon ray-casting algorithm, various practical problems appear in 3D, especially, when the boundary of a geometric object is represented as a triangulated surface. When a larger number of points have to be tested regarding their positions on the considered geometric object, preprocessing may drastically reduce the testing time. This paper considers comparisons between three such methods: The first uses a k-dimensional tree (kd-tree), the second an octree, and the last is based on a three-dimensional uniform grid (3D grid). The core operation for all three methods is the ray-casting, by which the odd-even rule can be efficiently applied. Ray-casting can be susceptible to the rounding errors, which are also considered in this paper.