Layer-based representation of polyhedrons for point containment tests

This paper presents the layer-based representation of polyhedrons and its use for point-in-polyhedron tests. In the representation, the facets and edges of a polyhedron are sequentially arranged, and so, the binary search algorithm is efficiently used to speed up inclusion tests. In comparison with conventional representation for polyhedrons, the layer-based representation that we propose greatly reduces the storage requirement because it represents much information implicitly though it still has a storage complexity O(n). It is simple to implement and robust for inclusion tests because many singularities are erased in constructing the layer-based representation. By incorporating an octree structure for organizing polyhedrons, our approach can run at a speed comparable with Binary space partitioning (BSP)-based inclusion tests and, at the same time, greatly reduce storage and preprocessing time in treating large polyhedrons. We have developed an efficient solution for point-in-polyhedron tests, with the time complexity varying between O(n) and O(logn), depending on the polyhedron shape and the constructed representation, and less than O(log³n) in most cases. The time complexity of preprocess is between O(n) and O(n²), varying with polyhedrons, where n is the edge number of a polyhedron.