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<p><b>Abstract</b>—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 we propose greatly reduces the storage requirement because it represents much information implicitly, though it still has a storage complexity <i>O(n)</i>. It is simple to implement, and robust for inclusion tests because many singularities are erased in constructing the layer-based representation. Incorporating an octree structure for organizing polyhedrons, our approach can run at a speed comparable with 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 <i>O(n)</i> and <i>O(log n)</i>, depending on the polyhedron shape and the constructed representation, and less than <i>O(log^3 n)</i> in most cases. The time complexity of preprocess is between <i>O(n)</i> and <i>O(n^2)</i>, varying with polyhedrons, where <i>n</i> is the edge number of a polyhedron.</p>
computational geometry, polyhedron, point containment, solid representation
Jing Li, Hanqiu Sun, Enhua Wu, Wencheng Wang, "Layer-Based Representation of Polyhedrons for Point Containment Tests", IEEE Transactions on Visualization & Computer Graphics, vol. 14, no. , pp. 73-83, January/February 2008, doi:10.1109/TVCG.2007.70407
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