A theorem to determine the spatial containment of a point in a planar polyhedron
Computer Vision, Graphics, and Image Processing
Applications of spatial data structures: Computer graphics, image processing, and GIS
Applications of spatial data structures: Computer graphics, image processing, and GIS
A fast planar partition algorithm, I
Journal of Symbolic Computation
The point in polygon problem for arbitrary polygons
Computational Geometry: Theory and Applications
Computational Geometry in C
Signed Distance Computation Using the Angle Weighted Pseudonormal
IEEE Transactions on Visualization and Computer Graphics
Layer-Based Representation of Polyhedrons for Point Containment Tests
IEEE Transactions on Visualization and Computer Graphics
Technical section: 2D point-in-polygon test by classifying edges into layers
Computers and Graphics
Technical section: Point in solid strategies
Computers and Graphics
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In this paper, we revisit the point-in-polyhedron problem. After reviewing previous work, we develop further insight into the problem. We then claim that, for a given testing point and a three-dimensional polyhedron, a single determining triangle can be found which suffices to determine whether the point is inside or outside the polyhedron. This work can be considered to be an extension and implementation of Horn's work, which inspired us to propose a theorem for obtaining determining triangles. Building upon this theorem, algorithms are then presented, implemented, and tested. The results show that although our code has the same asymptotic time efficiency as commonly used octree-based ray crossing methods, in practice it is usually several times and sometimes more than ten times faster, while other costs such as preprocessing time and memory requirements remain the same. The ideas proposed in this paper are simple and general. They thus extend naturally to multi-material models, i.e., polyhedrons subdivided into smaller regions by internal boundaries.