A rational rotation method for robust geometric algorithms
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
Minkowski addition of polytopes: computational complexity and applications to Gro¨bner bases
SIAM Journal on Discrete Mathematics
Overlaying simply connected planar subdivisions in linear time
Proceedings of the eleventh annual symposium on Computational geometry
V-Clip: fast and robust polyhedral collision detection
ACM Transactions on Graphics (TOG)
H-Walk: hierarchical distance computation for moving convex bodies
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Robot Motion Planning
Collision prediction for polyhedra under screw motions
SM '03 Proceedings of the eighth ACM symposium on Solid modeling and applications
Accurate Minkowski Sum Approximation of Polyhedral Models
PG '04 Proceedings of the Computer Graphics and Applications, 12th Pacific Conference
Advanced programming techniques applied to Cgal's arrangement package
Computational Geometry: Theory and Applications
Exact and efficient construction of Minkowski sums of convex polyhedra with applications
Computer-Aided Design
Spatial Planning: A Configuration Space Approach
IEEE Transactions on Computers
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Minkowski sums of rotating convex polyhedra
Proceedings of the twenty-fourth annual symposium on Computational geometry
Algorithmica - Special Issue: European Symposium on Algorithms 2007, Guest Editors: Larse Arge and Emo Welzl
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We present a novel method for fast retrieval of exact Minkowski sums of pairs of convex polytopes in R^3, where one of the polytopes frequently rotates. The algorithm is based on pre-computing a so-called criticality map, which records the changes in the underlying graph structure of the Minkowski sum of the two polytopes, while one of them rotates. We give tight combinatorial bounds on the complexity of the criticality map when the rotating polytope rotates about one, two, or three axes. The criticality map can be rather large already for rotations about one axis, even for summand polytopes with a moderate number of vertices each. We therefore focus on the restricted case of rotations about a single, though arbitrary, axis. Our work targets applications that require exact collision detection such as motion planning with narrow corridors and assembly maintenance where high accuracy is required. Our implementation handles all degeneracies and produces exact results. It efficiently handles the algebra of exact rotations about an arbitrary axis in R^3, and it well balances between preprocessing time and space on the one hand, and query time on the other. We use Cgal arrangements and in particular the support for spherical Gaussian maps to efficiently compute the exact Minkowski sum of two polytopes. We conducted several experiments (i) to verify the correctness of the algorithm and its implementation, and (ii) to compare its efficiency with an alternative (static) exact method. The results are reported.