Convex partitions of polyhedra: a lower bound and worst-case optimal algorithm
SIAM Journal on Computing
Offsetting operations in solid modelling
Computer Aided Geometric Design
Art gallery theorems and algorithms
Art gallery theorems and algorithms
Triangles in space or building (and analyzing) castles in the Air
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Vertical decompositions for triangles in 3-space
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Strategies for polyhedral surface decomposition: an experimental study
Computational Geometry: Theory and Applications
Robot Motion Planning
Polygon decomposition for efficient construction of Minkowski sums
Computational Geometry: Theory and Applications - Special issue on: Sixteenth European Workshop on Computational Geometry (EUROCG-2000)
Accurate Minkowski Sum Approximation of Polyhedral Models
PG '04 Proceedings of the Computer Graphics and Applications, 12th Pacific Conference
Computational Geometry: Theory and Applications
Exact and efficient construction of planar Minkowski sums using the convolution method
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
A kinetic framework for computational geometry
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
Decompositions and Boundary Coverings of Non-convex Fat Polyhedra
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Covering Minkowski sum boundary using points with applications
Computer Aided Geometric Design
Contributing vertices-based Minkowski sum computation of convex polyhedra
Computer-Aided Design
Decompositions and boundary coverings of non-convex fat polyhedra
Computational Geometry: Theory and Applications
Configuration products in geometric modeling
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
Fast and robust retrieval of Minkowski sums of rotating convex polyhedra in 3-space
Proceedings of the 14th ACM Symposium on Solid and Physical Modeling
Contributing vertices-based Minkowski sum of a nonconvex--convex pair of polyhedra
ACM Transactions on Graphics (TOG)
Configuration products and quotients in geometric modeling
Computer-Aided Design
Dynamic Minkowski sums under scaling
Computer-Aided Design
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We present the first exact and robust implementation of the 3D Minkowski sum of two non-convex polyhedra. Our implementation decomposes the two polyhedra into convex pieces, performs pairwise Minkowski sums on the convex pieces, and constructs their union. We achieve exactness and the handling of all degeneracies by building upon 3D Nef polyhedra as provided by CGAL. The implementation also supports open and closed polyhedra. This allows the handling of degenerate scenarios like the tight passage problem in robot motion planning. The bottleneck of our approach is the union step. We address efficiency by optimizing this step by two means: we implement an efficient decomposition that yields a small amount of convex pieces, and develop, test and optimize multiple strategies for uniting the partial sums by consecutive binary union operations. The decomposition that we implemented as part of the Minkowski sum is interesting in its own right. It is the first robust implementation of a decomposition of polyhedra into convex pieces that yields at most O(r2) pieces, where r is the number of edges whose adjacent facets comprise an angle of more than 180 degrees with respect to the interior of the polyhedron.