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16
Exact and efficient construction of Minkowski sums of convex polyhedra with applications
 In Proc. 8th Workshop Alg. Eng. Exper. (Alenex’06
, 2006
"... We present an exact implementation of an efficient algorithm that computes Minkowski sums of convex polyhedra in R 3. Our implementation is complete in the sense that it does not assume general position. Namely, it can handle degenerate input, and it produces exact results. We also present applicati ..."
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Cited by 35 (9 self)
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We present an exact implementation of an efficient algorithm that computes Minkowski sums of convex polyhedra in R 3. Our implementation is complete in the sense that it does not assume general position. Namely, it can handle degenerate input, and it produces exact results. We also present applications of the Minkowskisum computation to answer collision and proximity queries about the relative placement of two convex polyhedra in R 3. The algorithms use a dual representation of convex polyhedra, and their implementation is mainly based on the Arrangement package of Cgal, the Computational Geometry Algorithm Library. We compare our Minkowskisum construction with the only three other methods that produce exact results we are aware of. One is a simple approach that computes the convex hull of the pairwise sums of vertices of two convex polyhedra. The second is based on Nef polyhedra embedded on the sphere, and the third is an outputsensitive approach based on linear programming. Our method is significantly faster. The results of experimentation with a broad family of convex polyhedra are reported. The relevant programs, source code, data sets, and documentation are available at
Boolean Operations on 3D Selective Nef Complexes: Optimized Implementation and Experiments
, 2005
"... Nef polyhedra in ddimensional space are the closure of halfspaces under boolean set operations. In consequence, they can represent nonmanifold situations, open and closed sets, mixeddimensional complexes and they are closed under all boolean and topological operations. We implemented a boundary ..."
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Cited by 9 (0 self)
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Nef polyhedra in ddimensional space are the closure of halfspaces under boolean set operations. In consequence, they can represent nonmanifold situations, open and closed sets, mixeddimensional complexes and they are closed under all boolean and topological operations. We implemented a boundary representation of threedimensional Nef polyhedra with efficient algorithms for boolean operations. These algorithms were designed for correctness and can handle all cases, in particular all degeneracies. The implementation is released as Open Source in the Cgal release 3.1. In this paper, we present experiments in order to (i) evaluate the practical runtime complexity, (ii) illustrate the effectiveness of several important optimizations, and (iii) compare our implementation with the Acis CAD kernel.
Complete, Exact and Efficient Implementation for Computing the Adjacency Graph of an Arrangement of Quadrics
, 2007
"... We present a complete, exact and efficient implementation to compute the adjacency graph of an arrangement of quadrics, i.e. surfaces of algebraic degree 2. This is a major step towards the computation of the full 3D arrangement. We enhanced an implementation for an exact parameterization of the in ..."
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Cited by 7 (2 self)
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We present a complete, exact and efficient implementation to compute the adjacency graph of an arrangement of quadrics, i.e. surfaces of algebraic degree 2. This is a major step towards the computation of the full 3D arrangement. We enhanced an implementation for an exact parameterization of the intersection curves of two quadrics, such that we can compute the exact parameter value for intersection points and from that the adjacency graph of the arrangement. Our implementation is complete in the sense that it can handle all kinds of inputs including all degenerate ones, i.e. singularities or tangential intersection points. It is exact in that it always computes the mathematically correct result. It is efficient measured in running times, i.e. it compares favorably to the only previous implementation.
On the exact maximum complexity of Minkowski sums of convex polyhedra
 PROCEEDINGS OF 23RD ANNUAL ACM SYMPOSIUM ON COMPUTATIONAL GEOMETRY (SOCG
, 2007
"... We present a tight bound on the exact maximum complexity of Minkowski sums of convex polyhedra in R³. In particular, we prove that the maximum number of facets of the Minkowski sum of two convex polyhedra with m and n facets respectively is bounded from above by f(m, n) = 4mn − 9m − 9n + 26. Given ..."
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Cited by 4 (3 self)
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We present a tight bound on the exact maximum complexity of Minkowski sums of convex polyhedra in R³. In particular, we prove that the maximum number of facets of the Minkowski sum of two convex polyhedra with m and n facets respectively is bounded from above by f(m, n) = 4mn − 9m − 9n + 26. Given two positive integers m and n, we describe how to construct two convex polyhedra with m and n facets respectively, such that the number of facets of their Minkowski sum is exactly f(m, n). We generalize the construction to yield a lower bound on the maximum complexity of Minkowski sums of many convex polyhedra in R3. That is, given k positive integers m1, m2,..., mk, we describe how to construct k convex polyhedra with corresponding number of facets, such that the number of facets of their Minkowski sum is � 1≤i<j≤k (2mi − 5)(2mj − 5) + � � k � 2 + 1≤i≤k mi. We also provide a conservative upper bound for the general case. Snapshots of several preconstructed convex polyhedra, the Minkowski sum of which is maximal, are available at http://www.cs.tau.ac.il / ~ efif/Mink. The polyhedra models and an interactive program that computes their Minkowski sums and visualizes them can be downloaded as well.
Arrangements on parametric surfaces II: Concretizations and applications
 IN COMPUTER SCIENCE
, 2010
"... We describe the algorithms and implementation details involved in the concretizations of a generic framework that enables exact construction, maintenance, and manipulation of arrangements embedded on certain twodimensional orientable parametric surfaces in threedimensional space. The fundamental ..."
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Cited by 4 (4 self)
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We describe the algorithms and implementation details involved in the concretizations of a generic framework that enables exact construction, maintenance, and manipulation of arrangements embedded on certain twodimensional orientable parametric surfaces in threedimensional space. The fundamentals of the framework are described in a companion paper. Our work covers arrangements embedded on elliptic quadrics and cyclides induced by intersections with other algebraic surfaces, and a specialized case of arrangements induced by arcs of great circles embedded on the sphere. We also demonstrate how such arrangements can be used to accomplish various geometric tasks efficiently, such as computing the Minkowski sums of polytopes, the envelope of surfaces, and Voronoi diagrams embedded on parametric surfaces. We do not assume general position. Namely, we handle degenerate input, and produce exact results in all cases. Our implementation is realized using Cgal and, in particular, the package that provides the underlying framework. We have conducted experiments on various data sets, and documented the practical efficiency of our approach.
Complete, Exact and Efficient Implementation for Computing the Adjacency Graph of an Arrangement of Quadrics
 in "15th Annual European Symposium on Algorithms  ESA 2007
"... We present a complete, exact and efficient implementation to compute the adjacency graph of an arrangement of quadrics, i.e. surfaces of algebraic degree 2. This is a major step towards the computation of the full 3D arrangement. We enhanced an implementation for an exact parameterization of the int ..."
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Cited by 2 (0 self)
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We present a complete, exact and efficient implementation to compute the adjacency graph of an arrangement of quadrics, i.e. surfaces of algebraic degree 2. This is a major step towards the computation of the full 3D arrangement. We enhanced an implementation for an exact parameterization of the intersection curves of two quadrics, such that we can compute the exact parameter value for intersection points and from that the adjacency graph of the arrangement. Our implementation is complete in the sense that it can handle all kinds of inputs including all degenerate ones, i.e. singularities or tangential intersection points. It is exact in that it always computes the mathematically correct result. It is efficient measured in running times, i.e. it compares favorably to the only previous implementation. Key words: Arrangement, intersection of surfaces, quadrics, pencils of quadrics, curve parameterization.
Implementation and Parallelization of a ReverseSearch Algorithm for Minkowski Sums
"... We present an implementation of a reversesearch algorithm of Fukuda for computing Minkowski sums of polytopes efficiently. The algorithm allows summing any number of polytopes in any dimension, and is complete in the sense that it does not assume general position. Its running time depends linearly ..."
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Cited by 1 (0 self)
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We present an implementation of a reversesearch algorithm of Fukuda for computing Minkowski sums of polytopes efficiently. The algorithm allows summing any number of polytopes in any dimension, and is complete in the sense that it does not assume general position. Its running time depends linearly on the size of the output. To the best of our knowledge, this is the only existing implementation that can efficiently compute Minkowski sums in higher dimensions. The implementation uses the exact arithmetic GMP, which ensures robustness of the program and exactness of the results. We furthermore present a parallel version of our implementation to demonstrate the simplicity and efficiency of performing the reverse search in parallel. The results of the performance tests show a nearlinear acceleration of our parallel implementation. 1
Geometry Freedom in Geometric Computation  Towards HigherOrder Genericity through Purely Combinatorial Geometric Algorithms
"... Any geometric algorithm can be dissected into combinatorial parts and geometric parts. The two parts intertwine in both the description and the implementation of the algorithm. We consider the benefits of relaxing this tight coupling. Coordinate freedom is widely considered the most important princi ..."
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Any geometric algorithm can be dissected into combinatorial parts and geometric parts. The two parts intertwine in both the description and the implementation of the algorithm. We consider the benefits of relaxing this tight coupling. Coordinate freedom is widely considered the most important principle in designing and implementing geometric systems. By ensuring that client code not manipulate individual coordinates and by developing two foundations for homogeneous and Cartesian coordinates, switching from one to the other can be easily performed after the system has been completed. We take another step and show that geometry freedom is possible. By removing the geometric classes from the implementation of a geometric algorithm, the algorithm becomes purely combinatorial. An arbitrary Euclidean or spherical geometry is then used as a parameter to the combinatorial algorithm to produce a geometric system in that geometry. Geometric freedom is helpful, for instance, when a geographic input is no longer constrained to a small area of Earth and one wishes to use spherical instead of Euclidean geometry. We apply geometry freedom to three classical problems. For the first two problems—convex hulls and Delaunay triangulations—the algorithms become generic with respect to the geometry. For the third—binary space partitioning—the algorithm becomes generic with respect to both the geometry and the dimension.