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28
Advanced programming techniques applied to Cgal’s arrangement package
 Computational Geometry: Theory and Applications
, 2005
"... Arrangements of planar curves are fundamental structures in computational geometry. Recently, the arrangement package of Cgal, the Computational Geometry Algorithms Library, has been redesigned and reimplemented exploiting several advanced programming techniques. The resulting software package, whi ..."
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Cited by 31 (15 self)
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Arrangements of planar curves are fundamental structures in computational geometry. Recently, the arrangement package of Cgal, the Computational Geometry Algorithms Library, has been redesigned and reimplemented exploiting several advanced programming techniques. The resulting software package, which constructs and maintains planar arrangements, is easier to use, to extend, and to adapt to a variety of applications. It is more efficient space and timewise, and more robust. The implementation is complete in the sense that it handles degenerate input, and it produces exact results. In this paper we describe how various programming techniques were used to accomplish specific tasks within the context of computational geometry in general and Arrangements in particular. These tasks are exemplified by several applications, whose robust implementation is based on the arrangement package. Together with a set of benchmarks they assured the successful application of the adverted programming techniques. 1
Sweeping and Maintaining Twodimensional Arrangements on Quadrics
"... We show how to compute and maintain the twodimensional arrangement on a quadric that is induced by intersection curves with other quadrics. The key idea is to parameterize the quadric by two variables, which then allows to implicitly compute the arrangement in a modified parameter space. We give ..."
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Cited by 14 (8 self)
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We show how to compute and maintain the twodimensional arrangement on a quadric that is induced by intersection curves with other quadrics. The key idea is to parameterize the quadric by two variables, which then allows to implicitly compute the arrangement in a modified parameter space. We give details of a possible parameterization and explain how to implement the needed geometric and topological predicates.
Hybrid motion planning using Minkowski sums
 IN PROC. ROBOTICS: SCI. SYS.
, 2008
"... Probabilistic and deterministic planners are two major approximatebased frameworks for solving motion planning problems. Both approaches have their own advantages and disadvantages. In this work, we provide an investigation to the following question: Is there a planner that can take the advantages ..."
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Cited by 12 (4 self)
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Probabilistic and deterministic planners are two major approximatebased frameworks for solving motion planning problems. Both approaches have their own advantages and disadvantages. In this work, we provide an investigation to the following question: Is there a planner that can take the advantages from both probabilistic and deterministic planners? Our strategy to achieve this goal is to use the pointbased Minkowski sum of the robot and the obstacles in workspace. Our experimental results show that our new method, called Msum planner, which uses the geometric properties of Minkowski sum to solve motion planning problems, provides advantages over the existing probabilistic or deterministic planners. In particular, Msum planner is significantly more efficient than the Probabilistic Roadmap Methods (PRMs) and its variants for problems that can be solved by reusing configurations.
Motion planning via manifold samples
 IN: ESA
, 2011
"... We present a general and modular algorithmic framework for path planning of robots. Our framework combines geometric methods for exact and complete analysis of lowdimensional configuration spaces, together with practical, considerably simpler samplingbased approaches that are appropriate for hig ..."
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Cited by 5 (2 self)
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We present a general and modular algorithmic framework for path planning of robots. Our framework combines geometric methods for exact and complete analysis of lowdimensional configuration spaces, together with practical, considerably simpler samplingbased approaches that are appropriate for higher dimensions. In order to facilitate the transfer of advanced geometric algorithms into practical use, we suggest taking samples that are entire lowdimensional manifolds of the configuration space that capture the connectivity of the configuration space much better than isolated point samples. Geometric algorithms for analysis of lowdimensional manifolds then provide powerful primitive operations. The modular design of the framework enables independent optimization of each modular component. Indeed, we have developed, implemented and optimized a primitive operation for complete and exact combinatorial analysis of a certain set of manifolds, using arrangements of curves of rational functions and concepts of generic programming. This in turn enabled us to implement our framework for the concrete case of a polygonal robot translating and rotating amidst polygonal obstacles. We demonstrate that the integration of several carefully engineered components leads to significant speedup over the popular PRM samplingbased algorithm, which represents the more simplistic approach that is prevalent in practice. We foresee possible extensions of our framework to solving highdimensional problems beyond motion planning.
A Simple Method for Computing Minkowski Sum Boundary in 3D Using Collision Detection
"... Abstract: Computing the Minkowski sum of two polyhedra exactly has been shown difficult. Despite its fundamental role in many geometric problems in robotics, to the best of our knowledge, no 3d Minkowski sum software for general polyhedra is available to the public. One of the main reasons is the d ..."
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Cited by 5 (2 self)
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Abstract: Computing the Minkowski sum of two polyhedra exactly has been shown difficult. Despite its fundamental role in many geometric problems in robotics, to the best of our knowledge, no 3d Minkowski sum software for general polyhedra is available to the public. One of the main reasons is the difficulty of implementing the existing methods. There are two main approaches for computing Minkowski sums: divideandconquer and convolution. The first approach decomposes the input polyhedra into convex pieces, computes the Minkowski sums between a pair of convex pieces, and unites all the pairwise Minkowski sums. Although conceptually simple, the major problems of this approach include: (1) The size of the decomposition and the pairwise Minkowski sums can be extremely large and (2) robustly computing the union of a large number of components can be very tricky. On the other hand, convolving two polyhedra can be done more efficiently. The resulting convolution is a superset of the Minkowski sum boundary. For nonconvex inputs, filtering or trimming is needed. This usually involves computing (1) the arrangement of the convolution and its substructures and (2) the winding numbers for the arrangement
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.
Fast and Robust Retrieval of Minkowski Sums of Rotating Convex Polyhedra in 3Space
, 2010
"... We present a novel method for fast retrieval of exact Minkowski sums of pairs of convex polytopes in R³, where one of the polytopes keeps rotating. The algorithm is based on precomputing a socalled criticality map, which records the changes in the underlying graphstructure of the Minkowski sum, w ..."
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Cited by 4 (1 self)
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We present a novel method for fast retrieval of exact Minkowski sums of pairs of convex polytopes in R³, where one of the polytopes keeps rotating. The algorithm is based on precomputing a socalled criticality map, which records the changes in the underlying graphstructure of the Minkowski sum, while one of the polytopes 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 collisiondetection 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 R³, 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 Gaussianmaps to efficiently compute the exact Minkowski sum of two polytopes. We conducted several experiments to verify the correctness of the algorithm and its implementation, and to compare its efficiency with an alternative (static) exact method. The results are reported.
Computing Exact Rational Offsets of Quadratic Triangular Bézier Surface Patches
"... The offset surfaces to nondevelopable quadratic triangular Bézier patches are rational surfaces. In this paper we give a direct proof of this result and formulate an algorithm for computing the parameterization of the offsets. Based on the observation that quadratic triangular patches are capable o ..."
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Cited by 4 (2 self)
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The offset surfaces to nondevelopable quadratic triangular Bézier patches are rational surfaces. In this paper we give a direct proof of this result and formulate an algorithm for computing the parameterization of the offsets. Based on the observation that quadratic triangular patches are capable of producing C 1 smooth surfaces, we use this algorithm to generate rational approximations to offset surfaces of general free–form surfaces.
Video: Exact Minkowski sums of convex polyhedra
 In Proceedings of 21st Annual ACM Symposium on Computational Geometry (SoCG
, 2005
"... 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 produces exact results. Our software also includes a ..."
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Cited by 3 (1 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 produces exact results. Our software also includes 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 a naïve approach that computes the convex hull of the pairwise sums of vertices of two convex polyhedra. Our method is significantly faster. The video demonstrates the techniques used on simple cases as well as on degenerate cases. The relevant programs, source code, data sets, and documentation are available at