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34
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 33 (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 16 (9 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 10 (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.
Exact Minkowksi sums of polyhedra and exact and efficient decomposition of polyhedra into convex pieces. Algorithmica 2008. Online first
"... Abstract. We present the first exact and robust implementation of the 3D Minkowski sum of two nonconvex 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 hand ..."
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Cited by 8 (0 self)
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Abstract. We present the first exact and robust implementation of the 3D Minkowski sum of two nonconvex 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. 1
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 6 (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.
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 5 (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.
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
Exact implementation of arrangements of geodesic arcs on the sphere with applications
 In Abstracts of 24th Eur. Workshop Comput. Geom
, 2008
"... Recently, the Arrangement 2 package of Cgal, the Computational Geometry Algorithms Library, has been greatly extended to support arrangements of curves embedded on twodimensional parametric surfaces. The general framework for sweeping a set of curves embedded on a twodimensional parametric surface ..."
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Cited by 5 (4 self)
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Recently, the Arrangement 2 package of Cgal, the Computational Geometry Algorithms Library, has been greatly extended to support arrangements of curves embedded on twodimensional parametric surfaces. The general framework for sweeping a set of curves embedded on a twodimensional parametric surface was introduced in [3]. In this paper we concentrate on the specific algorithms and implementation details involved in the exact construction and maintenance of arrangements induced by arcs of great circles embedded on the sphere, also known as geodesic arcs, and on the exact computation of Voronoi diagrams on the sphere, the bisectors of which are geodesic arcs. This class of Voronoi diagrams includes the subclass of Voronoi diagrams of points and its generalization, power diagrams, also known as Laguerre Voronoi diagrams. The resulting diagrams are represented as arrangements, and can be passed as input to consecutive operations supported by the Arrangement 2 package and its derivatives. The implementation is complete in the sense that it handles degenerate input, and it produces exact results. An example that uses real world data is included. Additional material is available at
Covering Minkowski sum boundary using points . . .
 COMPUTER AIDED GEOMETRIC DESIGN
, 2008
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