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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 41 (10 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
The VisibilityVoronoi complex and its applications
 In Proc. 21st Annu. ACM Sympos. Comput. Geom. (SCG
, 2005
"... We introduce a new type of diagram called the VV (c)diagram (the Visibility–Voronoi diagram for clearance c), which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 ..."
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Cited by 37 (4 self)
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We introduce a new type of diagram called the VV (c)diagram (the Visibility–Voronoi diagram for clearance c), which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 to ∞. This diagram can be used for planning naturallooking paths for a robot translating amidst polygonal obstacles in the plane. A naturallooking path is short, smooth, and keeps — where possible — an amount of clearance c from the obstacles. The VV (c)diagram contains such paths. We also propose an algorithm that is capable of preprocessing a scene of configurationspace polygonal obstacles and constructs a data structure called the VVcomplex. The VVcomplex can be used to efficiently plan motion paths for any start and goal configuration and any clearance value c, without having to explicitly construct the VV (c)diagram for that cvalue. The preprocessing time is O(n 2 log n), where n is the total number of obstacle vertices, and the data structure can be queried directly for any cvalue by merely performing a Dijkstra search. We have implemented a Cgalbased software package for computing the VV (c)diagram in an exact manner for a given clearance value, and used it to plan naturallooking paths in various applications.
Exact and Efficient Construction of Planar Minkowski Sums using the Convolution Method
"... The Minkowski sum of two sets A, B ∈ IR d, denoted A⊕B, is defined as {a + b  a ∈ A, b ∈ B}. We describe an efficient and robust implementation for the construction of Minkowski sums of polygons in IR 2 using the convolution of the polygon boundaries. This method allows for faster computation of th ..."
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Cited by 18 (0 self)
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The Minkowski sum of two sets A, B ∈ IR d, denoted A⊕B, is defined as {a + b  a ∈ A, b ∈ B}. We describe an efficient and robust implementation for the construction of Minkowski sums of polygons in IR 2 using the convolution of the polygon boundaries. This method allows for faster computation of the sum of nonconvex polygons in comparison to the widelyused methods for Minkowskisum computation that decompose the input polygons into convex subpolygons and compute the union of the pairwise sums of these convex subpolygon. Our source code, as well as the data sets we used in our experiments, can be downloaded from:
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 17 (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.
An Efficient Algorithm for the Stratification and Triangulation of Algebraic Surfaces
 COMPUTATIONAL GEOMETRY: THEORY AND APPLICATIONS 43 (2010) 257–278. SPECIAL ISSUE ON SOCG’08
, 2010
"... We present a method to compute the exact topology of a real algebraic surface S, implicitly given by a polynomial f ∈ Q[x,y,z] of arbitrary total degree N. Additionally, our analysis provides geometric information as it supports the computation of arbitrary precise samples of S including critical po ..."
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Cited by 10 (9 self)
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We present a method to compute the exact topology of a real algebraic surface S, implicitly given by a polynomial f ∈ Q[x,y,z] of arbitrary total degree N. Additionally, our analysis provides geometric information as it supports the computation of arbitrary precise samples of S including critical points. We compute a stratification ΩS of S into O(N 5) nonsingular cells, including the complete adjacency information between these cells. This is done by a projection approach. We construct a special planar arrangement AS with fewer cells than a cad in the projection plane. Furthermore, our approach applies numerical and combinatorial methods to minimize costly symbolic computations. The algorithm handles all sorts of degeneracies without transforming the surface into a generic position. Based on ΩS we also compute a simplicial complex which is isotopic to S. A complete C++implementation of the stratification algorithm is presented. It shows good performance for many wellknown examples from algebraic geometry.
Constructing TwoDimensional Voronoi Diagrams via DivideandConquer of Envelopes in Space
"... We present a general framework for computing twodimensional Voronoi diagrams of different site classes under various distance functions. The computation of the diagrams employs the Cgal software for constructing envelopes of surfaces in 3space, which implements a divideandconquer algorithm. A st ..."
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Cited by 9 (4 self)
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We present a general framework for computing twodimensional Voronoi diagrams of different site classes under various distance functions. The computation of the diagrams employs the Cgal software for constructing envelopes of surfaces in 3space, which implements a divideandconquer algorithm. A straightforward application of the divideandconquer approach for Voronoi diagrams yields highly inefficient algorithms. We show that through randomization, the expected running time is nearoptimal (in a worstcase sense). We believe this result, which also holds for general envelopes, to be of independent interest. We describe the interface between the construction of the diagrams and the underlying construction of the envelopes, together with methods we have applied to speed up the (exact) computation. We then present results, where a variety of diagrams are constructed with our implementation, including power diagrams, Apollonius diagrams, diagrams of line segments, Voronoi diagrams on a sphere, and more. In all cases the implementation is exact and can handle degenerate input.
Arrangements on parametric surfaces I: General framework and infrastructure
, 2010
"... Abstract. We introduce a framework for the construction, maintenance, and manipulation of arrangements of curves embedded on certain twodimensional orientable parametric surfaces in threedimensional space. The framework applies to planes, cylinders, spheres, tori, and surfaces homeomorphic to them ..."
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Cited by 6 (6 self)
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Abstract. We introduce a framework for the construction, maintenance, and manipulation of arrangements of curves embedded on certain twodimensional orientable parametric surfaces in threedimensional space. The framework applies to planes, cylinders, spheres, tori, and surfaces homeomorphic to them. We reduce the effort needed to generalize existing algorithms, such as the sweep line and zone traversal algorithms, originally designed for arrangements of bounded curves in the plane, by extensive reuse of code. We have realized our approach as the Cgal package Arrangement on surface 2. We define a compact interface for our framework; only the operations in the interface need to be implemented for a specific application. The companion paper [6] describes concretizations for several types of surfaces and curves embedded on them, and applications. This is the first implementation of a generic algorithm that can handle arrangements on a large class of parametric surfaces.
Abstract An Exact, Complete and Efficient Computation of Arrangements of Bézier Curves
"... Arrangements of planar curves are fundamental structures in computational geometry. The arrangement package of CGAL can construct and maintain arrangements of various families of curves, when provided with the representation of the curves and some basic geometric functionality on them. It employs th ..."
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Cited by 5 (0 self)
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Arrangements of planar curves are fundamental structures in computational geometry. The arrangement package of CGAL can construct and maintain arrangements of various families of curves, when provided with the representation of the curves and some basic geometric functionality on them. It employs the exact computation paradigm in order to handle all degenerate cases in a robust manner. We present the representations and algorithms that are needed for implementing arrangements of Bézier curves using exact arithmetic. The implementation is efficient and complete, handling all degenerate cases. In order to avoid the prohibitive running times incurred by an indiscriminate usage of exact arithmetic, we make extensive use of the geometric properties of Bézier curves for filtering. As a result, most operations are carried out using fast approximate methods, and only in degenerate (or neardegenerate) cases do we resort to the exact, and more computationally demanding, procedures. To the best of our knowledge this is the first complete implementation that can construct arrangements of Bézier curves of any degree, and handle all degenerate cases in a robust manner.
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.