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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 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
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
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 25 (3 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 16 (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:
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 6 (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.
An experimental study of point location in general planar arrangements
 In ALENEX/ANALCO
, 2006
"... We study the performance in practice of various pointlocation algorithms implemented in Cgal, including a newly devised Landmarks algorithm. Among the other algorithms studied are: a naïve approach, a “walk along a line ” strategy and a trapezoidaldecomposition based search structure. The current ..."
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Cited by 5 (3 self)
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We study the performance in practice of various pointlocation algorithms implemented in Cgal, including a newly devised Landmarks algorithm. Among the other algorithms studied are: a naïve approach, a “walk along a line ” strategy and a trapezoidaldecomposition based search structure. The current implementation addresses general arrangements of arbitrary planar curves, including arrangements of nonlinear segments (e.g., conic arcs) and allows for degenerate input (for example, more than two curves intersecting in a single point, or overlapping curves). All calculations use exact number types and thus result in the correct point location. In our Landmarks algorithm (a.k.a. Jump & Walk), special points, “landmarks”, are chosen in a preprocessing stage, their place in the arrangement is found, and they are inserted into a datastructure that enables efficient nearestneighbor search. Given a query point, the nearest landmark is located and then the algorithm “walks ” from the landmark to the query point. We report on extensive experiments with arrangements composed of line segments or conic arcs. The results indicate that the Landmarks approach is the most efficient when the overall cost of a query is taken into account, combining both preprocessing and query time. The simplicity of the algorithm enables an almost straightforward implementation and rather easy maintenance. The generic programming implementation allows versatility both in the selected type of landmarks, and in the choice of the nearestneighbor search structure. The end result is a highly effective pointlocation algorithm for most practical purposes. ∗ Work reported in this paper has been supported in part by the IST Programme of the EU as a Sharedcorst RTD
Efficient implementation of redblack trees with split and catenate operations
, 2005
"... We describe the implementation of the Multiset classtemplate within the support library of Cgal, the Computational Geometry Algorithms ’ Library. The interface of this class is inspired by the multiset classtemplate included in the standard template library (Stl). Both class templates are implemen ..."
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Cited by 3 (1 self)
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We describe the implementation of the Multiset classtemplate within the support library of Cgal, the Computational Geometry Algorithms ’ Library. The interface of this class is inspired by the multiset classtemplate included in the standard template library (Stl). Both class templates are implemented in terms of redblack trees, yet they differ from one another quite substantially. In this report we highlight the differences between the two class templates and show the advantages of our implementation. 1
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
Exact and Efficient Construction of Planar . . .
"... We describe a simple yet powerful approach for computing planar arrangements of circular arcs and line segments in a robust and exact manner. Constructing arrangements using this approach is about one order of magnitude faster compared to other techniques that employ the exact computation paradigm. ..."
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We describe a simple yet powerful approach for computing planar arrangements of circular arcs and line segments in a robust and exact manner. Constructing arrangements using this approach is about one order of magnitude faster compared to other techniques that employ the exact computation paradigm. We have successfully applied our technique for computing offsets of planar polygons and for performing Boolean operations on general polygons that contain circular edges.