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Boolean operations on 3D selective Nef complexes: Data structure, algorithms, and implementation
 IN PROC. 11TH ANNU. EURO. SYMPOS. ALG., VOLUME 2832 OF LNCS
, 2003
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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.
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.
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 7 (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.
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.