Abstract not found

We show how to compute and maintain the two-dimensional 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.

Abstract not found

Nef polyhedra in d-dimensional space are the closure of half-spacesunder boolean set operation. In consequence, they can represent non-manifold situations, open and closed sets, mixed-dimensionalcomplexes and they are closed under all boolean and topological operations, such as complement and boundary. They were intro-duced by W. Nef in his seminal 1978 book on polyhedra. We presented in previous work a new data structure for theboundary representation of three-dimensional Nef polyhedra with efficient algorithms for boolean operations. These algorithms weredesigned for correctness and can handle all cases, in particular all degeneracies. To this extent we rely on exact arithmetic to avoidwell known problems with floating-point arithmetic. In this paper, we present important optimizations for the algo-rithms. We describe the chosen implementations for the pointlocation and the intersection-finding subroutines, a kd-tree and afast box-intersection algorithm, respectively. We evaluate this optimized implementation with extensive experiments that supplementthe runtime analysis from our previous paper and that illustrate the effectiveness of our optimizations. We compare our implementa-tion with the A

Abstract. We introduce a framework for the construction, maintenance, and manipulation of arrangements of curves embedded on certain two-dimensional orientable parametric surfaces in three-dimensional 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.

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.