On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Fat triangles determine linearly many holes
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
On a class of O(n2) problems in computational geometry
Computational Geometry: Theory and Applications
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
The design and implementation of panar maps in CGAL
Journal of Experimental Algorithmics (JEA)
Robot Motion Planning
Two-Dimensional Arrangements in CGAL and Adaptive Point Location for Parametric Curves
WAE '00 Proceedings of the 4th International Workshop on Algorithm Engineering
Output-sensitive construction of the union of triangles
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
SGP '05 Proceedings of the third Eurographics symposium on Geometry processing
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We present a new incremental algorithm for constructing the union of n triangles in the plane. In our experiments, the new algorithm, which we call the Disjoint-Cover (DC) algorithm, performs significantly better than the standard randomized incremental construction (RIC) of the union. Our algorithm is rather hard to analyze rigorously, but we provide an initial such analysis, which yields an upper bound on its performance that is expressed in terms of the expected cost of the RIC algorithm. Our approach and analysis generalize verbatim to the construction of the union of other objects in the plane, and, with slight modifications, to three dimensions. We present experiments with a software implementation of our algorithm using the CGAL library of geometric algorithms.