TY - GEN
T1 - Constructing the exact Voronoi diagram of arbitrary lines in three-dimensional space
AU - Hemmer, Michael
AU - Setter, Ophir
AU - Halperin, Dan
N1 - Funding Information:
This work has been supported in part by the Israel Science Foundation (grant no. 236/06), by the German-Israeli Foundation (grant no. 969/07), and by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University.
PY - 2010
Y1 - 2010
N2 - We introduce a new, efficient, and complete algorithm, and its exact implementation, to compute the Voronoi diagram of lines in space. This is a major milestone towards the robust construction of the Voronoi diagram of polyhedra. As we follow the exact geometric-computation paradigm, it is guaranteed that we always compute the mathematically correct result. The algorithm is complete in the sense that it can handle all configurations, in particular all degenerate ones. The algorithm requires O(n3+ε ) time and space, where n is the number of lines. The Voronoi diagram is represented by a data structure that permits answering point-location queries in O(log2 n) expected time. The implementation employs the Cgal packages for constructing arrangements and lower envelopes together with advanced algebraic tools.
AB - We introduce a new, efficient, and complete algorithm, and its exact implementation, to compute the Voronoi diagram of lines in space. This is a major milestone towards the robust construction of the Voronoi diagram of polyhedra. As we follow the exact geometric-computation paradigm, it is guaranteed that we always compute the mathematically correct result. The algorithm is complete in the sense that it can handle all configurations, in particular all degenerate ones. The algorithm requires O(n3+ε ) time and space, where n is the number of lines. The Voronoi diagram is represented by a data structure that permits answering point-location queries in O(log2 n) expected time. The implementation employs the Cgal packages for constructing arrangements and lower envelopes together with advanced algebraic tools.
KW - CGAL
KW - Computational Geometry
KW - Lower Envelopes
KW - Point Location
KW - Robust Geometric Computing
KW - Voronoi Diagrams
UR - http://www.scopus.com/inward/record.url?scp=78249290515&partnerID=8YFLogxK
U2 - 10.1007/978-3-642-15775-2_34
DO - 10.1007/978-3-642-15775-2_34
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AN - SCOPUS:78249290515
SN - 3642157742
SN - 9783642157745
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 398
EP - 409
BT - Algorithms, ESA 2010 - 18th Annual European Symposium, Proceedings
Y2 - 6 September 2010 through 8 September 2010
ER -