Abstract
In recent years the practical computation of Delaunay triangulations, resp. Voronoi diagrams has received a lot of attention in the literature. While the Delaunay triangulation is an important basic tool in geometric optimization algorithms, it is nontrivial to achieve a numerically stable computer implementation. In this technical note we assume that all generating points are grid points of a regularM byM lattice in the plane. Depending onM we derive the necessary word length a binary computer must have for integer representation in order to obtain exact Delaunay triangulations. This analysis is carried out for theL 1-,L 2- andL ∞-metric.
Zusammenfassung
In den letzten Jahren hat die praktische Berechnung von Delaunay-Triangulationen bzw. Voronoi-Diagrammen große Aufmerksamkeit erfahren, da sie wichtige grundlegende Konzepte für geometrische Algorithmen darstellen. In dieser technischen Notiz betrachten wir das Problem ihrer numerisch stabilen Berechnung. Hierzu nehmen wir an, daß die generierenden Punkte Gitterpunkte eines quadratischenM×M-Gitters in der Ebene sind. Abhängig vonM bestimmen wir die notwendige Wortlänge zur Durchführung ganzzahliger Arithmetik, die es erlaubt, Delaunay-Triangulationen exakt zu berechnen. Die Analyse wird für dieL 1-,L 2- undL ∞-Metrik durchgeführt.
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References
Ohya, T., Iri, M., Murota, K.: Improvements of the incremental method for the Voronoi diagram with computational comparison of various algorithms. J. Operations Research Society of Japan27, 306–337 (1984).
Shamos, M. I., Hoey, D.: Closest point problems. Proc. 16th IEEE Ann. Symp. Found. Comput. Sci. 151–162 (1975).
Sugihara, K., Iri, M.: Geometric algorithms in finite-precision arithmetic. Research Memorandum RMI 88-10, Faculty of Engineering, University of Tokyo, 1988.
Sugihara, K.: A simple method for avoiding numerical errors and degeneracy in Voronoi diagram construction. Research Memorandum RMI 88-14, Faculty of Engineering, University of Tokyo, 1988.
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Jünger, M., Reinelt, G. & Zepf, D. Computing correct Delaunay triangulations. Computing 47, 43–49 (1991). https://doi.org/10.1007/BF02242021
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DOI: https://doi.org/10.1007/BF02242021