Abstract
Given a set ofn vertices in the plane together with a set of noncrossing, straight-line edges, theconstrained Delaunay triangulation (CDT) is the triangulation of the vertices with the following properties: (1) the prespecified edges are included in the triangulation, and (2) it is as close as possible to the Delaunay triangulation. We show that the CDT can be built in optimalO(n logn) time using a divide-and-conquer technique. This matches the time required to build an arbitrary (unconstrained) Delaunay triangulation and the time required to build an arbitrary constrained (non-Delaunay) triagulation. CDTs, because of their relationship with Delaunay triangulations, have a number of properties that make them useful for the finite-element method. Applications also include motion planning in the presence of polygonal obstacles and constrained Euclidean minimum spanning trees, spanning trees subject to the restriction that some edges are prespecified.
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L. P. Chew and R. L. Drysdale, Voronoi diagrams based on convex distance functions,Proceedings of the First Symposium on Computational Geometry, Baltimore (1985), pp. 235–244. (Revised version submitted toDiscrete and Computational Geometry.)
L. P. Chew, There is a planar graph almost as good as the complete graph,Proceedings of the Second Annual Symposium on Computational Geometry, Yorktown Heights (1986), pp. 169–177.
L. P. Chew, Planar graphs and sparse graphs for efficient motion planning in the plane, in preparation.
L. P. Chew, Guaranteed-quality triangular meshes, in preparation.
L. De Floriani, B. Falcidieno, and C. Pienovi, Delaunay-based representation of surfaces defined over arbitrarily shaped domains,Computer Vision, Graphics, and Image Processing,32 (1985), 127–140.
S. Fortune, A Sweepline algorithm for Voronoi diagrams,Algorithmica,2 (1987), 153–174.
D. G. Kirkpatrick, Efficient computation of continuous skeletons,Proceedings of the 20th Annual Symposium on the Foundations of Computer Science, IEEE Computer Society (1979), pp. 18–27.
D. T. Lee, Proximity and reachability in the plane, Technical Report R-831, Coordinated Science Laboratory, University of Illinois (1978).
D. T. Lee and A. K. Lin, Generalized Delaunay triangulation for planar graphs,Discrete and Computational Geometry,1 (1986), 201–217.
D. T. Lee and B. Schachter, Two algorithms for constructing Delaunay triangulations,International Journal of Computer and Information Sciences,9 (1980), 219–242.
F. P. Preparata and M. I. Shamos,Computational Geometry, Springer-Verlag, New York (1985).
C. K. Yap, AnO(n logn) algorithm for the Voronoi diagram of a set of simple curve segments. Technical Report, Courant Institute, New York University (October 1984).
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Communicated by Chee-Keng Yap.
An earlier version of the results presented here appeared in theProceedings of the Third Annual Symposium on Computational Geometry (1987).
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Paul Chew, L. Constrained delaunay triangulations. Algorithmica 4, 97–108 (1989). https://doi.org/10.1007/BF01553881
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DOI: https://doi.org/10.1007/BF01553881