Constrained delaunay triangulations
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.
Key wordsTriangulation Delaunay triangulation Constrained triangulation Algorithm Voronoi diagram
Unable to display preview. Download preview PDF.
- [CD85]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.)Google Scholar
- [Ch86]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.Google Scholar
- [Ch87a]L. P. Chew, Planar graphs and sparse graphs for efficient motion planning in the plane, in preparation.Google Scholar
- [Ch87b]L. P. Chew, Guaranteed-quality triangular meshes, in preparation.Google Scholar
- [Ki79]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.Google Scholar
- [Le78]D. T. Lee, Proximity and reachability in the plane, Technical Report R-831, Coordinated Science Laboratory, University of Illinois (1978).Google Scholar
- [PS85]F. P. Preparata and M. I. Shamos,Computational Geometry, Springer-Verlag, New York (1985).Google Scholar
- [Ya84]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).Google Scholar