Constrained Delaunay Triangulations and Algorithms

Two-dimensional constrained Delaunay triangulations (of a planar straight-linegraph) introduced independently by Lee et al. [2] and Chew [1] are well studied struc-tures and can be constructed in optimal time. However, the generalization of suchobjects to three and higher dimensions is much less discussed in literature.In this talk, we first define a constrained Delaunay triangulation (CDT) of a d-dimensional geometric object called piecewise linear system (PLS) for d ≥ 0. Ourdefinition is, in some sense, more general than that of Shewchuk’s [3] since it allowsSteiner points (points which are not given in the input). We show several fundamentalproperties of such objects which are very close to those of Delaunay triangulations.We then propose an algorithm for constructing a CDT of any d-dimensional PLS,and show the correctness of this algorithm. Next we realize this algorithm for con-structing CDTs of three-dimensional PLSs. An implementation of this algorithm canbe found in the publicly available program TetGen [4]. Finally we discuss the complex-ity issues of this algorithm together with some examples. References[1] P. L. Chew. Constrained Delaunay triangulation. Algorithmica, 4:97–108, 1989.[2] D. T. Lee and A. K. Lin. Generalized Delaunay triangulations for planar graphs.Discrete and Computational Geometry, 1:201–217, 1986.[3] J. R. Shewchuk. General-dimensional constrained Delaunay and constrained regulartriangulations i: Combinatorial properties. To appear in Discrete and ComputationalGeometry, 2007.[4] H. Si. TetGen. http://tetgen.berlios.de, 2007.