Two algorithms for constructing a Delaunay triangulation

This paper provides a unified discussion of the Delaunay triangulation. Its geometric properties are reviewed and several applications are discussed. Two algorithms are presented for constructing the triangulation over a planar set ofN points. The first algorithm uses a divide-and-conquer approach. It runs inO(N logN) time, which is asymptotically optimal. The second algorithm is iterative and requiresO(N2) time in the worst case. However, its average case performance is comparable to that of the first algorithm.

[1]  Georges Voronoi.  Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les parallélloèdres primitifs. , 1908 .

[2]  A. H. Thiessen.  PRECIPITATION AVERAGES FOR LARGE AREAS , 1911 .

[3]  D. J. Bogue,et al.  The structure of the metropolitan community , 1949 .

[4]  E. Gilbert.  Random Subdivisions of Space into Crystals , 1962 .

[5]  C. A. Rogers,et al.  Packing and Covering , 1964 .

[6]  T. Kiang.  RANDOM FRAGMENTATION IN TWO AND THREE DIMENSIONS. , 1966 .

[7]  M. Stearns.  Probability Distribution of a Network of Triangles , 1967 .

[8]  P. Switzer.  RECONSTRUCTING PATTERNS FROM SAMPLE DATA. , 1967 .

[9]  R. E. Miles.  Probability Distribution of a Network of Triangles (Mary Beth Stearns) , 1969 .

[10]  R. E. Miles.  On the homogeneous planar Poisson point process , 1970 .

[11]  P. Lloyd,et al.  Location in space: A theoretical approach to economic geography , 1974 .

[12]  J. Besag.  Spatial Interaction and the Statistical Analysis of Lattice Systems , 1974 .

[13]  Michael Ian Shamos,et al.  Closest-point problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[14]  D. H. McLain,et al.  Two Dimensional Interpolation from Random Data , 1976, Comput. J..

[15]  Der-Tsai Lee.  On Finding K Nearest Neighbors in the Plane , 1976 .

[16]  F. P. Preparata,et al.  Convex hulls of finite sets of points in two and three dimensions , 1977, CACM.

[17]  Errol L. Lloyd.  On triangulations of a set of points in the plane , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[18]  Malcolm A. Sabin,et al.  Piecewise Quadratic Approximations on Triangles , 1977, TOMS.

[19]  B. A. Lewis,et al.  Triangulation of Planar Regions with Applications , 1978, Comput. J..

[20]  Bruce J. Schachter,et al.  Decomposition of Polygons into Convex Sets , 1978, IEEE Transactions on Computers.

[21]  J. M. Nelson.  A triangulation algorithm for arbitrary planar domains , 1978 .

[22]  Der-Tsai Lee.  Proximity and reachability in the plane. , 1978 .

[23]  A. Rosenfeld,et al.  Random Mosaic Models for Textures , 1978 .

[24]  Robin Sibson,et al.  Locally Equiangular Triangulations , 1978, Comput. J..

[25]  Robert E. Tarjan,et al.  Triangulating a Simple Polygon , 1978, Inf. Process. Lett..

[26]  Robin Sibson,et al.  Computing Dirichlet Tessellations in the Plane , 1978, Comput. J..

[27]  J. Molnár.  Packing of congruent spheres in a strip , 1978 .

[28]  D. T. Lee,et al.  Two-Dimensional Voronoi Diagrams in the Lp-Metric , 1980, J. ACM.

[29]  F. Frances Yao,et al.  Computational Geometry , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[30]  Meike J. Wittmann,et al.  Mathematical Ecology , 2006 .