Complexity of the delaunay triangulation of points on surfaces the smooth case

It is well known that the complexity of the Delaunay triangulation of N points in R 3, i.e. the number of its faces, can be O (N2). The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms first construct the Delaunay triangulation of a set of points measured on a surface.In this paper, we bound the complexity of the Delaunay triangulation of points distributed on generic smooth surfaces of R 3. Under a mild uniform sampling condition, we show that the complexity of the 3D Delaunay triangulation of the points is O(N log N).

[1]  Jean-Daniel Boissonnat,et al.  A Linear Bound on the Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces , 2004, Discret. Comput. Geom..

[2]  Jean-Daniel Boissonnat,et al.  Smooth surface reconstruction via natural neighbour interpolation of distance functions , 2002, Comput. Geom..

[3]  Olivier Devillers,et al.  The Delaunay Hierarchy , 2002, Int. J. Found. Comput. Sci..

[4]  Jeff Erickson,et al.  Nice Point Sets Can Have Nasty Delaunay Triangulations , 2001, SCG '01.

[5]  Hyeon-Suk Na,et al.  On the average complexity of 3D-Voronoi diagrams of random points on convex polytopes , 2003, Comput. Geom..

[6]  Jean-Daniel Boissonnat,et al.  Geometric structures for three-dimensional shape representation , 1984, TOGS.

[7]  Tamal K. Dey,et al.  Curve and Surface Reconstruction , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[8]  Jeff Erickson,et al.  Dense Point Sets Have Sparse Delaunay Triangulations or “... But Not Too Nasty” , 2001, SODA '02.

[9]  Jeff Erickson,et al.  Dense point sets have sparse Delaunay triangulations , 2001, ArXiv.

[10]  Tamal K. Dey,et al.  Delaunay based shape reconstruction from large data , 2001, Proceedings IEEE 2001 Symposium on Parallel and Large-Data Visualization and Graphics (Cat. No.01EX520).

[11]  Laurent Schwartz,et al.  Analyse : Topologie générale et analyse fonctionnelle , 1993 .

[12]  Hyeon-Suk Na,et al.  The probabilistic complexity of the Voronoi diagram of points on a polyhedron , 2002, SCG '02.

[13]  E. V. Anoshkina,et al.  Ridges, Ravines and Singularities , 1997 .