Updating and constructing constrained delaunay and constrained regular triangulations by flips

I discuss algorithms based on bistellar flips for inserting and deleting constraining (d - 1)-facets in d-dimensional constrained Delaunay triangulations (CDTs) and weighted CDTs, also known as constrained regular triangulations. The facet insertion algorithm is likely to outperform other known algorithms on most inputs. The facet deletion algorithm is the first proposed for d > 2, short of recomputing the CDT from scratch. An incremental facet insertion algorithm that begins with an unconstrained Delaunay triangulation can construct the CDT of a ridge-protected piecewise linear complex with nv vertices in O(nv[d / 2] + 1 log nv) time. Hence, in odd dimensions, CDT construction by incremental facet insertion is within a factor of log nv of worst-case optimal. Perhaps the most important feature of these algorithms is that they are relatively easy to implement.

[1]  Franz Aurenhammer,et al.  Pseudotriangulations from Surfaces and a Novel Type of Edge Flip , 2003, SIAM J. Comput..

[2]  C. Lawson.  Software for C1 Surface Interpolation , 1977 .

[3]  Adrian Bowyer,et al.  Computing Dirichlet Tessellations , 1981, Comput. J..

[4]  B. Joe.  Three-dimensional triangulations from local transformations , 1989 .

[5]  Raimund Seidel,et al.  The Upper Bound Theorem for Polytopes: an Easy Proof of Its Asymptotic Version , 1995, Comput. Geom..

[6]  Jonathan Richard Shewchuk,et al.  Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates , 1997, Discret. Comput. Geom..

[7]  D. T. Lee,et al.  Generalized delaunay triangulation for planar graphs , 1986, Discret. Comput. Geom..

[8]  Olivier Devillers,et al.  Fast and Robust Triangle-Triangle Overlap Test Using Orientation Predicates , 2003, J. Graphics, GPU, & Game Tools.

[9]  Herbert Edelsbrunner,et al.  Sliver exudation , 2000, J. ACM.

[10]  Fujio Yamaguchi,et al.  Computer-Aided Geometric Design , 2002, Springer Japan.

[11]  Barry Joe.  Construction of K-Dimensional Delaunay Triangulations Using Local Transformations , 1993, SIAM J. Sci. Comput..

[12]  J. Radon.  Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten , 1921 .

[13]  Jesús A. De Loera,et al.  The Number of Geometric Bistellar Neighbors of a Triangulation , 1999, Discret. Comput. Geom..

[14]  Charles L. Lawson,et al.  Properties of n-dimensional triangulations , 1986, Comput. Aided Geom. Des..

[15]  Raimund Seidel,et al.  Voronoi diagrams and arrangements , 1985, SCG '85.

[16]  Jonathan Richard Shewchuk,et al.  Sweep algorithms for constructing higher-dimensional constrained Delaunay triangulations , 2000, SCG '00.

[17]  Christopher J. Van Wyk,et al.  Static analysis yields efficient exact integer arithmetic for computational geometry , 1996, TOGS.

[18]  Herbert Edelsbrunner,et al.  Incremental topological flipping works for regular triangulations , 1992, SCG '92.

[19]  Herbert Edelsbrunner,et al.  Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms , 1988, SCG '88.

[20]  L. Chew.  Building Voronoi Diagrams for Convex Polygons in Linear Expected Time , 1990 .

[21]  L. Paul Chew,et al.  Guaranteed-Quality Triangular Meshes , 1989 .

[22]  Olivier Devillers,et al.  Walking in a triangulation , 2001, SCG '01.

[23]  D. F. Watson.  Computing the n-Dimensional Delaunay Tesselation with Application to Voronoi Polytopes , 1981, Comput. J..

[24]  Jonathan Richard Shewchuk,et al.  What is a Good Linear Element? Interpolation, Conditioning, and Quality Measures , 2002, IMR.

[25]  Leonidas J. Guibas,et al.  Kinetic data structures: a state of the art report , 1998 .

[26]  Mariette Yvinec,et al.  Triangulations in CGAL , 2002, Comput. Geom..

[27]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1988, SCG '88.

[28]  Jonathan Richard Shewchuk,et al.  Constrained Delaunay Tetrahedralizations and Provably Good Boundary Recovery , 2002, IMR.

[29]  Herbert Edelsbrunner,et al.  Sliver exudation , 1999, SCG '99.

[30]  Jonathan Richard Shewchuk,et al.  A condition guaranteeing the existence of higher-dimensional constrained Delaunay triangulations , 1998, SCG '98.

[31]  Jonathan Richard Shewchuk,et al.  Mesh generation for domains with small angles , 2000, SCG '00.

[32]  Gary L. Miller,et al.  Control Volume Meshes Using Sphere Packing , 1998, IRREGULAR.