84,284 research outputs found
Well-Centered Triangulation
Meshes composed of well-centered simplices have nice orthogonal dual meshes
(the dual Voronoi diagram). This is useful for certain numerical algorithms
that prefer such primal-dual mesh pairs. We prove that well-centered meshes
also have optimality properties and relationships to Delaunay and minmax angle
triangulations. We present an iterative algorithm that seeks to transform a
given triangulation in two or three dimensions into a well-centered one by
minimizing a cost function and moving the interior vertices while keeping the
mesh connectivity and boundary vertices fixed. The cost function is a direct
result of a new characterization of well-centeredness in arbitrary dimensions
that we present. Ours is the first optimization-based heuristic for
well-centeredness, and the first one that applies in both two and three
dimensions. We show the results of applying our algorithm to small and large
two-dimensional meshes, some with a complex boundary, and obtain a
well-centered tetrahedralization of the cube. We also show numerical evidence
that our algorithm preserves gradation and that it improves the maximum and
minimum angles of acute triangulations created by the best known previous
method.Comment: Content has been added to experimental results section. Significant
edits in introduction and in summary of current and previous results. Minor
edits elsewher
Triangulation of Simple 3D Shapes with Well-Centered Tetrahedra
A completely well-centered tetrahedral mesh is a triangulation of a three
dimensional domain in which every tetrahedron and every triangle contains its
circumcenter in its interior. Such meshes have applications in scientific
computing and other fields. We show how to triangulate simple domains using
completely well-centered tetrahedra. The domains we consider here are space,
infinite slab, infinite rectangular prism, cube and regular tetrahedron. We
also demonstrate single tetrahedra with various combinations of the properties
of dihedral acuteness, 2-well-centeredness and 3-well-centeredness.Comment: Accepted at the conference "17th International Meshing Roundtable",
Pittsburgh, Pennsylvania, October 12-15, 2008. Will appear in proceedings of
the conference, published by Springer. For this version, we fixed some typo
Dense point sets have sparse Delaunay triangulations
The spread of a finite set of points is the ratio between the longest and
shortest pairwise distances. We prove that the Delaunay triangulation of any
set of n points in R^3 with spread D has complexity O(D^3). This bound is tight
in the worst case for all D = O(sqrt{n}). In particular, the Delaunay
triangulation of any dense point set has linear complexity. We also generalize
this upper bound to regular triangulations of k-ply systems of balls, unions of
several dense point sets, and uniform samples of smooth surfaces. On the other
hand, for any n and D=O(n), we construct a regular triangulation of complexity
Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available
at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm
Generating families of surface triangulations. The case of punctured surfaces with inner degree at least 4
We present two versions of a method for generating all triangulations of any
punctured surface in each of these two families: (1) triangulations with inner
vertices of degree at least 4 and boundary vertices of degree at least 3 and
(2) triangulations with all vertices of degree at least 4. The method is based
on a series of reversible operations, termed reductions, which lead to a
minimal set of triangulations in each family. Throughout the process the
triangulations remain within the corresponding family. Moreover, for the family
(1) these operations reduce to the well-known edge contractions and removals of
octahedra. The main results are proved by an exhaustive analysis of all
possible local configurations which admit a reduction.Comment: This work has been partially supported by PAI FQM-164; PAI FQM-189;
MTM 2010-2044
Curvature bounds for surfaces in hyperbolic 3-manifolds
We prove existence of thick geodesic triangulations of hyperbolic 3-manifolds
and use this to prove existence of universal bounds on the principal curvatures
of surfaces embedded in hyperbolic 3-manifolds.Comment: 21 pages, 9 figures, published version, added figures, fixed typo
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