2,149 research outputs found
Restricted Constrained Delaunay Triangulations
We introduce the restricted constrained Delaunay triangulation (restricted CDT), a generalization of both the restricted Delaunay triangulation and the constrained Delaunay triangulation. The restricted CDT is a triangulation of a surface whose edges include a set of user-specified constraining segments. We define the restricted CDT to be the dual of a restricted Voronoi diagram defined on a surface that we have extended by topological surgery. We prove several properties of restricted CDTs, including sampling conditions under which the restricted CDT contains every constraining segment and is homeomorphic to the underlying surface
Two triangulations methods based on edge refinement
In this paper two curvature adaptive methods of surface triangulation
are presented. Both methods are based on edge refinement to obtain a
triangulation compatible with the curvature requirements. The first
method applies an incremental and constrained Delaunay triangulation
and uses curvature bounds to determine if an edge of the triangulation
is admissible. The second method uses this function also in the edge
refinement process, i.e. in the computation of the location of a
refining point, and in the re-triangulation needed after the insertion
of this refining point. Results are presented, comparing both
approachesPostprint (published version
Kinetic and Dynamic Delaunay tetrahedralizations in three dimensions
We describe the implementation of algorithms to construct and maintain
three-dimensional dynamic Delaunay triangulations with kinetic vertices using a
three-simplex data structure. The code is capable of constructing the geometric
dual, the Voronoi or Dirichlet tessellation. Initially, a given list of points
is triangulated. Time evolution of the triangulation is not only governed by
kinetic vertices but also by a changing number of vertices. We use
three-dimensional simplex flip algorithms, a stochastic visibility walk
algorithm for point location and in addition, we propose a new simple method of
deleting vertices from an existing three-dimensional Delaunay triangulation
while maintaining the Delaunay property. The dual Dirichlet tessellation can be
used to solve differential equations on an irregular grid, to define partitions
in cell tissue simulations, for collision detection etc.Comment: 29 pg (preprint), 12 figures, 1 table Title changed (mainly
nomenclature), referee suggestions included, typos corrected, bibliography
update
Computing Three-dimensional Constrained Delaunay Refinement Using the GPU
We propose the first GPU algorithm for the 3D triangulation refinement
problem. For an input of a piecewise linear complex and a
constant , it produces, by adding Steiner points, a constrained Delaunay
triangulation conforming to and containing tetrahedra mostly of
radius-edge ratios smaller than . Our implementation of the algorithm shows
that it can be an order of magnitude faster than the best CPU algorithm while
using a similar amount of Steiner points to produce triangulations of
comparable quality
Constrained Construction of Planar Delaunay Triangulations without Flipping
The construction of Voronoi diagrams and Delaunay triangulations finds wide application in many branches of science. Delaunay triangulations have properties which make them more desirable than other triangulations for the same node set. Delaunay has characterized his triangulations by the empty circle property. The partitioning and flipping methods which have been developed for digital construction of Voronoi diagrams and Delaunay triangulations only make indirect use of this property. A novel method of construction is proposed, which is based directly on the empty circle property of Delaunay. The geometry of the steps of the algorithm is simple and can be grasped intuitively. The method can be applied to constrained triangulations, in which a triangulation domain and some of the edges are prescribed. A data structure for triangulations of concave and multiply-connected domains is presented which permits convenient specification of the constraints and the triangulation. The method is readily implemented, efficient and robust
Three-dimensional unstructured grid generation via incremental insertion and local optimization
Algorithms for the generation of 3D unstructured surface and volume grids are discussed. These algorithms are based on incremental insertion and local optimization. The present algorithms are very general and permit local grid optimization based on various measures of grid quality. This is very important; unlike the 2D Delaunay triangulation, the 3D Delaunay triangulation appears not to have a lexicographic characterization of angularity. (The Delaunay triangulation is known to minimize that maximum containment sphere, but unfortunately this is not true lexicographically). Consequently, Delaunay triangulations in three-space can result in poorly shaped tetrahedral elements. Using the present algorithms, 3D meshes can be constructed which optimize a certain angle measure, albeit locally. We also discuss the combinatorial aspects of the algorithm as well as implementational details
- …