Delaunay triangulation

About: Delaunay triangulation is a research topic. Over the lifetime, 5816 publications have been published within this topic receiving 126615 citations. The topic is also known as: Delone triangulation.

Node-nested multi-grid method with Delaunay coarsening

Abstract: For finite-element non-structured type meshes, the non-nested multigrid algorithms require to build a sequence of independent meshes. The paper proposes an automatic way to generate the coarse meshes given the finest one. The method first eliminates a set of points from the current mesh level and then uses the Delaunay-Voronoi algorithm to triangulate the remaining set of points. The algorithm is presented and it is shown that it owns good properties with respect to multigrid algorithms. Several examples of its application to bi-dimensional meshes are presented.

Three-Dimensional Delaunay Mesh Generation

Abstract: We propose an algorithm to compute a conforming Delaunay mesh of a bounded domain in ${\Bbb R}^3$ specified by a piecewise linear complex. Arbitrarily small input angles are allowed, and the input complex is not required to be a manifold. Our algorithm encloses the input edges with a small buffer zone, a union of balls whose sizes are proportional to the local feature sizes at their centers. In the output mesh, the radius-edge ratio of the tetrahedra outside the buffer zone is bounded by a constant independent of the domain, while that of the tetrahedra inside the buffer zone is bounded by a constant depending on the smallest input angle. Furthermore, the output mesh is graded. Our work is the first that provides quality guarantees for Delaunay meshes in the presence of small input angles.

Analysis of Two Triangle-Based Multi-Surface Registration Algorithms of Irregular Point Clouds

Abstract: The registration of multiple surface point clouds into a common reference frame is a well addressed topic, and the Iterative Closest Point (ICP) is – perhaps – the most used method when registering laser scans due to their irregular nature. In this paper, we examine the proposed Iterative Closest Projected Point (ICPP) algorithm for the simultaneous registration of multiple point clouds. First, a point to triangular patch (i.e. closest three points) match is established by checking if the point falls within the triangular dipyramid, which has the three triangular patch points as a base and a user-chosen normal distance as the height to establish the two peaks. Then, the point is projected onto the patch surface, and its projection is then used as a match for the original point. It is also shown through empirical experimentation that the Delaunay triangles are not a requirement for establishing matches. In fact, Delaunay triangles in some scenarios may force blunders into the final solution, while using the closest three points leads to avoiding some undesired erroneous points. In addition, we review the algorithm by which the ICPP is inspired, namely, the Iterative Closest Patch (ICPatch); where conjugate point-patch pairs are extracted in the overlapping surface areas, and the transformation parameters between all neighbouring surfaces are estimated in a pairwise manner. Then, using the conjugate point-patch pairs, and applying the transformation parameters from the pairwise registration as initial approximations, the final surface transformation parameters are solved for simultaneously. Finally, we evaluate the assumptions made and examine the performance of the new algorithm against the ICPatch.

JIGSAW-GEO (1.0): locally orthogonal staggered unstructured grid generation for general circulation modelling on the sphere

Abstract: An algorithm for the generation of non-uniform, locally-orthogonal staggered unstructured spheroidal grids is described. This technique is designed to generate very high-quality staggered Voronoi/Delaunay meshes appropriate for general circulation modelling on the sphere, including applications to atmospheric simulation, ocean-modelling and numerical weather prediction. Using a recently developed Frontal-Delaunay refinement technique, a method for the construction of high-quality unstructured spheroidal Delaunay triangulations is introduced. A locally-orthogonal polygonal grid, derived from the associated Voronoi diagram, is computed as the staggered dual. It is shown that use of the Frontal-Delaunay refinement technique allows for the generation of very high-quality unstructured triangulations, satisfying a-priori bounds on element size and shape. Grid-quality is further improved through the application of hill-climbing type optimisation techniques. Overall, the algorithm is shown to produce grids with very high element quality and smooth grading characteristics, while imposing relatively low computational expense. A selection of uniform and non-uniform spheroidal grids appropriate for high-resolution, multi-scale general circulation modelling are presented. These grids are shown to satisfy the geometric constraints associated with contemporary unstructured C-grid type finite-volume models, including the Model for Prediction Across Scales (MPAS-O). The use of user-defined mesh-spacing functions to generate smoothly graded, non-uniform grids for multi-resolution type studies is discussed in detail.

A triangulated spatial model for cartographic generalisation of areal objects

Abstract: Cartographic generalisation involves interaction between individual operators concerned with processes such as object elimination, detail reductions amalgamation, typification and displacement. Effective automation of these processes requires a means of maintaining knowledge of the spatial relationships between map objects in order to ensure that constraints of topology and of proximity are obeyed in the course of the individual generalisation transformations. Triangulated spatial models, based on the constrained Delaunay triangulation, have proven to be of particular value in representing the proximal and topological relations between map objects and hence in performing many of the essential tasks of fully automated cartographic generalisation. These include the identification of nearby objects; determination of the structure of space between nearby objects; execution of boundary simplification, merge and collapse operations; and the detection and resolution, by displacement, of topological inconsistencies arising from individual operators. In this paper we focus on the use of a triangulated model for operations specific to execution of merge operations between areal objects. The model is exploited to identify the regions of space between nearby objects and to execute merge operations in which the triangulation is used variously to adopt intervening space and to move adjacent rectangular objects to touch each other. Methods for updating the triangulation are described.