Topic
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.
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TL;DR: In this paper, a special case of the critical point theory of distance functions is studied, namely, the gradient flow associated with the distance function to a finite point set in R^3.
Abstract: We study a special case of the critical point (Morse) theory of distance functions namely, the gradient flow associated with the distance function to a finite point set in R^3 The fixed points of this flow are exactly the critical points of the distance function Our main result is a mathematical characterization and algorithms to compute the stable manifolds, ie, the inflow regions, of the fixed points It turns out that the stable manifolds form a polyhedral complex that shares many properties with the Delaunay triangulation of the same point set We call the latter complex the flow complex of the point set The flow complex is suited for geometric modeling tasks like surface reconstruction
100 citations
TL;DR: This paper provides a critical survey of multiresolution terrain models, describing and discussing models proposed in the literature and construction algorithms for all such models together with an analysis of their time and space complexities.
Abstract: Multiresolution terrain models describe a topographic surface at various levels of resolution. Besides providing a data compression mechanism for dense topographic data, such models enable us to analyze and visualize surfaces at a variable resolution. This paper provides a critical survey of multiresolution terrain models. Formal definitions of hierarchical and pyramidal models are presented. Multiresolution models proposed in the literature (namely, surface quadtree, restricted quadtree, quaternary triangulation, ternary triangulation, adaptive hierarchical triangulation, hierarchical Delaunay triangulation, and Delaunay pyramid) are described and discussed within such frameworks. Construction algorithms for all such models are given, together with an analysis of their time and space complexities.
100 citations
07 Jun 1998
TL;DR: If the bounding segments of these facets are subdivided so that the subsegments are strongly Delaunay, then a constrained tetrahedralization exists and fewer vertices are needed than in the most common practice in the literature.
Abstract: Let X be a complex of vertices and piecewise linear constraining facets embedded in Ed. Say that a simplex is strongly Delaunay if its vertices are in X and there exists a sphere that passes through its vertices but passes through and encloses no other vertex. Then X has a d-dimensional constrained Delaunay triangulation if each k-dimensional constraining facet in X with k d 2 is a union of strongly Delaunay k-simplices. This theorem is especially useful in E3 for forming tetrahedralizations that respect specified planar facets. If the bounding segments of these facets are subdivided so that the subsegments are strongly Delaunay, then a constrained tetrahedralization exists. Hence, fewer vertices are needed than in the most common practice in the literature, wherein additional vertices are inserted in the relative interiors of facets to form a conforming (but unconstrained) Delaunay tetrahedralization.
99 citations
01 Jul 1990
TL;DR: A new randomized incremental algorithm for the construction of planar Voronoi diagrams and Delaunay triangulations is given which obviates the need for building a separate point-location structure for nearest-neighbor queries.
Abstract: In this paper we give a new randomized incremental algorithm for the construction of planar Voronoi diagrams and Delaunay triangulations. The new algorithm is more “online” than earlier similar methods, takes expected time O(n log n) and space O(n), and is eminently practical to implement. The analysis of the algorithm is also interesting in its own right and can serve as a model for many similar questions in both two and three dimensions. Finally we demonstrate how this approach for constructing Voronoi diagrams obviates the need for building a separate point-location structure for nearest-neighbor queries.
99 citations
TL;DR: This paper connects the predominantly combinatorial work in classical computational geometry with the numerical interest in mesh generation with the two- and three-dimensional case and covers results obtained during the twentieth century.
Abstract: The Delaunay triangulation of a finite point set is a central theme in computational geometry. It finds its major application in the generation of meshes used in the simulation of physical processes. This paper connects the predominantly combinatorial work in classical computational geometry with the numerical interest in mesh generation. It focuses on the two- and three-dimensional case and covers results obtained during the twentieth century.
99 citations