Showing papers by "Herbert Edelsbrunner published in 2000"

Topological persistence and simplification

Abstract: We formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during growth as either a feature or noise, depending on its life-time or persistence within the filtration. We give fast algorithms for completing persistence and experimental evidence for their speed and utility.

Silver exudation

Abstract: A silver is a tetrahedon whose four vertices lie close to a plane and whose orthogonal projection to that plane is a convex quadrilateral with no short edge. Silvers are notoriously common in 3-dimensional Delaunay triangulations even for well-spaced point sets. We show that, if the Delaunay triangulation has the ratio property introduced in Miller et al. [1995], then there is an assignment of weights so the weighted Delaunay traingulation contains no silvers. We also give an algorithm to compute such a weight assignment.

Triangulations and meshes in computational geometry

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.

Smoothing and cleaning up slivers

Abstract: A sliver is a tetrahedron whose four vertices lie close to a plane and whose perpendicular projection to that plane is a convex quadrilateral with no short edge. Slivers axe both undesirable and ubiquitous in 3-dimensional Delaunay triangulations. Even when the point-set is well-spaced, slivers may result. This paper shows that such a point set permits a small perturbation whose Delaunay triangulation contains no slivers. It also gives deterministic algorithms that compute the perturbation of n points in time O(n logn) with one processor and in time O(log n) with O(n) processors.

Edgewise Subdivision of a Simplex

Abstract: In this paper we introduce the abacus model of a simplex and use it to subdivide a d-simplex into k d d-simplices all of the same volume and shape characteristics. The construction is an extension of the subdivision method of Freudenthal [3] and has been used by Goodman and Peters [4] to design smooth manifolds.

Fast software for box intersections

Abstract: We present fast implementations of a hybrid algorithm for reporting box and cube intersections. Our algorithm initially takes a divide-and-conquer approach and switches to simpler algorithms for low numbers of boxes. We use our implementations as engines to solve problems about geometric primitives. We look at two such problems in the category of quality analysis of surface triangulations.

Methods, apparatus and computer program products for automatically generating nurbs models of triangulated surfaces using homeomorphisms

Abstract: Embodiments automatically generate an accurate network of watertight NURBS patches from polygonal models of objects while automatically detecting and preserving character lines thereon. These embodiments generate from an initial triangulation of the surface, a hierarchy of progressively coarser triangulations of the surface by performing a sequence of edge contractions using a greedy algorithm that selects edge contractions by their numerical properties. Operations are also performed to connect the triangulations in the hierarchy using homeomorphisms that preserve the topology of the initial triangulation in the coarsest triangulation. A desired quadrangulation of the surface can then be generated by homeomorphically mapping edges of a coarsest triangulation in the hierarchy back to the initial triangulation. This quadrangulation is topologically consistent with the initial triangulation and is defined by a plurality of quadrangular patches. These quadrangular patches are linked together by a (U, V) mesh that is guaranteed to be continuous at patch boundaries. A grid is then preferably fit to each of the quadrangles in the resulting quadrangulation by decomposing each of the quadrangles into k 2 smaller quadrangles. A watertight NURBS model may be generated from the resulting quadrangulation.