Anisotropic voronoi diagrams and guaranteed-quality anisotropic mesh generation

Abstract

We introduce anisotropic Voronoi diagrams, a generalization of multiplicatively weighted Voronoi diagrams suitable for generating guaranteed-quality meshes of domains in which long, skinny triangles are required, and where the desired anisotropy varies over the domain. We discuss properties of anisotropic Voronoi diagrams of arbitrary dimensionality-most notably circumstances in which a site can see its entire Voronoi cell. In two dimensions, the anisotropic Voronoi diagram dualizes to a triangulation under these same circumstances. We use these properties to develop an algorithm for anisotropic triangular mesh generation in which no triangle has an angle smaller than 20 • , as measured from the skewed perspective of any point in the triangle.

Dense Glyph Sampling for Visualization

“…Due to the anisotropy of the metric, we use an anisotropic Voronoi diagram and an anisotropic centroid computation for the relaxation step. For the definition of the anisotropic Voronoi diagram and the centroid computation we built on the works of Labelle and Shewchuk [16] and Du et al [2]. Our method is a combination of these two methods, satisfying our demands.…”

“…The equations of motion are solved numerically to yield a force-balancing configuration. A geometric approach for anisotropic mesh generation was chosen by Du et al [2] and Labelle et al [16]. Both methods define a generalized Voronoi tessellation based on a non Euclidean metric using different distance approximations as basis for the final triangulation.…”

“…Anisotropic spot samples with certain characteristics, such as spatially varying density and size, have many applications in visualization and computer graphics ranging from glyph rendering and texture generation for visualization purposes [14,8,13,11,12,3], digital halftoning [19,15,25] to mesh generation [2,16,22]. While some of the desirable properties of the samples are similar across applications, the goals and appropriate sampling strategies are problem-dependent.…”

“…Next, we use a generalized anisotropic Lloyd relaxation to distribute the spot samples more evenly. Our anisotropic Lloyd relaxation is a straight-forward generalization of the isotropic version and is based on the work of Labelle and Shewchuk [16] and Du et al [2]. A parameter in the Voronoi cell definition controls the alignment of the ellipses.…”

Anisotropic Geodesics for Perceptual Grouping and Domain Meshing

“…Based on this geodesic distance, one can define the Riemannian Voronoi diagram of a point set [1] and its dual Delaunay complex to perform grouping and meshing tasks. Such geometrical structures are not easy to compute, and several approximate solutions have been proposed (see for instance [2], [3]). Here, we adopt an approach based on the Fast Marching algorithm to compute geodesic distances and Voronoi diagrams.…”