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.

On Synge-type angle condition for $d$-simplices

“…The dihedral angle α between two such facets is defined by means of the inner product of their outward Figure 1. Classification of degenerated tetrahedra according to [9] and [11].…”

A Flexible Content-Adaptive Mesh-Generation Strategy for Image Representation

“…That is, given a face of the triangulation with vertices (x i , y i ), (x j , y j ), and (x k , y k ), and the corresponding sample values z i = f (x i , y i ), z j = f (x j , y j ), and z k = f (x k , y k ), we form the unique planar interpolant passing through the points (x i , y i , z i ), (x j , y j , z j ), and (x k , y k , z k ). In step 2, the Delaunay triangulation [30] is employed, due largely to its good properties for approximation. In many applications, it is desirable that the triangulation of S be uniquely determined by S alone, as this avoids the need for additional side information during the triangulation process.…”

Anisotropic mesh adaptation for image representation

“…Triangular meshes have recently received considerable interest in adaptive sampling for image representation [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. One common approach is to find proper sample points then connect the points to form a mesh.…”