Incrementally Constructing and Updating Constrained Delaunay Tetrahedralizations with Finite Precision Coordinates

Abstract

Summary. Constrained Delaunay tetrahedralizations (CDTs) are valuable for generating meshes of nonconvex domains and domains with internal boundaries, but they are difficult to maintain robustly when finite-precision coordinates yield vertices on a line that are not perfectly collinear and polygonal facets that are not perfectly flat. We experimentally compare two recent algorithms for inserting a polygonal facet into a CDT: a bistellar flip algorithm of Shewchuk (Proc. 19th Annual Symposium on Computational Geometry, June 2003) and a cavity retriangulation algorithm of Si and Gärtner (Proc. Fourteenth International Meshing Roundtable, September 2005). We modify these algorithms to succeed in practice for polygons whose vertices deviate from exact coplanarity.

Fast segment insertion and incremental construction of constrained Delaunay triangulations

“…Shewchuk's algorithm for polygon insertion, which is based on bistellar flips, incurs two additional costs: an O(log n)-time cost per flip to perform priority queue (binary heap) operations that ensure that flips occur in the correct order; and the fact that the number of bistellar flips could far exceed the number of necessary structural changes because some tetrahedra are created only to be immediately deleted again during a single polygon insertion operation, though the total number of flips never exceeds O(n 2 ). Experiments [43] suggest that it is uncommon for the number of flips to exceed the number of deleted tetrahedra by more than a small constant, and our intuition is that such circumstances are analogous to the circumstances in which a Delaunay triangulation has a superlinear size-possible for inputs exhibiting a certain regular structure, but not the norm. For many, probably most, PLCs that arise in practice, g(n) ∈ O(n), so we anticipate that randomized incremental polygon insertion implemented with bistellar flips will often run in O(n log n log k log m) time in practice.…”

A 2D advancing-front Delaunay mesh refinement algorithm

“…As a result, constrained Delaunay meshes are smaller and easier to generate than truly Delaunay meshes. A number of incremental techniques to compute the constrained or truly Delaunay tetrahedralization [35,39,43,44] are known.…”