General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

Abstract

Two-dimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have "nicely shaped" triangles that optimize several criteria, and they are easy to construct and update. The present work generalizes constrained Delaunay triangulations (CDTs) to higher dimensions and describes constrained variants of regular triangulations, here christened weighted CDTs and constrained regular triangulations. CDTs and weighted CDTs are powerful and practical models of geometric domains, especially in two and three dimensions.The main contributions are rigorous, theory-tested definitions of constrained Delaunay triangulations and piecewise linear complexes (geometric domains that incorporate nonconvex faces with "internal" boundaries), a characterization of the combinatorial properties of CDTs and weighted CDTs (including a generalization of the Delaunay Lemma), the proof of several optimality properties of CDTs when they are used for piecewise linear interpolation, and a simple and useful condition that guarantees that a domain has a CDT. These results provide foundations for reasoning about CDTs and proving the correctness of algorithms. Later articles in this series discuss algorithms for constructing and updating CDTs.

Lattice Cleaving: A Multimaterial Tetrahedral Meshing Algorithm with Guarantees

“…One strategy is to cut (or cleave) the cells of the input lattice to match the surface, an idea popularized by the well-known marching cubes algorithm for isosurfacing [31], and the related dual contouring method [27]. Constrained delaunay triangulation [15], [42] can then be applied to generate volume-filling tetrahedra [51]. The different configurations of surface/cell intersections are typically represented by stencils with the appropriate topology.…”

3D boundary recovery by constrained Delaunay tetrahedralization

“…A two‐dimensional pure CDT is the same as the one defined by Lee and Lin 18 and Chew 40. Shewchuk's definition of a CDT 41 is also a pure CDT.…”

Variational generation of prismatic boundary‐layer meshes for biomedical computing

“…Our approach to generating a boundary constrained tetrahedral core, where applicable, is due to Si [50]. We therefore outline our approach very generally and cite the excellent references that address this topic [51, 52, 53, 50]. Our approach to generating a non-boundary constrained tetrahedral mesh is presented in [12].…”