Sliver exudation

Abstract

A sliver is a tetrahedron whose four vertices lie close to a plane and whose orthogonal projection to that plane is a convex quadrilateral with no short edge. Slivers 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 triangulation contains no slivers. We also give an algorithm to compute such a weight assignment.

Manifold Reconstruction Using Tangential Delaunay Complexes

“…Our approach is based on two main ideas : the notion of tangential Delaunay complex defined in [20,6,19], and the technique of sliver removal by weighting the sample points [13]. The tangential complex is obtained by gluing local (Delaunay) triangulations around each sample point.…”

“…Input to our algorithm is a point sample coming from an unknown manifold. Our approach is based on two main ideas : the notion of tangential Delaunay complex defined in [6,19,20], and the technique of sliver removal by weighting the sample points [13]. Differently from previous methods, we do not construct any subdivision of the embedding d-dimensional space.…”

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

“…7 Every generic CDT (recall Definition 19) is a constrained regular triangulation. This fact is a consequence of the Delaunay Lemma and the fact that in a generic CDT, constrained regularity and constrained semiregularity are the same.…”

“…6 The vertex v might lie in several d-simplices of T (on a shared boundary), and Property D explicitly applies to only one of them. However, the lifted surface T + is continuous where simplices of T meet, so Property D holds for all the simplices in T that contain v. 7 Obviously, there is always an assignment of weights to the vertices of X missing from T that satisfies Property D of the Delaunay Lemma. Just make their weights really small.…”

“…For the goal of approximating ∇f (p) in three or more dimensions, the weighted CDT is sometimes far from optimal even for simple functions like the paraboloid f (p) = |p| 2 . Still, the CDT is a good starting point for mesh improvement algorithms [6,7,10,11,16,30,46,48] that create a triangulation that is appropriate for approximating both f (p) and ∇f (p).…”