There are simple and robust refinements (almost) as good as Delaunay

Abstract

A new edge-based partition for triangle meshes is presented, the Seven Triangle Quasi-Delaunay partition (7T-QD). The proposed partition joins together ideas of the Seven Triangle Longest-Edge partition (7T-LE), and the classical criteria for constructing Delaunay meshes. The new partition performs similarly compared to the Delaunay triangulation (7T-D) with the benefit of being more robust and with a cheaper cost in computation. It will be proved that in most of the cases the 7T-QD is equal to the 7T-D. In addition, numerical tests will show that the difference on the minimum angle obtained by the 7T-QD and by the 7T-D is negligible.

Current Options for Visualization of Local Deformation in Modern Shape Analysis Applied to Paleobiological Case Studies

“…The concept of "localness" inevitably implies the choice of a series of evaluation points within the geometry subjected to a deformation. One of the most effective approaches, widely applied over a large spectrum of applications, is the Delaunay triangulation (Márquez et al, 2014;Dryden and Mardia, 2016). The construction of the triangulation proceeds iteratively by choosing the centroids of the initial sets of defined triangles as a new set of triangle's vertices.…”

“…In the case of two aligned shapes or of a PCA performed on a collection of shapes either, the notion of deformation always pertains to a pair of shapes, i.e., a source (X: the "undeformed" shape that can be a real shape or a sample's consensus) and a target (Y: representing the deformation of the source, i.e., a real shape or a shape predicted by an ordination axis). Recently, several contributions focused on the visualization of deformations by using different kinds of local measures from infinitesimal local area changes (Márquez et al, 2012(Márquez et al, , 2014 to velocity fields of local deformations (Kratz et al, 2013). Locally (see below), tensors are used to quantify local deformation.…”