Affine invariant triangulations

TL;DR

This paper explores affine invariant triangulation methods, demonstrating their properties, extensions to other geometric graphs, and proposing new affine invariant sorting techniques for points and polygons.

Contribution

It revisits and geometrically interprets Nielson's affine invariant triangulation, extending it to related structures and introducing new affine invariant sorting methods.

Findings

The affine invariant triangulation retains properties like being 1-tough and containing a perfect matching.

It can be viewed as a Delaunay triangulation of a transformed point set.

The method extends to other geometric graphs such as MST and Gabriel graph.

Abstract

We study affine invariant 2D triangulation methods. That is, methods that produce the same triangulation for a point set $S$ for any (unknown) affine transformation of $S$. Our work is based on a method by Nielson [A characterization of an affine invariant triangulation. Geom. Mod, 191-210. Springer, 1993] that uses the inverse of the covariance matrix of $S$ to define an affine invariant norm, denoted $A_{S}$, and an affine invariant triangulation, denoted ${DT}_{A_{S}}[S]$. We revisit the $A_{S}$-norm from a geometric perspective, and show that ${DT}_{A_{S}}[S]$ can be seen as a standard Delaunay triangulation of a transformed point set based on $S$. We prove that it retains all of its well-known properties such as being 1-tough, containing a perfect matching, and being a constant spanner of the complete geometric graph of $S$. We show that the $A_{S}$-norm extends to a hierarchy of related geometric structures such as the minimum spanning tree, nearest neighbor graph, Gabriel graph, relative neighborhood graph, and higher order versions of these graphs. In addition, we provide different affine invariant sorting methods of a point set $S$ and of the vertices of a polygon $P$ that can be combined with known algorithms to obtain other affine invariant triangulation methods of $S$ and of $P$.