par Bose, Prosenjit ;Cano, Pilar;Silveira, Rodrigo R.I.
Référence Computer aided geometric design, 91, 102039
Publication Publié, 2021-11
Article révisé par les pairs
Résumé : 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 (1993) that uses the inverse of the covariance matrix of S to define an affine invariant norm, denoted AS, and an affine invariant triangulation, denoted DTAS[S]. We revisit the AS-norm from a geometric perspective, and show that DTAS[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 AS-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. Our sorting methods provide a novel alternative to computing affine invariant geometric structures that are computationally simpler than using the AS-norm. In this work we focus on the theoretical part of affine invariant methods and as such do not provide experimental results.