Metric methods in surface triangulation. (English) Zbl 1259.65047

Saucan, E.; Appleboim, E.

Hancock, Edwin R. (ed.) et al., Mathematics of surfaces XIII. 13th IMA international conference York, UK, September 7–9, 2009. Proceedings. Berlin: Springer (ISBN 978-3-642-03595-1/pbk). Lecture Notes in Computer Science 5654, 335-355 (2009).

Summary: We consider the problem of better approximating surfaces by triangular meshes. The approximating triangulations are regarded as finite metric spaces and the approximated smooth surfaces are viewed as their Haussdorff-Gromov limit. This allows us to define in a more natural way the relevant elements, constants and invariants, such as principal directions and Gauss curvature, etc. By a “natural way” we mean intrinsic, discrete, metric definitions as opposed to approximating or paraphrasing the differentiable notions. Here we consider the problem of determining the Gauss curvature of a polyhedral surface, by using the metric curvatures in the sense of Wald, Menger and Haantjes. We present three modalities of employing these definitions for the computation of Gauss curvature.
For the entire collection see [Zbl 1173.68011].

MSC:

65D18	Numerical aspects of computer graphics, image analysis, and computational geometry
65D17	Computer-aided design (modeling of curves and surfaces)