Coarse-grained curvature tensor on polygonal surfaces

TL;DR

This paper introduces a new coarse-grained curvature tensor for polygonal surfaces, which converges quickly to the smooth surface curvature tensor and captures principal curvatures and directions accurately.

Contribution

It defines a local coarse-grained curvature tensor on polygonal surfaces using integral geometry, with an efficient algorithm and convergence properties.

Findings

Converges rapidly to smooth surface curvature tensor

Captures principal curvatures and directions accurately

Compatible with existing computational frameworks

Abstract

Using concepts from integral geometry, we propose a definition for a local coarse-grained curvature tensor that is well-defined on polygonal surfaces. This coarse-grained curvature tensor shows fast convergence to the curvature tensor of smooth surfaces, capturing with accuracy not only the principal curvatures but also the principal directions of curvature. Thanks to the additivity of the integrated curvature tensor, coarse-graining procedures can be implemented to compute it over arbitrary patches of polygons. When computed for a closed surface, the integrated curvature tensor is identical to a rank-2 Minkowski tensor. We also provide an algorithm to extend an existing C++ package, that can be used to compute efficiently local curvature tensors on triangulated surfaces.