Computation of Circular Area and Spherical Volume Invariants via Boundary Integrals
Numerical Analysis
2019-05-07 v1 Computational Geometry
Computer Vision and Pattern Recognition
Differential Geometry
Abstract
We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the Divergence Theorem to express the area and volume integrals as line and surface integrals, respectively, against particular kernels; our results also extend to higher dimensional hypersurfaces. The resulting surface integrals are computable analytically on a triangulated mesh. This gives a simple computational algorithm for computing the spherical volume invariant for triangulated surfaces that does not involve discretizing the ambient space. We discuss potential applications to feature detection on broken bone fragments of interest in anthropology.
Cite
@article{arxiv.1905.02176,
title = {Computation of Circular Area and Spherical Volume Invariants via Boundary Integrals},
author = {Riley O'Neill and Pedro Angulo-Umana and Jeff Calder and Bo Hessburg and Peter J. Olver and Chehrzad Shakiban and Katrina Yezzi-Woodley},
journal= {arXiv preprint arXiv:1905.02176},
year = {2019}
}