Curvature corrected estimates for geodesic arc-length

TL;DR

This paper introduces relations between geodesic arc-lengths constrained to subspaces, with applications in curvature estimation on meshes and improved area error bounds for Schwarz lanterns.

Contribution

It provides new formulas linking geodesic lengths in subspaces and demonstrates their utility in curvature and area estimation problems.

Findings

Enhanced convergence of Gaussian curvature estimates on meshes

Improved error bounds for Schwarz lantern area calculations

Validated relations through practical examples

Abstract

We will develop simple relations between the arc-lengths of a pair of geodesics that share common end-points. The two geodesics differ only by the requirement that one is constrained to lie in a subspace of the parent manifold. We will present two applications of our results. In the first example we explore the convergence of Gaussian curvature estimates on a simple triangular mesh. The second example demonstrates an improved error estimate for the area of a Schwarz lantern.