Distributed mean curvature on a discrete manifold for Regge calculus

TL;DR

This paper introduces a natural way to define pointwise mean curvature on discrete manifolds using hybrid cells, bridging Regge Calculus and Discrete Differential Geometry for better geometric understanding.

Contribution

It proposes hybrid cells formed from the simplicial lattice and its dual as a natural structure for distributing local integrated curvature in discrete manifolds.

Findings

Hybrid cells form a complete tessellation of the manifold.

They contain a geometric orthonormal basis.

They provide a pointwise mean curvature as a fractional change of the normal vector.

Abstract

The integrated mean curvature of a simplicial manifold is well understood in both Regge Calculus and Discrete Differential Geometry. However, a well motivated pointwise definition of curvature requires a careful choice of volume over which to uniformly distribute the local integrated curvature. We show that hybrid cells formed using both the simplicial lattice and its circumcentric dual emerge as a remarkably natural structure for the distribution of this local integrated curvature. These hybrid cells form a complete tessellation of the simplicial manifold, contain a geometric orthonormal basis, and are also shown to give a pointwise mean curvature with a natural interpretation as a fractional rate of change of the normal vector.