Curvature of an Arbitrary Surface for Discrete Gravity and for $d=2$ Pure Simplicial Complexes

TL;DR

This paper introduces an efficient algorithm for computing the discrete curvature of arbitrary 2D surfaces embedded in 3D objects, with applications in discrete gravity and network geometry.

Contribution

It presents a novel method to numerically compute local metrics and curvature on 2D simplicial complexes with an $ ext{O}(N ext{log} N)$ algorithm, linking surface orientation and spin-connection solutions.

Findings

Provides an $ ext{O}(N ext{log} N)$ algorithm for curvature computation.

Enables numerical analysis of 2D surface curvature in 3D objects.

Facilitates applications in discrete gravity and network geometry.

Abstract

We propose a computation of curvature of arbitrary two-dimensional surfaces of three-dimensional objects, which is a contribution to discrete gravity with potential applications in network geometry. We begin by linking each point of the surface in question to its four closest neighbors, forming quads. We then focus on the simplices of $d=2$, or triangles embedded in these quads, which make up a pure simplicial complex with $d=2$. This allows us to numerically compute the local metric along with zweibeins, which subsequently leads to a derivation of discrete curvature defined at every triangle or face. We provide an efficient algorithm with $\mathcal{O}(N \log{N})$ complexity that first orients two-dimensional surfaces, solves the nonlinear system of equations of the spin-connections resulting from the torsion condition, and returns the value of curvature at each face.