Discrete Differential Geometry for $C^{1,1}$ Hyperbolic Surfaces of Non-Constant Curvature

TL;DR

This paper introduces a discrete differential geometry framework for modeling non-constant negative curvature surfaces, with applications to biological phenomena and a novel iterative solution method for implicit immersion equations.

Contribution

It develops a new discrete differential geometry approach for $C^{1,1}$ hyperbolic surfaces of non-constant curvature, including an iterative solution and fast marching methods.

Findings

Successfully models non-constant curvature surfaces.

Demonstrates surface generation with branch points.

Provides a practical iterative solution method.

Abstract

We develop a discrete differential geometry for surfaces of non-constant negative curvature, which can be used to model various phenomena from the growth of flower petals to marine invertebrate swimming. Specifically, we derive and numerically integrate a version of the classical Lelieuvre formulas that apply to immersions of $C^{1,1}$ hyperbolic surfaces of non-constant curvature. In contrast to the constant curvature case, these formulas do not provide an explicit method for constructing an immersion but rather describe an immersion via an implicit set of equations. We propose an iterative method for resolving these equations. Because we are interested in scenarios where the curvature is a function of the intrinsic material coordinates, in particular, on the geodesic distance from an origin or from an edge, we suggest a fast marching method for computing geodesic distance on manifolds. We apply our methods to generate surfaces of non-constant curvature and demonstrate how one can introduce branch points to account for the multi-generational buckling and subwrinkling observed in many applications.