Generation of tubular and membranous shape textures with curvature functionals

TL;DR

This paper introduces a new curvature-based modeling approach for tubular and membranous shapes, providing a flexible framework that can be applied in biology, computer graphics, and architecture, supported by an efficient GPU algorithm.

Contribution

It generalizes the Helfrich energy to asymmetric curvature functionals and develops a novel phase-field method with proven Gamma-limsup convergence for shape optimization.

Findings

Validated on Willmore energy minimizers

Developed an efficient GPU implementation

Revealed a wide variety of shape textures

Abstract

Tubular and membranous shapes display a wide range of morphologies that are difficult to analyze within a common framework. By generalizing the classical Helfrich energy of biomembranes, we model them as solutions to a curvature optimization problem in which the principal curvatures may play asymmetric roles. We then give a novel phase-field formulation to approximate this geometric problem, and study its Gamma-limsup convergence. This results in an efficient GPU algorithm that we validate on well-known minimizers of the Willmore energy; the software for the implementation of our algorithm is freely available online. Exploring the space of parameters reveals that this comprehensive framework leads to a wide continuum of shape textures. This first step towards a unifying theory will have several implications, in biology for quantifying tubular shapes or designing bio-mimetic scaffolds, but also in computer graphics or architecture.