Surfaces from Circles

TL;DR

This paper explores discretizations of surface theory using M"obius invariant elements like circles and spheres, focusing on properties like Willmore energy and conformal parametrizations, and demonstrating convergence to smooth surfaces.

Contribution

It introduces a new discrete Willmore energy based on circle patterns and proves its convergence to the smooth functional, advancing surface discretization methods.

Findings

Discrete Willmore energy converges to the smooth version.

Circle pattern-based discretizations preserve conformal properties.

Special classes like minimal surfaces are effectively constructed discretely.

Abstract

In the search for appropriate discretizations of surface theory it is crucial to preserve such fundamental properties of surfaces as their invariance with respect to transformation groups. We discuss discretizations based on M\"obius invariant building blocks such as circles and spheres. Concrete problems considered in these lectures include the Willmore energy as well as conformal and curvature line parametrizations of surfaces. In particular we discuss geometric properties of a recently found discrete Willmore energy. The convergence to the smooth Willmore functional is shown for special refinements of triangulations originating from a curvature line parametrization of a surface. Further we treat special classes of discrete surfaces such as isothermic and minimal. The construction of these surfaces is based on the theory of circle patterns, in particular on their variational description.