[Paper Review] A conformal energy for simplicial surfaces

Alexander I. Bobenko|arXiv (Cornell University)|Jun 7, 2004

Advanced Combinatorial Mathematics3 references4 citations

TL;DR

This paper introduces a Möbius-invariant conformal energy for simplicial surfaces, defined via the external intersection angles of circumcircles of adjacent triangles. It serves as a discrete analogue of the smooth Willmore functional, with minimizers corresponding to convex polyhedral surfaces inscribed in a sphere, and enables a new bending energy for discrete thin-shell models.

ABSTRACT

A new functional for simplicial surfaces is suggested. It is invariant with respect to Moebius transformations and is a discrete analogue of the Willmore functional. Minima of this functional are investigated. as an application a bending energy for discrete thin-shells is derived.

Motivation & Objective

• To develop a geometrically meaningful discrete analogue of the smooth Willmore functional for simplicial surfaces.
• To ensure invariance under Möbius transformations, a key symmetry of the smooth Willmore energy.
• To provide a variational framework for discrete surfaces that supports optimization and fairing applications.
• To derive a physically motivated bending energy for discrete thin shells based on the conformal energy.
• To establish connections between discrete conformal energy and curvature line nets in the smooth limit.

Proposed method

• Define the local conformal energy at each vertex as the sum of external intersection angles of circumcircles of adjacent triangles minus 2π.
• Construct the global conformal energy as half the sum over all vertices, equivalent to ∑β(e) − π|V| over all edges.
• Use Möbius invariance and geometric duality to prove non-negativity and characterization of minimizers.
• Leverage the limit of small dihedral angles to derive a bending energy proportional to l/L ⋅ θ², where l is edge length and L is distance between circumcenters.
• Apply the energy to derive a discrete bending model for thin-shell deformation, with empirical validation in prior work.
• Use the conformal energy to define optimal triangulations via combinatorial optimization, minimizing W(S) for fixed vertex sets.

Experimental results

Research questions

• RQ1 Can a discrete functional be constructed that is Möbius-invariant and approximates the smooth Willmore energy?
• RQ2 What is the geometric characterization of simplicial surfaces that minimize the conformal energy?
• RQ3 How does the conformal energy relate to the smooth Willmore functional in the limit of fine triangulation?
• RQ4 Can the conformal energy be used to derive a physically plausible bending energy for discrete thin shells?
• RQ5 What combinatorial triangulation minimizes the conformal energy for a fixed set of vertices on a sphere?

Key findings

• The conformal energy W(S) is non-negative and vanishes if and only if the surface is a convex polyhedron inscribed in a sphere.
• The conformal energy is invariant under Möbius transformations, preserving the key symmetry of the smooth Willmore functional.
• For a surface with all vertices on a 2-sphere, W(S) = 0 if and only if the triangulation is the Delaunay triangulation (convex hull boundary).
• In the limit of small dihedral angles, the conformal energy yields a bending energy proportional to l/L ⋅ θ², where L is the distance between circumcenters.
• The conformal energy approximates the smooth Willmore energy under refinement when vertices are sampled along curvature lines, but not under equilateral refinement.
• Minimizing the conformal energy over combinatorial types yields an optimal triangulation for fixed vertex data, with applications in surface fairing and reconstruction.