On a new conformal functional for simplicial surfaces

TL;DR

This paper introduces a new quadratic conformal functional for simplicial surfaces, demonstrating its effectiveness in minimizing conformal energies and applications like conformal surface deformation.

Contribution

The paper proposes a novel smooth quadratic conformal functional and its weighted version, with analysis of their minimizers and applications in geometry processing.

Findings

Minimizers of the functional tend to be inscribed polyhedra.

The functional effectively minimizes conformal energies.

Applications include conformal deformation of surfaces to spheres.

Abstract

We introduce a smooth quadratic conformal functional and its weighted version $$W_2=\sum_e \beta^2(e)\quad W_{2,w}=\sum_e (n_i+n_j)\beta^2(e),$$ where $\beta(e)$ is the extrinsic intersection angle of the circumcircles of the triangles of the mesh sharing the edge $e=(ij)$ and $n_i$ is the valence of vertex $i$. Besides minimizing the squared local conformal discrete Willmore energy $W$ this functional also minimizes local differences of the angles $\beta$. We investigate the minimizers of this functionals for simplicial spheres and simplicial surfaces of nontrivial topology. Several remarkable facts are observed. In particular for most of randomly generated simplicial polyhedra the minimizers of $W_2$ and $W_{2,w}$ are inscribed polyhedra. We demonstrate also some applications in geometry processing, for example, a conformal deformation of surfaces to the round sphere. A partial theoretical explanation through quadratic optimization theory of some observed phenomena is presented.