# A Conformal Functional for Simplicial Surfaces

Alexander I. Bobenko, Martin P. Weidner

### Description

Consider a smooth quadratic conformal functional and its weighted version

$$W_2 = \sum_{e}\beta^2(e) \qquad W_{2,w} = \sum_{e} (n_i + n_j)\beta^2(e)$$

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 $\beta$.

By investigation of 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. Shown are also some applications in geometry processing.

### References

• Alexander I. Bobenko and Martin P. Weidner.
On a new conformal functional for simplicial surfaces.
preprint, 2015.
arXiv:1505.08054.

#### Prof. Dr. Alexander I. Bobenko   +

Projects: A01, A02, B02, C01, CaP, Z
University: TU Berlin, Institut für Mathematik, MA 881
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31424655
E-Mail: bobenko[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~bobenko/