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