  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.
