  A variant of the Circle Packing Theorem states that the combinatorial class
of any convex polyhedron contains elements midscribed to the unit sphere
centered at the origin, and that these representatives are unique up to
M\"obius transformations of the sphere. Motivated by this result, various
papers investigate the problem of centering spherical configurations under
M\"obius transformations. In particular, Springborn proved that for any
discrete point set on the sphere there is a M\"obius transformation that maps
it into a set whose barycenter is the origin, which implies that the
combinatorial class of any convex polyhedron contains an element midsribed to a
sphere with the additional property that the barycenter of the points of
tangency is the center of the sphere. This result was strengthened by Baden,
Krane and Kazhdan who showed that the same idea works for any reasonably nice
measure defined on the sphere. The aim of the paper is to show that
Springborn's statement remains true if we replace the barycenter of the
tangency points by many other polyhedron centers. The proof is based on the
investigation of the topological properties of the integral curves of certain
vector fields defined in hyperbolic space. We also show that most centers of
Koebe polyhedra cannot be obtained as the center of a suitable measure defined
on the sphere.
