  Given two subsets of R^d, when does there exist a projective transformation
that maps them to two sets with a common centroid? When is this transformation
unique modulo affine transformations? We study these questions for 0- and
d-dimensional sets, obtaining several existence and uniqueness results as well
as examples of non-existence or non-uniqueness.
  If both sets have dimension 0, then the problem is related to the analytic
center of a polytope and to polarity with respect to an algebraic set. If one
set is a single point, and the other is a convex body, then it is equivalent by
polar duality to the existence and uniqueness of the Santal\'o point. For a
finite point set versus a ball, it generalizes the M\"obius centering of
edge-circumscribed convex polytopes and is related to the conformal barycenter
of Douady-Earle. If both sets are $d$-dimensional, then we are led to define
the Santal\'o point of a pair of convex bodies. We prove that the Santal\'o
point of a pair exists and is unique, if one of the bodies is contained within
the other and has Hilbert diameter less than a dimension-depending constant.
The bound is sharp and is obtained by a box inside a cross-polytope.
