  We answer two questions of Beardon and Minda that arose from their study of
the conformal symmetries of circular regions in the complex plane. We show that
a configuration of closed balls in the $N$-sphere is determined up to
M\"{o}bius transformations by the signed inversive distances between pairs of
its elements, except when the boundaries of the balls have a point in common,
and that a configuration of points in the $N$-sphere is determined by the
absolute cross-ratios of 4-tuples of its elements. The proofs use the
hyperboloid model of hyperbolic $(N+1)$-space.
