@article{9ae271b2b93e49138b962a023d8c97fa, title = "Rigidity of configurations of balls and points in the N-sphere", abstract = "We answer two questions of Beardon and Minda which 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 Mbius 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 up to Mbius transformations by the absolute cross-ratios of 4-tuples of its elements. The proofs use the hyperboloid model of hyperbolic (N 1)-space.", author = "ET Crane and I Short", note = "Publisher: Oxford University Press", year = "2011", month = "6", doi = "10.1093/qmath/hap044", language = "English", volume = "62", pages = "351 -- 362", journal = "Quarterly Journal of Mathematics", issn = "0033-5606", publisher = "Oxford University Press", number = "2", }