Rigidity of configurations of balls and points in the N-sphere

Crane, Edward and Short, Ian (2011). Rigidity of configurations of balls and points in the N-sphere. The Quarterly Journal of Mathematics, 62(2) pp. 351–362.

DOI: https://doi.org/10.1093/qmath/hap044

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 Möbius 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 Möbius transformations by the absolute cross-ratios of 4-tuples of its elements. The proofs use the hyperboloid model of hyperbolic (N + 1)-space.

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