Hyperbolic geometry and the Hillam-Thron theorem

Short, Ian (2006). Hyperbolic geometry and the Hillam-Thron theorem. Geometriae Dedicata, 119(1) pp. 91–104.

DOI: https://doi.org/10.1007/s10711-006-9053-4

Abstract

Every open ball within $\mathbb{R}\frac{N}{\infty}$ has an associated hyperbolic metric and Möbius transformations act as hyperbolic isometries from one ball to another. The Hillam–Thron Theorem is concerned with images of balls under Möbius transformation, yet existing proofs of the theorem do not make use of hyperbolic geometry. We exploit hyperbolic geometry in proving a generalisation of the Hillam–Thron Theorem and examine the precise configurations of points and balls that arise in that theorem.

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