Beyond the Descartes circle theorem

Details Beyond the Descartes circle theorem Beyond the Descartes circle theorem

2001-01-09

arxiv.org/abs/math/0101066v1

Amer. Math. Monthly 109 (2002), 338--361.

Mathematics

Metric Geometry

25 pages, 6 figures, Latex

Scientific paper

The Descartes circle theorem states that if four circles are mutually tangent with disjoint intersion, then their curvatures (or "bends) b_j = 1/r_j satisfy the relation (b_1 + b_2 + b_3 + b_4)^2 = 2(b_1^2 + b_2^2 + b_3^2 + b_4^2). We show that similar relations hold involving the centers of the circles in such a configuration, coordinatized as complex numbers, yielding a complex Descartes theorem. These relations have matrix generalizations to the n-dimensional case, in each of Euclidean, spherical and hyperbolic geometries, and they include a Descartes circle theorem for spherical and hyperbolic space.

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