Beyond the Descartes circle theorem

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dc.creatorLagarias, Jeffrey C.
dc.creatorMallows, Colin L.
dc.creatorWilks, Allan R.
dc.date2001-01-09
dc.date.accessioned2026-07-25T23:04:59Z
dc.descriptionThe 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.
dc.description25 pages, 6 figures, Latex
dc.identifierhttps://arxiv.org/abs/math/0101066
dc.identifierhttp://arxiv.org/abs/math/0101066
dc.identifierAmer. Math. Monthly 109 (2002), 338--361.
dc.identifier.urihttps://dspace.dare.co.zw/handle/123456789/95324
dc.subjectMetric Geometry
dc.subject52C26 (Primary), 11H55, 51M10, 53A35 (Secondary)
dc.titleBeyond the Descartes circle theorem
dc.typetext

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