Area in non-Euclidean geometry

Abstract : We start by recalling the classical theorem of Girard on the area of a spherical triangle in terms of its angle sum, and its analogue in hyperbolic geometry. We then use a formula of Euler for the area of a spherical triangle in terms of side lengths and its analogue in hyperbolic geometry in order to give an equality for the distance between the midpoints of two sides of a spherical (respectively hyperbolic) triangle, in terms of the third side. These equalities give quantitative versions of the positivity (respectively negativity) of the curvature in the sense of Busemann. We present several other results related to area in non-Euclidean geometry.
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Contributor : Athanase Papadopoulos <>
Submitted on : Sunday, May 12, 2019 - 6:37:09 AM
Last modification on : Monday, May 13, 2019 - 6:54:14 AM

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Norbert A'Campo, Athanase Papadopoulos. Area in non-Euclidean geometry. In: Eighteen Essays in Non-Euclidean Geometry (V. Alberge and A. Papadopoulos, ed.), European Mathematical Soiety Publishing House, Zurich, p. 3-25., 2019, 978-3-03719-196-5. ⟨10.4171/196⟩. ⟨hal-02126615⟩

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