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Universal hyperbolic geometry. I: Trigonometry. (English) Zbl 1277.51018

For a joint review of parts I and II see [KoG 14, 3–24 (2010; Zbl 1217.51009)].
Author’s abstract: Hyperbolic geometry is developed in a purely algebraic fashion from first principles, without a prior development of differential geometry. The natural connection with the geometry of Lorentz, Einstein and Minkowski comes from a projective point of view, with trigonometric laws that extend to ‘points at infinity’, here called ‘null points’, and beyond to ‘ideal points’ associated to a hyperboloid of one sheet. The theory works over a general field not of characteristic two, and the main laws can be viewed as deformations of those from planar rational trigonometry. There are many new features; this paper gives 92 foundational theorems.

MSC:

51M10 Hyperbolic and elliptic geometries (general) and generalizations
14N05 Projective techniques in algebraic geometry

Citations:

Zbl 1217.51009
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References:

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