Universal hyperbolic geometry I: trigonometry |
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Authors: | N. J. Wildberger |
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Affiliation: | 1. School of Mathematics and Statistics, UNSW, Sydney, 2052, Australia
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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. |
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