Geometriae Dedicata

, Volume 163, Issue 1, pp 215–274

Universal hyperbolic geometry I: trigonometry

Original Paper

DOI: 10.1007/s10711-012-9746-9

Cite this article as:
Wildberger, N.J. Geom Dedicata (2013) 163: 215. doi:10.1007/s10711-012-9746-9


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.


Hyperbolic geometryProjective geometryRational trigonometryRelativistic geometryNull points

Mathematics Subject Classification (2000)


Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUNSWSydneyAustralia