Einstein's theory of relativity, as interpreted by Minkowski, introduced a new type of geometry based on the quadratic form x2+y2+z2-t2. Surprisingly, in the hundred years since, pure geometry based on this quadratic form has seen relatively little development---we seem reluctant to leave the familiar Euclidean setting, despite encouragement from our physicist friends.
In this talk we will show how relativistic geometry allows a fresh development of hyperbolic geometry, using the projective view of Cayley, Beltrami and Klein, and the theory of Rational Trigonometry, suitably modified.
This results in a simpler and more elegant hyperbolic trigonometry, increased accuracy for computations, and many new theorems. Additionally the connections with spherical/elliptic geometry become more natural, and the theory extends to general fields.