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.