Universal hyperbolic geometry gives a purely algebraic approach to the subject that connects naturally with Einstein's special theory of relativity. In this paper, we give an overview of some aspects of this theory relating to triangle geometry and in particular the remarkable new analogues of midpoints called sydpoints. We also discuss how the generality allows us to consider hyperbolic geometry over general fields, in particular over finite fields.Keywords: rational trigonometry; universal hyperbolic geometry; sydpoints; finite fields
Two Famous Questions and a Projective/Algebraic Look at Hyperbolic GeometryWhile physicists have long pondered the question of the physical nature of the "continuum", mathematicians have struggled to similarly understand the corresponding mathematical structure. In the last decade, we have seen the emergence of rational trigonometry [1,2] as a viable alternative to traditional geometry, built not over a continuum of "real numbers", but rather algebraically over a general field, so also over the rational numbers, or over finite fields.Universal hyperbolic geometry (UHG) extends this understanding to the projective setting, yielding a new and broader approach to the Cayley-Klein framework (see [3]) for the remarkable geometry discovered now almost two centuries ago by Bolyai, Gauss and Lobachevsky as in [4][5][6]. See also [7,8] for the classical and modern use of projective metrical structures in geometry. In this paper, we will give an outline of this new approach, which connects naturally to the relativistic geometry of Lorentz, Einstein and Minkowski and also allows us to consider hyperbolic geometries over general fields, including finite fields.To avoid technicalities and make the subject accessible to a wider audience, including physicists, we aim to describe things both geometrically in a projective visual fashion, as well as algebraically in a linear algebraic setting.
The Polarity of a Conic Discovered by ApolloniusWe augment the projective plane, which we may regard as a two-dimensional affine plane and a line at infinity, with a fixed conic. This conic is called the absolute in Cayley-Klein geometry. In this universal hyperbolic geometry (UHG), developed in [9-12], we take it to be a circle, typically in blue, and call it the null circle.The polarity associated with a conic was investigated by Apollonius and gives a duality between points a and lines A = a ⊥ in the space, which we also write as a = A ⊥ . Given a point a, consider any two lines through a, which meet the conic at two points each as in Figure 1. The other two diagonal points of this cyclic quadrilateral defines the dual line A. Remarkably, this construction does not depend on the choice of lines through a, as Apollonius realized.