Notes on non-Euclidean geometry

Abstract

These are notes on plane hyperbolic geometry. The presentation is rather unusual since it is model-free. The methods can be used in the same way for spherical geometry. In particular, we derive the trigonometric formulae without using any Euclidean model. After a brief introduction to the axioms and the basic notions, we study in detail the geometry of triangles and of quadrilaterals. We discuss the notions of area and of dissection, and we derive the trigonometric formulae. Then we present two models of the hyperbolic plane, namely, a disk model whose automorphisms are projective maps, and a model that arises from algebra, whose points are the prime ideals of the ring of polynomials with one real variable. We study parallelism, horocycles, parabolic transformations and the Lobachevsky parallelism function. Finally, we describe a 3-dimensional space which contains the hyperbolic plane, the Euclidean plane and the elliptic space of spherical geometry. In this 3-dimensional space, we can make continuous transitions between objects in the three 2-dimensional geometries (hyperbolic, Euclidean and spherical) and we can watch the transformation of their geometrical properties.