Differential Hyperbolic Geometry

Abstract

In this paper, we develop two model spaces for hyperbolic geometry using differential calculus. Our approach is to first develop the Euclidean model space R[2] and then mirror the development for hyperbolic geometry. The differential approach is advantageous because it provides a metric for each geometry. This enables us to develop the geometric isometries. A geometric isometry of two dimensional space X is a one-to-one function from X into X that preserves distance and preserves angles. Felix Klein in 1872 pioneered the viewpoint that a geometry is reflected by its isometries. Thus, by developing the concept of a hyperbolic metric we can find the hyperbolic isometries and hence better understand hyperbolic geometry. [From Introduction]