Non-Euclidean Geometry

In this country, the typical high school graduate has had at least some exposure to Euclidean geometry, but most lay-people are not aware that any other geometries exist. In this paper we provide an overview of the basics of hyperbolic geometry, one of many Non-Euclidean geometries, that should be accessible to anyone whose mathematical background includes geometry, trigonometry, and the calculus. We will begin with a brief history of geometry and the two hundred years of uncertainty about the independence of Euclid's fifth postulate, the resolution of which led to the development of several Non-Euclidean geometries. After an axiomatic development of neutral (absolute) and hyperbolic geometries, we will introduce the three major models of hyperbolic geometry, the Klein Disk, Poincare Disk and Upper Half-Plane Models. The Upper Half-Plane Model will aid us in our exploration of triangles, trigonometry, and circles in hyperbolic geometry, concluding with a discussion of hyperbolic in-circles and circum-circles. The development of dynamic geometry software has greatly facilitated exploration and hypothesis testing in hyperbolic models. The program Cabri II has been included, as well as macro files containing the basic constructions in each of the three major models. Detailed explanations of these constructions are included in the Appendix.