Visualizing Non-Euclidean Geometry A Bolyai Bicentennial Survey

The setting Like many fundamental mathematical discoveries, non-euclidean geometry was rst received as a bizarre oddity, but with time it has entered the mainstream of scientic thought. Indeed, there is growing acceptance of the idea that the universe we live in, is locally or globally non-euclidean. This makes the development of techniques for visualizing non-euclidean geometry of more than purely academic interest. This article will attempt to sketch what sorts of techniques have been developed, and what challenges still remain. The focus in the article is on hyperbolic geometry, although elliptic geometry is given attention also. After quickly reviewing the various mathematical models that have been developed for these geometries, we consider some simple examples of how graphics can aid the understanding of plane hyperbolic geometry. Then we will turn our attention to the more challenging case of three dimensions, and investigate in more depth two case studies of visualization projects in this area, one hyperbolic and one spherical. Here we indicate connections with the growing disciplines of 3D image synthesis and photorealistic rendering. Our exposition makes no attempt to be complete or rigorous, but aims to be a trail guide with references to sources for readers requiring more rigorous details.