In most virtual reality applications, 3-d space is a passive, ambient continuum in which the objects of study are placed. When the 3-d space itself is the object of study, as with mathematical manifolds, VR is especially important as a visualization medium. We describe the visualization of such spaces in the CAVE virtual environment.

We describe a software implementation for interactive visualization of a wide class of discrete groups.In addition to familiar Euclidean space, these groups act on the curved geometries of hyperbolic and spherical space.We construct easily computable models of our geometric spaces based on projective geometry; and establish algorithms for visualization of three-dimensional manifolds based upon the close connection between discrete groups and manifolds.We describe an object-oriented implementation of these concepts, and several novel visualization applications.As a visualization tool, this software breaks new ground in two directions: interactive exploration of curved spaces, and of topological manifolds modeled on these spaces.It establishes a generalization of the application of projective geometry to computer graphics, and lays the groundwork for visualization of spaces of non-constant curvature.

We briefly discuss hyperbolic geometry, one of the most useful and important kinds of non-Euclidean geometry.Rigid motions of hyperbolic space may be represented by 4 x 4 homogeneous transformations in exactly the same way as rigid motions of Euclidean space.This is a happy situation for those of us interested in visualizing what life in hyperbolic space might be like, because it means we can use existing graphics hardware and software libraries to animate scenes in hyperbolic space.We present formulas for computing reflections, translations, and rotations in hyperbolic space.These are a bit more complicated than the corresponding formulas for Euclidean geometry, which emphasizes our need for graphics libraries which allow completely arbitrary 4 X 4 transformations.The use of 4 x 4 transformations to represent isometries of hyperbolic space is not new; it has been used since the discovery of non-Euclidean geometry in the 19-century.The new part of our work is the application of this theory to real-time 3D computer graphics technology, which for the first time ever is allowing mathematicians to interactively explore hyperbolic geometry.