Discrete groups and visualization of three-dimensional manifolds

Abstract

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. CR Categories and Subject Descriptors: I.3.3 [Picture/Image Generation] display algorithms I.3.5 [Computational Geometry and Object Modeling Graphics]: geometric algorithms, hierarchy and geometric transformations, I.3.7 [Three dimensional Graphics and Realism] color, shading, shadowing, and texture Additional Key Words and Phrases: discrete group, tessellation, quotient space, projective geometry, hyperbolic geometry, spherical geometry, curvature, geodesic.