Visualizing hyperbolic space: unusual uses of 4x4 matrices

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. The Geometry Center is funded by the National Science Foundation, the Department of Energy, Minnesota Technology, Inc., and the University of Minnesota. The authors may be reached at: The Geometry Center, 1300 South Second Street, Minneapolis, MN 55407. (612) 626-0888. Email: mbpcllgeom.umn. edu, gnnn@geom. nmn. edu. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. a 1992 ACM 0-89791-471-6/92/0003/0209...$1.50 Introduction The use of 4 x 4 matrices to represent affine transformations of Euclidean 3-space is well-known in computer graphics. Most graphics languages include provisions for specifying 4 x 4 transformations, and most interactive graphics workstations have the ability to multiply 4 x 4 matrices in hardware. These capabilities were designed with Euclidean geometry in mind, because we think of the space in which we live as Euclidean 3-space. There are, however, alternate systems of geometry which are of interest in mathematics and physics research and education. One of the most important of these is hyperbolic geometry. Hyperbolic space arises naturally, even more so than Euclidean geometry, in the study and classification of 3-manifolds. It is also frequently taught in introductory geometry courses because it is in some sense the simplest and most elegant type of non-Euclidean geometry. Learning hyperbolic geometry forces one to challenge many assumptions which are usually taken for granted, in the process strengthening one’s geometric reasoning skills. The “space” of hyperbolic geometry consists of the interior of the unit ball in R3; the boundary of the ball, the unit sphere, is “at infinity”. Distance is redefined to approach infinity as we move closer to this sphere. From a hyperbolic point of view, therefore, we can never actually reach the boundary sphere. We can think of hyperbolic space as consisting of points, lines, planes, surfaces, etc, just as in Euclidean space. In hyperbolic space, however, some of the rules of geometry are different. Specifically, Euclid’s fifth postulate is not valid: in the hyperbolic plane there are many lines through a given point which do not intersect a given line. Another non-Euclidean property is that the sum of the angles in a planar polygon is always less than 180 degrees. It is possible, for example, to have a “regular right pentagon” (all five sides are equal and all five angles are 90 degrees). Figure 1 shows a tesselation (tiling) of hyper-