“Discrete groups and visualization of three-dimensional manifolds” by Gunn

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.

1. Alan E Beardon. The Geometry of Discrete Groups. Springer-Verlag, 1983.
2. Carl B. Boyer. A History of Mathematics. Princeton University Press, 1968.
3. Manfredo P. Do Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976.
4. A. Cayley. A sixth memoir upon quantics. Philosophical Transactions of the Royal Society of London, 149:61-90, 1859.
5. H.M.S. Coxeter. Non-Euclidean Geometry. University of Toronto Press, 1965.
6. H.M.S. Coxeter. Regular Polytopes. Dover, 1973.
7. H.M.S. Coxeter. Projective Geometry. Springer Verlag, 1987.
8. D. B. A. Epstein, Jim Cannon, Derek Holt, Silvio Levy, Mike Patterson, and William Thurston. Word Processing in Groups. Jones and Bartlett, 1991.
9. Helaman Ferguson, Alyn Rockwood, and Jordan Cox. Topological design of sculptural surfaces. Computer Graphics, 26:149-156, July, 1992. Proceedings of SIGGRAPH 1992.
10. James Foley, Andries van Dam, Steven Feiner, and John Hughes. Computer Graphics: Principles and Practice. Addison-Wesley, 1990.
11. S. Gabriel and J. Kajiya. Spline interpolation in curved space. In State of the Art Image Synthesis, 1985. Course notes for SIGGRAPH 1985.
12. Charlie Gunn and Delle Maxwell. Not Knot. Jones and Bartlett, 1991.
13. Chaflie Gunn. A computer implementation of the twodimensional euclidean crystallographic groups. Master’s thesis, UNC, Chapel Hill, 1983.
14. Chaflie Gunn. Visualizing hyperbolic geometry. In Computer Graphics and Mathematics, pages 299-313. Eurographics, Springer Veflag, 1992.
15. Ping-Kang Hsiung and Robert H.E Dunn. Visualizing relativistic effects in spacetime. In Supercomputing 89. IEEE/ACM, Nov, 1989.
16. Silvio Levy. Automatic generation of hyperbolic tilings. Leonardo, 35:349-354, 1992.
17. E.H. Lockwood and R. H. Macmillan. Geometric symmetry. Cambridge University Press, 1978.
18. Tamara Munzner, Stuart Levy, Mark Phillips, Nathaniel Thurston, and Celeste Fowler. Geomview an interactive viewing program for sgi workstations, [email protected]
19. James Munkres. Topology: A First Course, chapter 8. Prentice-Hall, 1975.
20. Mark Phillips and Charlie Gunn. Visualizing hyperbolic space: Unusual uses of 4×4 matrices. In 1992 Symposium on Interactive 3D Graphics, pages 209- 214. ACM SIGGRAPH, ACM, 1992.
21. R.L.E. Schwarzenberger. N-Dimensional Crystallography. Pitman Publishing, 1980. chapters 13-16.
22. William Thurston. Three dimensional manifolds, kleinian groups and hyperbolic geometry. BAMS, 19:417-431, 1982.
23. Steve Upstill. The Renderman Companion. Addison- Wesley, 1989. chapters 13-16.
24. Jeff Weeks. snappea- a macintosh application for computing 3-manifolds. [email protected]
25. JeffWeeks. The Shape of Space. MarcelDekker, 1985.
26. Frederick Woods. Higher Geometry. Dover, 1961 (1922).