Visualizing hyperbolic space
Mark B. Phillips, Charlie Gunn · 1992
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.