A Library for Visualizing Three-Dimensional Non-Euclidean Spaces

Non-Euclidean spaces and geometries are applied throughout many areas of knowledge, namely, physics and mathematics. Some of their properties, however, may be hard to comprehend and perceive, especially by people who are unfamiliar with the subject area. Using computer graphics to visualize such spaces would help to demonstrate various properties of such spaces, and we would also be able to allow users to examine them interactively. A library that is developed by us and that allows visualizing dynamic and interactive scenes in various three-dimensional non-Euclidean spaces is presented in this paper. For the purpose of non-Euclidean visualization, the method based on exploiting non-Euclidean spaces’ transformation groups has been chosen and is described in the paper. This method allows us to define various transformations of objects present in non-Euclidean spaces using matrices similar to those traditionally used in computer graphics to visualize the Euclidean space. One advantage of it is being able to exploit the capabilities of graphics processing units (GPUs). For that purpose, the Direct3D 11 graphics API has been utilized. Additionally, to simplify navigation in such spaces, we implemented the distance fog effect, which makes it easier to differentiate between nearer and farther objects. The library allows visualizing dynamic and interactive scenes in the spherical, elliptic, hyperbolic and toric spaces. Fields of its possible application are briefly discussed, namely, using the library to visualize the Friedmann model of general relativity.