Topological Zoo. Mathematical visualization in virtual environment

Differential geometry and topology are the areas of higher mathematics, which particularly require visual representation of studying objects for better understanding of the material. On the other hand, the complex structure of such objects makes their visualization a challenge for modern graphical systems. In this paper we describe some methods useful for adequate representation of topologically non-trivial objects in virtual environment systems, particularly the acceleration of standard mechanisms for rendering of transparent surfaces, applicable for complex cases of self-intersecting time-dependent surfaces, not necessarily admitting continuous fields of normal vectors. We also use a simple technique of image-based rendering, which places stereo-photographs of real environments as static background for 3D virtual scenes. The methods are implemented in educational application, which demonstrates the main properties of elementary topological objects (Moebius band, Klein bottle, projective plane etc.) and gives a possibility to explore interactively the complex constructions, arising in higher mathematics.