Visualization of Complex Physical Phenomena and Mathematical Objects in Virtual Environment

In this paper we present a several topics in visualization and animation of topologically non-trivial objects: the open strings in 3D space; a process of un-coupling of 2 coupled handles on a sphere in 3D space by means of a homotopy in the class of embedding; the projective plane and the projective 3D space. Various properties of these objects are visualized: their non-orientability and non-trivial connectivity, and two representations of the projective plane in 3D space are examined: the cross-cap and the Boy immersion.Visualization in string theory and some results obtained from visual study of strings demonstrates the benefits of this approach.Problems are discussed that arise in computer assisted visualization of topologically non-trivial objects, and in their integration in virtual environments, in particular: fast rendering of self-crossing transparent surfaces, and smooth mapping of colors and textures onto topologically complex surfaces.