Using the Mathematica environment for generating efficient 3D graphics

Mathematica is an integrated environment for symbolic transformation of mathematical formulas. This environment has applications in scientific computing, scientific visualization and education. Mathematica provides the ability to describe visualized objects in form of mathematical formulas and expressions. Such descriptions are more clear and concise than low-level C or C++ code. Many visualization systems require input in the form of (sometimes huge) data files, which is a disadvantage for highly interactive and animated 3D graphics applications. This is also the case for graphics expressed in Mathematica which are computed interpretively and saved in a static data form before display. This causes low graphic performance. In this paper we describe an approach which uses object geometry descriptions in the form of efficient program code instead of huge data files. We have built a tool that produces 3D visualizations of geometrical objects and object trajectories from mathematical specifications expressed as parametric functions in Mathematica. A compiler has been developed which generates efficient C++ code from such functions and symbolic expressions. This code is linked together with a powerful 3D browsing environment and uses OpenGL with possible hardware support. All the computations are performed within the visualizing application. Object geometry, color, etc. can be changed dynamically during animations. Thus the flexibility of interactive exploration of 3D scenes and animation become available for the end-user