A Construction for Computer Visualization of Certain Complex Curves

Computer graphics has proven to be a very attractive tool for investigating low-dimensional algebraic manifolds and gaining intuition about their properties [9]. In principle, a computer image of any manifold described by algebraic equations can be produced by numerically solving the equations [2] to generate a fixed tessellation, or by using equivalent ray-tracing techniques [5]. However, for high-performance interactive manipulation of a manifold, it is much simpler and more practical to have a parametric representation instead of an implicit equation that must be solved numerically; a significant additional feature of many parametric representations is that they embody symmetry information that can be used to further enhance the visualization process. Numerical solutions typically do not naturally emphasize natural structures of manifolds, are poorly behaved near singularities and selfintersections, and are more difficult to explore using other visualization tools such as submanifold selection, coordinate transformations, and deformations.