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A Construction for Computer Visualization of Certain Complex Curves

Andrew J. Hanson
Organization: Indiana University
Journal / Anthology

Notices of the American Mathematical Society
Year: 1994
Volume: 41
Issue: 9
Page range: 1156-1163

Computer graphics has proven to be a very attractive tool for investigating low-dimensional algebraic manifolds and gaining intuition about their properties. In principle, a computer image of any manifold described by algebraic equations can be produced by numerically solving the equations to generate a fixed tessellation or by using equivalent ray-tracing techniques. 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 self-intersections, and are more difficult to explore using other visualization tools such as submanifold selection, coordinate transformation, and deformations. Therefore, in order to use mathematical visualization systems such as Mathematica, Maple, or Macsyma, or high-performance interactive systems such as Geomview or MeshView, one would much prefer an explicit parametric representation of a manifold's geometry. Driven by this motivation, we found an extremely useful construction for parametric models of large families of complex curves, that is, 2-manifolds representing the solutions of single equations in two complex variables, or equivalently, corresponding pairs of equations in four real variables.

*Mathematics > Geometry > Plane Geometry
*Mathematics > Topology