A Visual Introduction to Cubic Surfaces Using the Computer Software Spicy

Duco van Straten and Oliver Labs
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Duco van Straten: University of Mainz, Algebraic Geometry Group
Oliver Labs: University of Mainz, Algebraic Geometry Group

A chapter in Algebra, Geometry and Software Systems, 2003, pp 225-238 from Springer

Abstract: Abstract At the end of the 19th century geometers like Clebsch, Klein and Rodenberg constructed plaster models in order to get a visual impression of their surfaces, which are so beautiful from an abstract point of view. But these were static visualizations. Using the computer program Spicy 1, which was written by the second author, one can now draw algebraic curves and surfaces depending on parameters interactively. Using this software and Coble's explicit equations for the cubic surface that arises as the blowing-up of the projective plane in six points it was possible for the first time to visualize how some of the 27 lines upon the cubic surface coalesce when the surface develops a double point. When the user drags one of the six points, the equation and a ray-traced image of the cubic surface are computed using external programs. As the whole process takes less than half a second, one nearly gets the impression of a continuously changing surface.

Date: 2003
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DOI: 10.1007/978-3-662-05148-1_12

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