Rational Parametrizations of Real Cubic Surfaces

Real cubic algebraic surfaces may be described by either implicit or parametric equations. Each of these representations has strengths and weaknesses and have been used extensively in computer graphics. Applications involving both representations include the efficient computation of surface intersections, and triangulation of curved surfaces. One particularly useful representation is the rational parametrization, where the three spatial coordinates are given by rational functions of two parameters. Rational parametrizations speed up many computations, and their relatively simple structure allows one to control and avoid singularities in the parametrization. These parametrizations take on different forms for different classes of cubic surfaces. Classification of real cubic algebraic surfaces into five families for the nonsingular case is based on the configuration of twentyseven lines on them. We provide a method of extracting all these lines and from there a rational parametrization of each of these families. The parametrizations of the real cubic surface components are constructed using a pair of real skew lines for those three families which have them, and remarkably using a complex conjugate pair of skew lines, in a fourth family. The parametrization is based on the fact that a real line generally intersects a cubic surface at three points. Points on the surface are obtained by intersecting the surface with lines that pass through points on the two skew lines. We also analyze the image of the derived rational parametrization for both real and complex parameter values, together with "base" points where the parametrizations are ill-defined.