On the Existence and the Coefficients of the Implicit Equation of Rational Surfaces

The existence of the implicit equation of rational surfaces can be proved by three techniques: elimination theory, undetermined coefficients, and the theory of field extensions. The methods of elimination theory and undetermined coefficients also reveal that the implicit equation can be written with coefficients from the coefficient field of the parametric polynomials. All three techniques can be implemented as implicitization algorithms. For each method, the theoretical limitations of the proof and the practical advantages and disadvantages of the algorithm are discussed. Our results are important for two reasons. First, we caution that elimination theory cannot be generalized in a straightforward manner from rational plane curves to rational surfaces to show the existence of the implicit equation; thus other rigorous methods are necessary to bypass the vanishing of the resultant in the presence of base points. Second, as an immediate consequence of the coefficient relationship, we see that the implicit representation involves only rational (or real) coefficients if a parametric representation involves only rational (or real) coefficients. The existence of the implicit equation means every rational surface is a subset of an irreducible algebraic surface. The subset relation can be proper and this may cause problems in certain applications in computer aided geometric design. This anomaly is illustrated by an example.