Simple Methods For Drawing Rational Surfaces as Fouror Six

In this paper, we give several simple methods for drawing a whole rational surface (without base points) as several B ezier patches. The rst two methods apply to surfaces speciied by triangular control nets and partition the real projective plane RP 2 into four and six triangles respectively. The third method applies to surfaces speciied by rectangular control nets and partitions the torus RP 1 RP 1 into four rectangular regions. In all cases, the new control nets are obtained by sign ipping and permutation of indices from the original control net. The proofs that these formulae are correct involve very little computations and instead exploit the geometry of the parameter space (RP 2 or RP 1 RP 1). We illustrate our method on some classical examples. We also propose a new method for resolving base points using a simple \blowing up" technique involving the computation of \resolved" control nets.