Approximating Closed Rational Surfaces

In this chapter, the problem of drawing a closed rational surface is considered. As in the case of rational curves, the key to drawing a closed rational surface is to partition the parameter domain into simple connected regions Ri such as triangles or rectangles, in such a way that there is some prespecified region R0 and some projectivities such that every other region is the image of the region R0 under one of the projectivities. Furthermore, if the patch associated with the region R0 is given by a control net N0, we want the control net Ni associated with the region Ri to be computable very quickly from N0. However, since there are two kinds of rational surfaces, rectangular rational surfaces F : Ã× Ã → Ẽ and triangular rational surfaces F : P̃ → Ẽ , the situation is more complicated than in the case of curves.