Rational Parametrization

Chapter 4 is the central chapter of the book. In this chapter, we focus on rational or parametric curves, we study different problems related to this type of curves and we show how to algorithmically parametrize a rational curve. The chapter consists of three conceptual blocks. The first one (Sect. 4.1) is devoted to the notion of rational parametrization of a curve, and to the study of the class of rational curves, i.e., curves having a rational parametrization. In the second block of the chapter (Sects. 4.2–4.5), we assume that a rational parametrization of a curve is provided and we consider various problems related to such a rational parametrization. The material in this part of the chapter follows the ideas in [SeW01a]. In Sect. 4.2 the injectivity of the parametrization is studied, in Sect. 4.3 we analyze the number of times the points on the curve are traced via the parametrization, in Sect. 4.4 the inversion problem for proper parametrizations is studied, and in Sect. 4.5 the implicitization question is addressed. The third block of the chapter (Sects. 4.6–4.8) deals with the problem of algorithmically deciding whether a given curve is rational, and in the affirmative case, of actually computing a rational parametrization of the curve. The material in this part of the chapter follows the ideas in [SeW91], which are based on [AbB87a, AbB87b, AbB88, AbB89]. In Sect. 4.6 we study the simple case of curves parametrizable by lines, in Sect. 4.7 these ideas are extended to the general case, and in Sect. 4.8, once the theoretical and algorithmic ideas have been developed, we show how to carry out all these algorithms symbolically.