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In Chaps. 4 and 5 we have studied different problems related to rational curves, assuming that the original curve was given implicitly. In this chapter we consider similar problems but now from another point of view, namely, we assume the curve is given in parametric form. But this parametric representation might not be optimal with respect to various criteria. So we want to transform such a rational parametrization into a better one. This new statement of the problem is specially interesting in some practical applications in CAGD, where objects are often given and manipulated parametrically. More precisely, we will focus on three different types of criteria: properness of parametrizations in Section 6.1, polynomiality of parametrizations in Section 6.2, and normality of parametrizations in Section 6.3. There are other criteria that might be considered. For instance, as we did in Chap. 5, one may ask for the algebraic optimality of the parametrization. For this case we refer to [ARS97], [ARS99], [ARS04], [RSV04], [SeV01], [SeV02].
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