Tracing index of rational curve parametrizations

Abstract

A rational parametrization of an algebraic curve establishes a rational correspondence of this curve with the affine or projective line. This correspondence is a birational equivalence if the parametrization is proper. So, intuitively speaking, a rational parametrization determines a linear tracing of the curve, when the parameter takes values in the algebraic closure of the ground field. Such a parametrization might trace the curve once or several times. We formally introduce the concept of the tracing index of a parametrization, we show how to compute it, and we relate it to the degree of rational reparametrizations as well as to the degree of the curve. In addition, we show how to apply these results to the case of real curves, where we introduce the notion of real tracing index.

Structured matrices in the application of bivariate interpolation to curve implicitization

“…the degree in x, the degree in y) instead of the total degree. The following result (see [15] and [21])…”

Using implicit equations of parametric curves and surfaces without computing them: Polynomial algebra by values

“…The resultant of F (t, s) and G(t, s), with respect to t, is called the D-resultant or Taylor resultant of P (t) and Q(t) (see [15] and [1]). In addition, we suppose that the parametrization of the curve is proper [33,34]. Recall that a parametrization is said to be proper if it is injective for almost all the points of the curve, which implies that there is at most a finite number of points of the curve generated by more than one value of the parameter t. If the parametrization is not proper, then B(s) is identically zero.…”

A Grobner Bases Approach to the Detection of Improperly Parameterized Rational Curve

“…The above question is simple and can be answered by just checking for the genus of the curve. If the genus of a curve is zero, then it can be parameterized otherwise no parameterization exists for the curve [3][4][5][6][7]. Furthermore, it is only possible to use rational polynomial parametric equations to give an exact representation of a curve iff its genus is zero [8,9].…”