Manocha, D. and Canny, J. F. (February 1990) Rational Curves with Polynomial Parametrization, Technical Report no. UCB/CSD 90/560, Computer Science Division, University of California, Berkeley.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....d degQn . Since a = 1 and b = 2, the equation P d Q d = 0 has one real root and no other complex root. With a linear parameter change, we can achieve that this root is 0, so that P d = s m and Q d = s n . But then, P and Q can be written as polynomials in 1 s . b) c) See (Abhyankar, 1990; Manocha and Canny, 1991). It also follows by (H aux 1) in the last section of this paper. 2 Theorem 4.4. P exi 3) Let C be an algebraic curve that has no trigonometric parameterization. Then C has a partial trigonometric parameterization iff it has a polynomial parameterization. Proof. By plugging an arbitrary ....

Manocha, D., Canny, J. (1991). Rational curves with polynomial parametrization. Computer Aided Geometric Design, pages 12--19.

....w(t) of degree n has a single root of multiplicity n. Lemma I: Given a polynomial w(t) of degree n, the necessary and sufficient condition that it has a single root of multiplicity n is degree(GCD(w(t) w 0 (t) n Gamma 1; where w 0 (t) is the first derivative of w(t) Proof: [MC90b] Q.E.D. 6 We already know that degree of w 0 (t) n Gamma 1: Therefore, the lemma implies that w(t) has a single root of multiplicity n if and only if w 0 (t) divides w(t) Based on these results a simple and robust algorithm to decide whether w(t) has a single root of multiplicity n ....

Manocha, D. and Canny, J. F. (February 1990) Rational Curves with Polynomial Parametrization, Technical Report no. UCB/CSD 90/560, Computer Science Division, University of California, Berkeley.

....curve may have more than one place X Y Figure 1: A cubic curve with a loop at a singular point and has one place at every non singular point. In particular, the curve has two or more places at a node or loop and one place at a cusp. More on places and their representation as branches is given in [22, 1, 23, 21]. A rational parametric curve P(t) is properly parametrized if it has one to one relationship between the parameter t and points on the curve, except for a finite number of exceptional points. Let S be one of these exceptional points. In other words, there is more than one value of the parameter ....

D. Manocha and J.F. Canny. Rational curves with polynomial parametrization. Computer-Aided Design, 23(9):645--652, 1991.

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