Implicitization of rational curves and polynomial surfaces

Bull. Korean Math. Soc. 2007 Vol. 44, No. 1, 13-29 Printed March 1, 2007

Jian-ping Yu and Yong-li Sun University of Science and Technology Beijing, Beijing University of Chemical Technology

Abstract : In this paper, we first present a method for finding the implicit equation of the curve given by rational parametric equations. The method is based on the computation of Gr\"{o}bner bases. Then, another method for implicitization of curve and surface is given. In the case of rational curves, the method proceeds via giving the implicit polynomial $f$ with indeterminate coefficients, substituting the rational expressions for the given curve and surface into the implicit polynomial to yield a rational expression $\displaystyle\frac{g}{h}$ in the parameters. Equating coefficients of $g$ in terms of parameters to $0$ to get a system of linear equations in the indeterminate coefficients of polynomial $f$, and finally solving the linear system, we get all the coefficients of $f$, and thus we obtain the corresponding implicit equation. In the case of polynomial surfaces, we can similarly as in the case of rational curves obtain its implicit equation. This method is based on characteristic set theory. Some examples will show that our methods are efficient.