Implicitization of rational parametric curves and surfaces

Abstract

In this paper we use Gröbner bases for the implicitization of rational parametric curves and surfaces in 3D-space. We prove that the implicit form of a curve or surface given by the rational parametrization $$x_1 : = \frac{{p_1 }}{{q_1 }}x_2 : = \frac{{p_1 }}{{q_2 }}x_3 : = \frac{{p_1 }}{{q_3 }}$$ where the p's and q's are univariate polynomials in y 1 or bivariate polynomials in y 1 , y 2 over a field K, can always be found by computing $$GB(\{ q_{^{_1 } } \cdot x_1 - p_{1,} q_2 \cdot x_2 - p_{2,} q_3 \cdot x_3 - p_{3,} \} ) \cap K[x_1 ,x_2 ,x_3 ],$$ where GB is the Gröbner basis with respect to the lexical ordering with x1≺x2≺x3≺y1≺y2, if for every i, j∈{1,2,3} with i≠j the polynomials p i , q i ,p j ,q j have no common zeros. This result leads immediately to an implicitization algorithm for arbitrary rational parametric curves.Furthermore, we present an algorithm for the implicitization of arbitrary rational parametric surfaces and prove its termination and correctness.