Implicitization of Rational Parametric Curves and Surfaces

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 x1 := p1 q1 x2 := p2 q2 x3 := p3 q3 , where the p’s and q’s are univariate polynomials in y1 or bivariate polynomials in y1, y2 over a field K, can always be found by computing GB({q1 · x1 − p1, q2 · x2 − p2, q3 · x3 − p3}) ∩K[x1, x2, x3], 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 6= j the polynomials pi, qi, pj , qj 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.