Implicitizing Rational Curves by the Method of Moving Algebraic Curves

Abstract

A function F (x, y, t) that assigns to each parameter t an algebraic curve F (x, y, t) = 0 is called a moving curve. A moving curve F (x, y, t) is said to follow a rational curveA new technique for finding the implicit equation of a rational curve based on the notion of moving conics that follow the curve is investigated. For rational curves of degree 2n with no base points the method of moving conics generates the implicit equation as the determinant of an n × n matrix, where each entry is a quadratic polynomial in x and y, whereas standard resultant methods generate the implicit equation as the determinant of a 2n × 2n matrix where each entry is a linear polynomial in x and y. Thus implicitization using moving conics yields more compact representations for the implicit equation than standard resultant techniques, and these compressed expressions may lead to faster evaluation algorithms. Moreover whereas resultants fail in the presence of base points, the method of moving conics actually simplifies, because when base points are present some of the moving conics reduce to moving lines.

Tracing index of rational curve parametrizations

“…Moreover, if properness is guaranteed, Theorem 7 shows that the implicit equation can be computed by a single resultant. This result can be found in Sederberg et al (1997), or can be easily deduced from Lemma 9. In addition to these results, in Theorem 8 we see that when computing the resultant the implicit equation appears to the power of the tracing index.…”

“…This approach is specially usefull for parametric varieties in K n . Also, for surfaces, different approaches can be found in González-Vega (1997), Sederberg et al (1997). However, for the case of plane curves, the implicit equation can be found by means of gcd's and resultants.…”

Structured matrices in the application of bivariate interpolation to curve implicitization

“…In this sense, we can read in [19] how the reduction of the order of the resultant matrix from 2n to n may lead to faster computations.…”

A Direct Approach to Computing theμ-basis of Planar Rational Curves

“…This yields f (x, y, w) in the form of the determinant of an n × n matrix if Bezout's resultant is used, or a 2n × 2n matrix if Sylvester's resultant is used (Sederberg et al, 1997). The latter considers the ideal c(t)x−a(t), c(t)y −b(t) ⊂ K [x, y, t] where K is a computable field of characteristic zero.…”