Implicitizing rational curves by the method of moving algebraic curves

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 curve x = x(t)/w(t), y = y(t)/w(t) if F(x(t)/w(t), y(t)/w(t), t) is identically zero. A 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 x n matrix, where each entry is a quadratic polynomial in x and, whereas standard resultant methods generate the implicit equation as the determinant of a 2n x 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. (C) 1997 Academic Press Limited.