The μ-basis of a planar rational curve—properties and computation

Abstract

A moving line L( x, y; t)=0 is a family of lines with one parameter t in a plane. A moving line L( x, y; t)=0 is said to follow a rational curve P (t) if the point P (t 0) is on the line L( x, y; t 0)=0 for any parameter value t 0. A μ-basis of a rational curve P (t) is a pair of lowest degree moving lines that constitute a basis of the module formed by all the moving lines following P (t) , which is the syzygy module of P (t) . The study of moving lines, especially the μ-basis, has recently led to an efficient method, called the moving line method, for computing the implicit equation of a rational curve [3,6]. In this paper, we present properties and equivalent definitions of a μ-basis of a planar rational curve. Several of these properties and definitions are new, and they help to clarify an earlier definition of the μ-basis [3]. Furthermore, based on some of these newly established properties, an efficient algorithm is presented to compute a μ-basis of a planar rational curve. This algorithm applies vector elimination to the moving line module of P (t) , and has O( n 2) time complexity, where n is the degree of P (t) . We show that the new algorithm is more efficient than the fastest previous algorithm [7].