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].