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

A moving line Lðx; y; tÞ 1⁄4 0 is a family of lines with one parameter t in a plane. A moving line Lðx; y; tÞ 1⁄4 0 is said to follow a rational curve PðtÞ if the point Pðt0Þ is on the line Lðx; y; t0Þ 1⁄4 0 for any parameter value t0. A l-basis of a rational curve PðtÞ is a pair of lowest degree moving lines that constitute a basis of the module formed by all the moving lines following PðtÞ, which is the syzygy module of PðtÞ. The study of moving lines, especially the lbasis, 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 l-basis of a planar rational curve. Several of these properties and definitions are new, and they help to clarify an earlier definition of the l-basis [3]. Furthermore, based on some of these newly established properties, an efficient algorithm is presented to compute a lbasis of a planar rational curve. This algorithm applies vector elimination to the moving line module of PðtÞ, and has OðnÞ time complexity, where n is the degree of PðtÞ. We show that the new algorithm is more efficient than the fastest previous algorithm [7]. 2003 Elsevier Science (USA). All rights reserved.