9.
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(
t0) is on the line
L(
x,
y;
t0)=0 for any parameter value
t0. 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 and 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(
n2) 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].
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