Graphics Reference
In-Depth Information
Proof.
Exercise.
Definition.
If ord(h) = 1, then we say that the parameterizations (X
h
(t),Y
h
(t),Z
h
(t))
and (X(t),Y(t),Z(t)) are
equivalent
.
10.12.3. Proposition.
Being equivalent is an equivalence relation on the set of
parameterizations for a plane curve. Equivalent parameterizations have the same
center.
Proof.
Exercise.
Another way of looking at the relation “equivalent” is that it corresponds to an
automorphism of k[[t]] over k.
Definition.
A parameterization of the form (X(t),Y(t),Z(t)), with X(t), Y(t), Z(t) Œ
k[[t
m
]] and m > 1, is called
reducible
; otherwise it is called
irreducible
.
We shall be interested in irreducible parameterization because reducible ones can
be simplified using the substitution s = t
m
.
Up to now we have used homogeneous coordinates. Now let us translate every-
thing to affine coordinates. Assume that Z(t) π 0 in a parameterization (X(t),Y(t),Z(t)).
Let
(
)
=
(
1
fXY
,
FXY
,
,
be the affine equation of the curve
C
. We can think of
()
()
Xt
Zt
()
=
Xt
()
()
Yt
Zt
()
=
Yt
as an affine parameterization. It satisfies
(
()
()
)
=
fXt Yt
,
0
(10.78)
Conversely, given a
n
y
X
(t
) and
Y
(t) satisfying equation (10.78), we get a projective
parameterization ( (t), (t),1).
Note that in the interesting special case where ord(
Z(
t)) = 0
it
follows from
Theorem B.11.13 that 1/Z(t) belongs to k[[t]] so that both (t) and (t) also belong
to k[[t]]. Since we can always assume that one of ord(X(t)), ord(Y(t)), or ord(Z(t)) is
zero, there is always one coordinate system with respect to which the affine curve is
parameterized by formal power series.
X
Y
X
Y
Definition.
An (
affine
)
parameterization
of
C
(with respect to a given coordinate
system) is a pair (X(t),Y(t)) of rational functions X(t), Y(t) Πk((t)) satisfying
