Graphics Reference
In-Depth Information
Proof.
By translating and rotating the curve if necessary, we may assume that the
simple point is the origin and that it has a nonvertical tangent. Let h(X) be the power
series guaranteed by Lemma 10.12.7 and set
()
=
()
=+ +...
2
u
t
t
and
v
t
c t
c t
0
0
1
2
using the notation of that Lemma. It follows that (u
0
(t),v
0
(t)) determines a place of
the curve with center the origin. On the other hand, let (u(t),v(t)) be an arbitrary irre-
ducible parameterization of the curve with center the origin. By Theorem 10.12.4 we
may assume that ord(u), ord(v) > 0. Now
(
)
=
2
. .
.
fXcX cX
,
+
+
0
1
2
implies that
(
)
(
(
)
=- -
2
)
fXY
,
Y cX cX
-
...
gXY
,
1
2
for some polynomial g(X,Y) in k[[X]][Y]. The polynomial g(X,Y) has a nonzero con-
stant term since f has linear terms. Therefore,
(
)
(
(
)
=- -
2
)
=
fuv
,
v cu cu
-
...
guv
,
0
1
2
in k[[t]]. Because g(X,Y) has a nonzero constant term it is a unit in k[[t]]. Therefore
we must have
2
vcucu
=+ +
...,
1
2
which implies that ord v = 1, since the parameterization is irreducible. It follows that
(u(t),v(t)) and (u
0
(t),v
0
(t)) are equivalent parameterizations and hence determine the
same place. This proves the theorem.
Next, we handle singular points
p
of a plane curve. The idea will be to transform
the curve to a new curve so that if
q
is a point on the new curve corresponding to the
point
p
on the original curve, then
q
is no longer singular. By Theorem 10.12.8 the
new curve will have a place with center
q
and this place can be transformed back to
a place with center
p
on the original curve. As an example of the kind of transfor-
mation we have in mind, consider
UX V
Y
X
=
,
=
.
(10.79)
Definition.
The transformation defined by equations (10.79) is called a
(local) quad-
ratic transformation
with
center
(0,0).
First of all, here are some simple observations about the map in (10.79). As a
mapping from the plane to the plane, it is of course not defined for points on the y-
axis; however, consider a line Y = mX. The transformation (10.79) will send the points
of that line other than
0
to the line V = m. See Figure 10.16. What this means is that
if the origin is a singular point for a plane curve
C
and if the tangent lines to the curve
at the origin have slopes m
1
, m
2
,..., m
r
, then (0,m
i
) will be the points on the trans-
formed curve
C
¢ corresponding to the origin on
C
and
C
¢ will have tangent lines V =
