Graphics Reference
In-Depth Information
The new definition of a place leads to a development of results in algebraic geometry
that parallels what we can do with our definition. For more on this approach see
Volume II of [ZarS60]. Valuation rings are also discussed in [Kend77].
10.13
Rational and Birational Maps
The material in this section is necessary background material for what we have to say
about when an implicitly defined plane curve can be parameterized in the next section.
We start with an example.
10.13.1. Example.
Consider the affine conic
C
defined by
(
)
=- +-=
2
2
fXY
,
X
XY Y
30
X
.
(10.80)
See Figure 10.18. The point
p
0
= (x
0
,y
0
) = (4,2) lies on
C
. The line
L
through
p
0
with
slope t has equation
Yt X
=+
2
(
-
4 .
)
(10.81)
To find the intersections of
L
with
C
, we simply need to substitute the right hand
side of equation (10.81) into (10.80) and solve for X. We already have one
intersection of
L
with
C
and it is easy to check that the second intersection (x
1
,y
1
) is
given by
2
14 4
1
25 2
1
-+
-+
t t
tt
x
=
,
1
2
2
-+
-+
t t
tt
y
=
.
1
2
The approach used in Example 10.13.1 works to parameterize any nondegenerate
affine conic curve
C
defined by
(1,2)
(x
0
,y
0
) = (4,2)
L
(x
1
,y
1
)
C
(1,-1)
Figure 10.18.
The conic defined by
equation (10.80).
