Graphics Reference
In-Depth Information
Figure 10.6.
How conics can intersect.
more uniformity. Of course, in practice one probably wants answers with real
numbers, but we shall see over and over again in this chapter that, in so far as it is
possible to give such answers, they are obtained by first dealing with the problems
over the complex numbers. Before we get down to business though, we need to clarify
the connection between affine and projective algebraic geometry and also cover some
algebraic preliminaries.
10.3
More on Projective Space
This section discusses some important aspects of projective space that were not
addressed before because they were not needed until now. We also show how to pass
back and forth between projective and affine space.
When k = R or C , then the space P n (k) is actually a differentiable manifold of
dimension n or 2n depending on whether k is R or C , respectively. In fact, P n ( C ) is
what is called an n-dimensional complex manifold , but since its complex manifold
structure will not play any role in this topic, we will have nothing further to say about
complex manifolds as such. It is important to note though that P n (k) looks locally just
like k n . More precisely, we can use equations similar to equations (8.45) and (8.46) in
Section 8.13 to show that the space can be covered by coordinate neighborhoods
( U i ,j i ), where, for i = 1, 2,..., n + 1,
= [
{
]
}
U i
cc
,
,...,
c
c
π
0
(10.18)
12
n
+
1
i
and
n
j i
: U Æ
k
i
is defined by
c
c
c
c
c
c
c
c
) = Ê
Ë
ˆ
¯
1
i
-
1
i
+
1
n
+
1
(
[
]
j i
cc
,
,...,
c
,...,
,
,...,
.
(10.19)
12
n
+
1
i
i
i
i
By identifying k n with U i via the homeomorphism j i , one can think of U i as consist-
ing of the “affine” points of P n (k) with respect to the coordinate neighborhood ( U i ,j i ).
The set P n (k) - U i , which is the hyperplane V(f), where
Search WWH ::




Custom Search