Graphics Reference
In-Depth Information
k
d
f
:
V
Æ
(Theorem 10.16.9(2)). Such a map can be obtained by a succession of projections from
a point that leads to a projection of
V
with center a linear subspace of
P
n
(
C
). It is easy
to see that the integer d is characterized by the property that one can find an (n - d
- 1)-dimensional plane
X
in
P
n
(
C
) (the center of the projection) that does not inter-
sect
V
, but every (n - d)-dimensional plane in
P
n
(
C
) intersects
V
.
10.17
The Grassmann Varieties
Grassmann manifolds were defined in Section 8.14. We take another look at these
special manifolds, but from the point of view of algebraic geometry this time. The
Grassmann manifolds G
n
(
R
n+k
) can actually be thought of as subvarieties of projec-
tive space and are an important example of higher-dimensional varieties. An overview
of Grassmann varieties in the context of algebraic geometry can be found in [KleL72].
See also [Harr92] and [Shaf94].
Note that if
V
is an n-dimensional linear subspace of
R
n+k
and if B = (
v
1
,
v
2
,...,
v
n
)
is a basis for
V
, then the wedge product
v
1
Ÿ
v
2
Ÿ ...Ÿ
v
n
defines an element in the
N-dimensional vector space L
n
(
R
n+k
), where
nk
n
+
Ê
Ë
ˆ
¯
.
N
=
We shall identify L
n
(
R
n+k
) with
R
N
using the canonical basis
ee
ŸŸŸ
...
e
,
1
£<<<£+
i
i
...
i
nk
.
i
i
i
12
n
1
2
n
Changing the basis B for
V
will change the wedge product by a scalar multiple. There-
fore, the map
(
)
Æ
nk
+
N
-
1
m :G
n
R
P
defined by
()
=ŸŸŸ
[
]
m
Vvv
...
v
,
1
2
n
where (
v
1
,
v
2
,...,
v
n
) is a basis for
V
, is well defined. Now
Â
vv
ŸŸŸ =
...
v
a
e e
Ÿ ŸŸ
...
e
.
1
2
n
i i
...
i
i
i
i
12
n
1
2
n
1
£< <
ii
...
i nk
£+
12
n
Definition.
The homogeneous coordinates a
i
1
i
2
… i
n
are called the
Plücker coordinates
of
V
.
One can show that the Plücker coordinates are just the n ¥ n minors of the n ¥ (n
+ k) matrix whose rows are the vectors
v
i
.
