Graphics Reference
In-Depth Information
A (
projective
)
hyperplane
in
P
n
is any subset of
P
n
of the form
Definition.
{
[
]
(
)
π
}
XX
,
,
◊◊◊
,
X
aX aX
+
+ ◊◊◊+
a X
=
0
,
for fixed
aa
,
,...,
a
0
.
12
n
+
1 11 22
n
+
1
n
+
1
12
n
+
1
A
k-dimensional projective plane
in
P
n
is any subset of
P
n
of the form
Definition.
...
{
[
]
XX
,
,...,
X
aX
+
aX
+
+
a
X
=
0
0
,
12
n
+
1
11
22
11
,
n
+
n
+
1
...
aX
+
aX
+
+
a
X
=
,
21
1
22
2
2
,
n
+
1
n
+
1
◊
◊
◊
...
}
aXaX
+
+
+
a
X
=
0
nk
-
,
11
nk
-
,
22
nkn
-
,
+
1
n
+
1
where (a
i1
,a
i2
,...,a
i,n+1
), i = 1,...,n - k, are a fixed set of n - k linearly independent
vectors. A one-dimensional projective plane in
P
n
is called a (
projective
)
line
.
One can show that a projective hyperplane in
P
n
is just an (n - 1)-dimensional
projective plane. A k-dimensional projective plane should be thought of as an imbed-
ded
P
k
(see Corollary 3.5.2 below). The ideal points in
P
n
form a hyperplane defined
by the equation
X
n+
=
0.
1
A projective line either consists entirely of ideal points or is an ordinary lines in
R
n
together with the single ideal point associated to the family of lines in
R
n
parallel to
that ordinary line.
Define a map
n
n
p :
P
-
ideal points
Æ
R
.
x
x
x
x
x
]
Æ
Ê
Ë
ˆ
¯
1
2
n
[
xx
,
,...,
x
,
,...,
12
n
+
1
x
n
+
1
n
+
1
n
+
1
The map p is called the
standard projection of
P
n
onto
R
n
.
Definition.
Note that the map p is not defined on all of
P
n
. It corresponds to finding the inter-
section of the line through the origin and (x
1
, x
2
,..., x
n+1
) in
R
n+1
with the plane x
n+1
= 1.
Like in the projective plane, there are many ways to coordinatize the points of
P
n
.
One can also define the cross-ratio of four points on a projective line.
A
projective transformation
or
projectivity
of
P
n
Definition.
is any one-to-one and
onto map T :
P
n
Æ
P
n
that preserves collinearity and the cross-ratio of points.
One can prove that projective transformation of
P
n
are defined by means of
homogeneous equations in n + 1 variables and can be described by means of
(n + 1) ¥ (n + 1) nonsingular matrices.
Definition.
Two figures
F
and
F
¢ are
projectively equivalent
if there is an projective
transformation T with T(
F
) =
F
¢.
