Graphics Reference
In-Depth Information
(
)
2
2
2
2
2
xyz
+-=
0
r
x
+-=
y
1
0
using Cartesian coordinates
.
This shows that every such conic is projectively equivalent to the unit circle and we
are done.
To show that the conic y = x
2
is projectively equivalent to the unit
3.6.1.2. Example.
circle.
Solution.
Passing to homogeneous coordinates, the conic is defined by the equation
2
x z
-=
0
(3.64)
with associated symmetric matrix
Ê
ˆ
10 0
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
1
2
A =
00
-
.
1
2
0
-
0
Ë
¯
Using elementary matrices, we shall now show that A is congruent to a diagonal
matrix. First of all, if E is the elementary matrix E
23
(-1), then
Ê
ˆ
10 0
01
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
1
2
EAE
T
A
=
=
-
.
1
1
2
0
-
0
Ë
¯
Next, let F be the elementary matrix E
32
(1/2). Then
Ê
ˆ
10 0
01 0
Á
Á
Á
˜
˜
˜
T
A A
=
=
.
2
1
1
4
00
-
Ë
¯
Finally, if G is the elementary matrix E
33
(2), then
10 0
01 0
00 1
Ê
ˆ
Á
Á
˜
AGAG
T
=
=
˜
.
3
2
Ë
¯
-
It follows that if
