Graphics Reference
In-Depth Information
4
5
3
5
cos
q
=
and
sin
q
=
,
and the equation for the rotation R about the origin through the angle -q is
4
5
3
5
x
¢=
x
+
y
3
5
4
5
y
¢=-
x
+
y
.
Let
C
be the curve defined by (3.60). Then the curve
C
≤=RT(
C
) has equation
2
2
440
xy
+-=
(3.62)
and we are done.
Next, we use the eigenvalue approach. The eigenvalues are the roots of equation
(3.49), which reduces to
2
(
)
(
)
=
x
-
125
x
+
2500
=
x
-
25
x
-
100
0
in this case. To solve for the eigenvectors for eigenvalues 25 and 100 we must solve
(
)
=
(
)
=
(
)
2
(
)
2
xy
25
I
-
B
0
and
xy
100
I
-
B
0
,
that is,
-
27
36
48
36
Ê
Ë
ˆ
¯
)
Ê
Ë
ˆ
¯
(
)
=
(
)
(
=
(
)
x
y
00
and
x
y
00,
36
-
48
36
27
respectively. The solutions for the first equation are x = (4/3)y, so that
u
1
= (4/5,3/5) is
a unit eigenvector for eigenvalue 25. The solutions for the second equation are x =
(-3/4)y, so that
u
2
= (-3/5,4/5) is a unit eigenvector for eigenvalue 100. The sub-
stitution specified by equation (3.51) would then again give us equation (3.62). It
corresponds to the same rotation R described above.
One point that one needs to be aware of when using the eigenvalue approach is
that there is some leeway as to our choice of eigenvectors. Our only real constraint is
that the orthonormal basis (
u
1
,
u
2
) induce the standard orientation of the plane
because we want a rigid motion, specifically, a rotation. On the other hand, (
u
2
,-
u
1
)
would have been a legitimate alternative choice. This would have reduced our conic
equation to
2
2
4
x+-=.
4
0
But then, there are always basically two standard forms to which a general conic equa-
tion can be reduced. Which one we get depends on our choice of which axis we call
the x- and y-axis.
