Graphics Reference
In-Depth Information
Alternatively, rotate about the origin through the angle -q, where q is defined by
equation (3.55). This changes equation (3.35) into
af hg
ab
+
+
2
(3.59)
Iy
¢+
2
yc
¢+
=
0.
2
2
If there are real roots to this equation, then we get a factorization
(
)
(
)
=
yyyy
1
¢-
¢-
0,
2
which corresponds to one or two straight lines.
3.6.5. Example.
To transform
2
2
52
x
-
72
xy
+
73
y
+
8
x
-
294
y
+
333
=
0
(3.60)
into standard form.
Solution.
We have
52
-
36
4
Ê
ˆ
52
-
36
Ê
Ë
ˆ
¯
Á
Á
˜
˜
A
=
-
36
73
-
147
and
B
=
.
-
36
73
Ë
¯
4
-
147
333
Therefore, I = 125, D=det (A) =-250000, and D = det (B) = 2500 and we fall into Case
1. In fact, D > 0, Dπ0, and ID<0 means that we have an ellipse. After translating the
center of the ellipse, defined by equation (3.46), to the origin, we can finish the reduc-
tion of equation (3.60) in two ways: we can rotate through the angle specified by equa-
tions (3.36), or we could use the eigenvector approach indicated by equation (3.51).
Of course, we could also just simply use equations (3.48) and (3.50), but, although
this may seem simpler, the advantage of the other two methods is that they also give
us the coordinate transformation that transforms the standard coordinate system
into the one in which the curve has the standard form. One often needs to know this
transformation.
We start with the angle approach. Using equation (3.46), the center (x
0
,y
0
) of the
curve turns out to be (2,3). Substituting x + 2 and y + 3 for x and y in equation (3.60) gives
2
2
52
x
-
72
xy
+
73
y
-
100
=
0
.
(3.61)
We now want to rotate the coordinate axes through an angle -q, where, using formula
(3.36b),
q=-
4
3
3
4
tan
or
.
We arbitrarily choose 3/4, so that
