Graphics Reference
In-Depth Information
Figure 6.26.
Transforming a line.
6.10.1
Example.
Let
L
be the line in the plane defined by
x--10
(6.44)
and let T be the rotation about the origin through the angle of ninety degrees. The
equations for T and T
-1
are
-1
Tx
:
¢=-
y
T
:
x
¢=
y
yx
¢=
y x
¢=-
.
If
L
¢=T (
L
), then from what we just said above, the equation for
L
¢ is gotten from
equation (6.44) by substituting y and -x for x and y, respectively. This gives
-
()
-=+-=
yx
1
xy
1
0
(6.45)
That this is the correct answer can easily be checked. See Figure 6.26. Simply take
two points on
L
and find the equation of the line through the image of these two points
under T. For example, the points (0,-1) and (1,0) map to (1,0) and (0,1), respectively,
and equation (6.45) contains these two points.
6.11
E
XERCISES
Section 6.5
6.5.1
Find the intersection of the segments [(2,1),(6,-2)] and [(-1,-3),(7,1)].
6.5.2
Find a formula for the intersection of a ray with a segment in the plane.
6.5.3
Find the intersection of the line
L
defined by
x
=+
=
=- +
2
3
t
yt
z
52
t
