Graphics Reference
In-Depth Information
3
5
4
5
Ê
ˆ
-
Á
Á
˜
˜
4
5
3
5
Ë
¯
is its inverse. Therefore, F
-1
is the map defined by
)
(
)
-
1
(
)
=
(
(
)
-
TT
(2.22)
Fxy
,
xy
,
puu
,
12
or, in terms of equations,
(
)
+
(
)
xuxmuyn
11
¢=
-
-
12
(
)
+
(
)
yuxmuyn
¢=
-
-
.
(2.23)
21
22
Equations (2.20)-(2.23) are fundamental and worth remembering. They summarize
the main relationship between frames and motions.
We finish this discussion of frames with several examples.
2.2.8.2. Example.
To find the equations for the rotation about the origin which
rotates the point
A
= (2,0) into
B
= (1,÷
-
).
Solution.
All we have to do is to normalize
B
to
1
2
3
2
=
Ê
Ë
ˆ
¯
u
1
,
and combine this vector with the orthogonal vector
3
2
1
2
Ê
Ë
ˆ
¯
u
2
=-
,
(chosen so that the pair induces the standard orientation) to get the frame F = (
u
1
,
u
2
).
This frame defines the desired rotation.
2.2.8.3. Example.
To find a motion M that sends the origin to the point
A
= (3,0)
and the directed x-axis to the directed line
L
1
shown in Figure 2.14.
Solution.
Define a frame F = (
u
1
,
u
2
,
A
) by
AB
AB
1
5
2
5
=
Ê
Ë
ˆ
¯
u
=
,
,
1
2
5
1
5
Ê
Ë
ˆ
¯
u
2
=-
,
.
Then M = F does the job. In fact, M is a rigid motion.
