Graphics Reference
In-Depth Information
Proof.
This also follows from Theorem 2.2.7.1 since by that theorem c =-kb and
d = ka for some k =±1 and a
2
+ b
2
= 1.
Claims 1 and 2 can be summarized by saying that there is a one-to-one corre-
spondence between frames and motions. The special case where
p
=
0
shows that
there is a one-to-one correspondence between orthonormal bases and motions that
fix
0
.
2.2.8.1. Example.
Consider the rotation R about the origin defined by
3
5
4
5
x
¢=
x
-
y
4
5
3
5
y
¢=
x
+
y
.
3
5
4
5
4
5
3
5
Ê
Ë
ˆ
¯
=
Ê
Ë
ˆ
¯
The vectors
u
1
=-
,
and
u
2
,
clearly form an orthonormal basis.
Definition.
The motion T
F
, usually simply denoted by F, is called
the motion defined
by F
.
Using “F” to denote both the frame F and the motion T
F
should not cause any
confusion since it will always be clear from the context as to whether we are talking
about the frame or the map.
The observations above lead to a simple way to get the inverse of a motion.
Consider equations (2.20) again. Let R be the motion
u
u
)
Ê
Ë
ˆ
¯
1
(
)
=
(
Rxy
,
xy
,
2
and T, the translation
()
=+
(
,.
T
qq
m n
(Note that R is actually a rotation if the frame is oriented.) Then, as maps, F = TR
and F
-1
= R
-1
T
-1
. But it is easy to check that
u
u
10
01
Ê
Ë
ˆ
)
=
Ê
Ë
ˆ
¯
1
¯
(
)
=
(
TT
uu
u u
•
i
j
12
2
which shows that the inverses of the matrices
u
u
Ê
Ë
ˆ
¯
1
(
)
TT
and
uu
12
2
are just their transposes. Considering Example 2.2.8.1 again, note that the transpose of
