Graphics Reference
In-Depth Information
(
)
(
)
x
-
a
s
in
q
+-
y
b
c
os
q
+ b = x s
in
q
+
y
c
os
q
+
d
for a and b. The details are left as an exercise.
2.2.6.2. Theorem.
The
set
of all translations and rotations of the plane is a sub-
group of the group of all motions. The set of rotations by itself is not a group.
Proof.
To prove the theorem one uses Lemma 2.2.6.1 to show that the composites
of translations and rotations about an arbitrary point are again either a translation
or a rotation.
Definition.
A motion of the plane that is a composition of translations and/or
rotations is called a
rigid motion
or
displacement
.
Rigid motions are closely related to orientation-preserving maps. We defined that
concept in Section 1.6 for linear transformations and we would now like to extend
the definition to motions. Motions are not linear transformations, but by Theorem
2.2.4.2 they differ from one by a translation. Intuitively, we would like to say that a
motion M of the plane is “orientation preserving” if for every three noncollinear points
A
,
B
, and
C
the ordered pairs of basis vectors (
AB
,
AC
) and (M(
A
)M(
B
),M(
A
)M(
C
))
determine the same orientation of
R
2
. See Figure 2.11. This definition would be messy
to work with and so we take a different approach.
Let M be a motion in
R
n
. By Theorem 2.2.4.2 we can write M uniquely in the form
M = TM
0
, where T is a translation and M
0
is a motion that fixes the origin. Theorem
2.2.4.1 implies that M
0
is a linear transformation.
Definition.
The motion M is said to be
orientation preserving
if M
0
is. Otherwise, M
is said to be
orientation reversing
.
2.2.6.3. Theorem.
(1) A motion M is orientation preserving if and only if M
-1
is orientation
preserving.
(2) The composition MM¢ of two motions M and M¢ is orientation preserving if and
only if either both are orientation preserving or both are orientation reversing.
(3) The composition M
1
M
2
...M
k
of motions M
i
is orientation preserving if and
only if the number of orientation-reversing motions M
i
is even.
Proof.
The proof is left as an exercise. It makes heavy use of Theorem 2.2.4.2 to
switch translations from one side of a motion that fixes the origin to the other.
C
M(B)
B
M(C)
A
M(A)
Figure 2.11.
An orientation-preserving motion.
