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In-Depth Information
Proof.
The proof of this corollary is similar to the proof of Corollary 2.2.5.5.
2.2.6.8. Corollary.
If two orientation-reversing motions of the plane agree on two
distinct points, then they must be identical.
Proof.
If M and M¢ are the two orientation-reversing motions, then M¢ M
-1
is a rigid
motion that fixes two distinct points and hence is the identity map. It follows that
M = M¢.
2.2.6.9. Theorem.
A rigid motion of the plane that has a fixed point
p
is a rotation
about
p
.
Proof.
Exercise.
2.2.7
Summary for Motions in the Plane
We have defined motions and have shown that a motion of the plane is completely
specified by what it does to three noncollinear points and that it can be described in
terms of three very simple motions, namely, translations, rotations, and reflections.
To understand such motions it suffices to have a good understanding of these three
primitive types.
Planar motions are either orientation preserving or orientation reversing with
rigid motions being the orientation-preserving ones. Reflections are orientation
reversing. Another way to describe a planar motion is as a rigid motion or the com-
position of a rigid motion and a single reflection. In fact, we may assume that the
reflection, if it is needed, is just the reflection about the x-axis.
Combining various facts we know, it is now very easy to describe the equation of
an arbitrary motion of the plane.
2.2.7.1. Theorem.
Every motion M of the plane is defined by equations of the form
x x yc
¢=
+
+
(
)
+
y
¢=±-
bx
+
ay
d,
(2.19)
where a
2
+ b
2
= 1. Conversely, every such pair of equations defines a motion.
Proof.
Let M(
0
) = (c,d) and define a translation T by T(
P
) =
P
+ (c,d). Let M¢
=
T
-1
M. Then M = TM¢ and M¢ fixes the origin.
Case 1.
M is orientation preserving.
In this case M¢ is orientation preserving and must be a rotation about the origin
through some angle q (Theorem 2.2.6.9). Let a = cos q and b =-sin q. Clearly the equa-
tion for M has the desired form.
Case 2.
M is orientation reversing.
