Graphics Reference
In-Depth Information
Figure 2.7.
Motions preserve angles.
M(C)
M(B)
C
M
q
B
q
A
M(A)
and
() ()
=
()
()
MM
AC
M M
A C
.
0
0
The theorem now follows from Lemma 2.2.4.3 applied to M
0
.
2.2.4.5. Corollary
.
Motions preserve angles.
See Figure 2.7. There is a converse to the results proved above.
2.2.4.6. Theorem.
A map that preserves the length of vectors and the angles
between them also preserves distance, that is, it is a motion.
Proof.
Exercise 2.2.4.1.
2.2.5
Some Existence and Uniqueness Results
Let
P
1
,
P
2
,...,
P
k
and
P
1
¢,
P
2
¢,...,
P
k
¢ , k ≥ 1, be two sequences of points in the plane.
We would like to determine when there is a motion M that sends
P
i
to
P
i
¢. Since
motions always preserve distances, a minimal requirement is that |
P
i
P
j
| = |
P
i
¢
P
j
¢| for
all i and j. Is this enough though?
Case 1.
k = 1.
There is no problem in this case. For example, the translation T(
Q
) =
Q
+
P
1
P
1
¢
would do the job. In fact, so would M = RT, where R is any rotation about
P
1
¢. In other
words, there are an infinite number of distinct motions that send
P
1
to
P
1
¢.
Case 2.
k = 2.
Assume, without loss of generality, that
P
1
π
P
2
. Consider the translation T defined
in Case 1 that sends
P
1
to
P
1
¢. By hypothesis, |
P
1
¢T(
P
2
)| = |
P
1
¢
P
2
¢|. Let a be the angle
between the vectors
P
1
¢T(
P
2
) and
P
1
¢
P
2
¢ and let R be the rotation about the point
P
1
¢
through the angle a. See Figure 2.8. It is easy to show that the motion M = RT does
what we want, as does the motion M¢
= SRT, where S is the reflection about the line
through
P
1
¢
and
P
2
¢. M and M¢ are clearly distinct.
Case 3.
k = 3.
