Graphics Reference
In-Depth Information
chapter on projective transformations. References to where proofs may be found are
given in those cases where difficult results are stated but not proved.
Finally, we want to emphasize one point. The single most important topic in
this chapter is that of frames. Frames are so simple (they are just orthonormal bases),
yet if the reader masters their use, then dealing with transformations will be a
snap!
2.2
Motions
A transformation M :
R
n
Æ
R
n
Definition.
is called a
motion
or
isometry
or
congru-
ent transformation
of
R
n
if
() ()
=
M
pq q
,
for every pair of points
p
,
q
Œ
R
n
.
In simple terms, motions are
distance-preserving maps
. If one concentrates on
that aspect, then the term “isometry” is the one that mathematicians normally use
when talking about distance-preserving maps between arbitrary spaces. The term
“motion” is popular in the context of
R
n
.
2.2.1. Theorem.
(1) Motions preserve the betweenness relation.
(2) Motions preserve collinearity and noncollinearity.
(3) Motions send lines to lines.
Proof.
To prove (1), let M be a motion and let
C
be a point between two points
A
and
B
. Let (
A
¢,
C
¢,
B
¢) = M(
A
,
C
,
B
). We must show that
C
¢
is between
A
¢
and
B
¢.
Now
AB =AB
=AC
¢¢
+
¢¢+ ¢¢
CB
=AC
CB
.
The first and third equality above follows from the definition of a motion. The second
follows from Proposition 1.2.3. Using Proposition 1.2.3 again proves (1). Parts (2) and
(3) of the theorem clearly follow from (1).
2.2.2. Lemma.
Let M be a motion. If
(
)
CA AB
=+
t
=-
1
t
A B
+
t
,
then
()
=
()
+
() ()
=-
(
)
(
)
+
()
M
CA AB
M
tM
M
1
t M
A B
tM
.
