Graphics Reference
In-Depth Information
2.2.6.4. Theorem.
(1) Translations and rotations of the plane are orientation-preserving motions.
(2) Reflections are orientation-reversing motions.
Proof.
The fact that translations are orientation-preserving motions follows imme-
diately from the definition since the identity map is certainly orientation preserving.
To prove that rotations are orientation preserving, it suffices to show, by Theorem
2.2.6.3, that any rotation R about the origin is orientation preserving since an arbi-
trary rotation is a composition of translations and a rotation about the origin. The
fact that such an R is orientation preserving follows from Theorems 1.6.6 and 2.2.2.1
and the fact that the matrix for the linear transformation R has determinant +1. This
proves (1).
To prove (2) note that the reflection S
x
about the x-axis is a linear transformation
with equation (2.12) that clearly has determinant -1 and hence is orientation revers-
ing. Next, Theorem 2.2.3.6 showed that an arbitrary reflection can be written in the
form T
-1
R
-1
S
x
RT, where T is a translation and R is a rotation about the origin. Prop-
erty (2) now follows from (1) and Theorem 2.2.6.3.
We can also justify Theorem 2.2.6.4(2) geometrically based on the intuitive idea
mentioned earlier that a motion of the plane is orientation reversing if for some 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 opposite orientations for
R
2
. To see this we shall
use the same notation as in the definition of a reflection in Section 2.2.3. If
P
is a
point not on
L
, then clearly
AQ
and
QP
form a basis for
R
2
and
()()
=
TT
A
Q
AQ
=
1
◊
AQ
+
0
◊
QP
()()
=¢ =◊
+
()
◊
TT
Q
P
QP
0
AQ
1
QP
.
The determinant of the matrix of coefficients that relates the original basis to the
transformed one is -1. This means that the two bases are in opposite orientation
classes.
2.2.6.5. Theorem.
A motion of the plane is orientation preserving if and only if it
is a rigid motion.
Proof.
Exercise.
Although it takes three points to specify a general motion of the plane, two points
suffice in the special case of rigid motions.
2.2.6.6. Theorem.
If M is a rigid motion of the plane and if M fixes two distinct
points, then M is the identity.
Proof.
This theorem is an immediate consequence of Theorems 2.2.5.7 and 2.2.6.4.
2.2.6.7. Corollary.
Two rigid motions of the plane that agree on two distinct points
must be identical.
