Affine Geometry - Computer Graphics and Geometric Modeling - page 86

Let S be the reflection about the x-axis, that is, S(x,y) = (x,-y). Since M¢ is orien-
tation reversing, it follows that the motion R = SM¢ is orientation preserving, but R
also fixes the origin. Therefore, R must be a rotation about the origin through some
angle q. Note that SR = SSM¢ = M¢. Define a and b as in Case 1. It is again easy to see
that the equation for M = TM¢ = TSR has the desired form.
This proves the first part of Theorem 2.2.7.1. The second part is Exercise 2.2.7.1.
See also the next example.

2.2.7.2. Example. Let us show directly, without using Theorem 2.2.7.1, that the
transformation M defined by the equations

3
2

1
2

x ¢= x + y + 5

1
2

3
2

y ¢= x - y + 7

is a motion.

Solution. Define a translation T by T(x,y) = (x,y) + (5,7). Let R be the rotation about
the origin through the angle -p/6 and let S be the reflection about the x-axis. It is easy
to see that M = TSR and hence is a motion since it is a composite of motions.

Theorem 2.2.7.1 shows that motions can be represented by five real numbers (the
a, b, c, d, and ±1 depending on the sign). Rigid motions can be represented by four
real numbers. Chapter 20 in [AgoM05] describes a very compact way to represent
motions in terms of quaternions. The fact that a motion is defined by five numbers
leads to another way to solve for a motion when it is given in terms of some points
and their images. One simply solves the equations in Theorem 2.2.7.1 for the unknown
coefficients. Solving for five unknowns turns out to be not as complicated as it may
sound in this case.
Next, we would like to give a more complete geometric characterization of
motions than that given in Corollary 2.2.5.6.

2.2.7.3. Lemma. Every orientation-reversing motion M that fixes the origin is a
reflection about a line through the origin.

Proof. Let p be a nonzero point. If M fixes p , then Theorem 2.2.5.7 implies that M
is the reflection about the line through the origin and p and we are done. Assume
therefore that p ¢ = M( p ) π p . Let q be the midpoint of the segment [ p , p ¢] and let S be
the reflection about the line through the origin and q . Clearly, S and M agree on 0
and p . By Corollary 2.2.6.8 they are the same map.

Definition. A glide reflection is the composite of a reflection about a line L followed
by a translation with nonzero translation vector parallel to L . (See Figure 2.12.)

2.2.7.4. Theorem. Every orientation-reversing motion M is a reflection or a glide

reflection.

Proof. If M fixes the origin, then the theorem is true by Lemma 2.2.7.3. Assume
therefore that M( 0 ) = p is distinct from the origin and let T be the translation that