Download 10. Isometries continued. An isometry is the mathematical

10. Isometries continued.
An isometry is the mathematical manifestation of rigidly moving figures in
the plane. Is there a simple way to describe all possible isometries? The
following results allow us to do so.
Theorem 10.1: Let S: P P and T: P P be two isometries such that
S(P) = T(P), S(Q) = T(Q), and S(R) = T(R) for three non collinear points P,
Q, and R. Then S = T.
Proof: To be provided in class.
We have seen that if an isometry fixes three non collinear points then it
must be the identity. What can we say about an isometry that fixes two
points?
Theorem 10.2: Let T: P P be an isometry with T different from the
identity. If T fixes two points then T is a reflection.
Proof: To be provided in class.
!
!
Let C be a point in the plane and θ an angle measure. Define the function
R" : P P as follows: R" (C) = C, and for any other point P,
R" (P) = P’ where P’ is the unique point on the circle with center C and
radius d(C, P) with the measure of the angle "PCP # equal to θ. R" is
called the rotation
about C through angle θ. Rotations are isometries, as
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is easy to imagine and almost as easy to prove.
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Theorem 10.3 Let T: P P be an isometry with T different from the
identity. If T fixes exactly one point, then T is a rotation.
Recall that SketchPad Tour 5 demonstrated that the composition of two
reflections in a pair of intersecting lines corresponded to a rotation about
the point of intersection through an angle twice the angle between the
lines. Conversely, given any rotation about C through angle θ, if l and m
are any two lines that intersect at C such that the smallest angle
between them is θ/2, then the composition (in the correct order) of the
associated reflections is the given rotation.
Theorem 10.4: T: P P be an isometry that fixes no points. Then T is
the composition of at most three reflections.
Proof: To be provided in class.
The sequence of statements above demonstrates that any isometry of
the plane can be expressed as the composition of at most three
reflections. Therefore, to classify the isometries, we need only examine
all possible combinations of at most three reflections:
1 Reflection: Duh.
2 reflections: Either the lines l and m are parallel or they intersect
(exclude the case where they are equal). If they intersect, then
composing the corresponding reflections leads to a rotation. If they are
parallel, the composition is a translation, as will be described in lecture.
3 Reflections: The three reflection case is a little more complex, since
there are a number of ways that you can draw three distinct lines in the
plane. However, an examination of each case revels that the only possible
outcome from the composition of three lines is either a reflection or a
glide reflection, which can be described as a reflection about a line
followed by a translation along the reflection line (or the other way
around) We will look at several examples in class, but you can already
investigate this new type of isometry using SketchPad. We will see that,
although the glide reflection doesn’t seem quite as natural as the other
types of isometries, it really is.
So, in summary, there are exactly four distinct types of isometries of the
plane, reflections, rotations, translations, and glide reflections.