244
THE ALGEBRA OF ISOMETRIES
The proof that a= b- 1 can be obtained by multiplying the equation a· b = eon the
right by b- 1 .
D
The following theorem can be proven in much the same way as Theorem 8.2.5, but
we can give a more satisfying proof by using the fact that the isometries of the plane
form a group. This proof illustrates how nicely algebra fits with geometry.
Theorem 8.2.8. An isometry of the plane is completely determined by its action on
three noncollinear points.
Proof. Let A, B, and C be three non collinear points in the plane, and let S and T be
two isometries such that
S(A) = T(A),
S(B)
= T(B),
and
S(C) = T(C).
We want to show that S = T. Now, since the isometries of the plane form a group,
we know that T has an inverse r- 1 so that
T- 1 S(A) =A,
T- 1 S(B)
= B,
and
T- 1 S(C) =C.
Thus, the isometry r- 1 S fixes each of the points A, B, and C, and by Theorem
8.2.5, this means that r- 1 S must be the identity. By the previous theorem, S must
be equal to the inverse of r- 1 ; that is,
s = (r-1rl = r.
D
Theorem 8.2.9. Every isometry of the plane that is not the identity can be decomposed into the product of at most three reflections.
Proof. Let T be a given isometry, and let A, B, and C be three noncollinear points.
From the iron grate example, we know how to map A, B, and C to T(A), T(B), and
T( C), respectively, by a sequence of at most three reflections.
Suppose that it actually took three reflections, say,
and
in that order. Then
RnRmRz(A) =A'= T(A),
RnRmRz(B) = B' = T(B),
RnRmRz(C) = C' = T(C).
THE PRODUCT OF REFLECTIONS
245
Thus, RnRmRl and T both have exactly the same effect on A, B, and C, so
RnRmRl = T, by Theorem 8.2.8.
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Theorem 8.2.9, together with the fact that a reflection in a line is an opposite isometry,
now allows us to show that every isometry in the plane is either a direct isometry or
an opposite isometry.
Theorem 8.2.10. Every isometry in the plane is either a direct isometry or an
opposite isometry.
Proof. If the isometry T is the identity or the product of two reflections, it is a direct
isometry. If T is a reflection or the product of three reflections, it is an opposite
isometry.
D
8.3
The Product of Reflections
In this section, we use reflections to show that there are only four different types of
plane isometries: reflections (in a line), rotations, translations, and glide reflections.
We begin by examining the product of two reflections.
Example 8.3.1. Let land m be two distinct parallel lines. Show that RmRl = T XY,
where XY is a directed segment perpendicular to l and m and twice the distance
from l tom.
h
l
m
A+----+-_...
:._.A'
B+---+-_.
.. _.B'
,.
c
,,.. •C'
Solution. Let h be a line parallel to l and m such that l is midway between h and m,
as in the figure above.
Let A and B be points on h and let C be a point on l. R1 maps A and B to points A'
and B' on m and leaves C where it is.