214
THE EUCLIDEAN TRANSFORMATIONS OR ISOMETRIES
we must have
L.POQ =
e- L.QOP' =
Since
OP=OP'
and
L.P'OQ'.
OQ=OQ',
then by the SAS congruency theorem we have
L::.OPQ
and therefore PQ
=
L::.OP'Q',
P' Q'.
D
Composition of lsometries
What happens if an isometry T is applied to the plane and then followed by another
isometry S? When a transformation T is followed by another one S, the combined
result is called the composition ofthe two transformations and is written SoT. Notice
that the first transformation is on the right, while the second is on the left. 7
Suppose that we start with points P and Q at distance d from each other. When T is
applied, these points are mapped into P' and Q', and
dist(P', Q')
= dist(P, Q) =d.
When S is applied to P' and Q', these points are mapped to P" and Q", respectively,
and
dist(P", Q") = dist(P', Q') = d.
The combined effect is that P and Q are mapped to P" and Q", and the distance is
preserved; that is, S o T is itself an isometry.
Since we can create new isometrics by composing known isometrics, it seems like
there is an unlimited supply of different types of isometrics. For example, we could
create a new isometry by first doing a rotation, then a reflection about some line, then
a reflection about another line, then a translation. We will see later that we cannot
really get too much that is new, and, in fact, there are only four different types of
isometrics in the plane. In addition to rotations, reflections, and translations, the only
other type is a glide reflection.
7 This is the conventional notation in geometry. It is a common convention in algebra texts to write the
first transformation on the left.
MAPPINGS AND TRANSFORMATIONS
A.........
........
__..-..---------
------ B
------------------
215
------1R1
.,'x
.....
\
\\
\
\
''
'
A glide reflection Gz,AB is simply a translation TAB followed by a reflection Rz
about a line l that is parallel to AB. We will prove that this is the only other additional
isometry later.
It is obvious that all isometries have inverses and that the inverses themselves must
also be isometries. Not quite so obvious is the fact that an isometry also preserves
straight lines.
Theorem 7.2.3. (lsometries Preserve Straight Lines)
(I) Let P, Q, and R be three points, and let P', Q', and R' be their images under
an isometry. The points P, Q, and R are collinear, with Q between P and R,
if and only if the points P', Q', and R' are collinear, with Q' between P' and
R'.
(2) Let l be a straight line, and let l' be the image of l under an isometry. Then l'
is a straight line.
Proof. Here we write lAB I for dist(A, B).
(1) We will show that if Q is between P and R, then Q' must be between P' and
R' (the proof of the converse may be obtained by interchanging P, Q, and R
with P', Q', and R').
R
p
....--
P'
_____ .... -------
----------Q·:---------;;.
BASIC ALGEBRAIC PROPERTIES
237
Note that the right bisector of C' C" passes through B' because B' C' = B' C" . After
the second step, the grate is in position A" B' C". The third and final step is to reflect
A" B' C" about the line B' C".
The Composition of Transformations
Earlier we defined the composition of two isometries. This terminology is used
for any collection of transformations. If T and S are two transformations, the
composition or product ofT and S is denoted by either S o T or ST. In terms of the
individual points in the plane, ST acts as follows. First, the point X is mapped by T
to T(X), then T(X) is mapped by S to S (T(X)). In other words, for each point X
in the plane:
ST(X ) = S (T(X)).
To move the grate to its desired position, we applied the product R nR mR t of
three reflections. Again, we emphasize that we follow the "right-to-left" rule.
When evaluating RnRmRt(X), first find Rt(X) , then R m (Rt(X)), and finally
Rn (Rm (Rt(X) )) .
Equal Transformations
In the iron grate problem, we used the product of three reflections to shift the grate
to its desired position. If we had a better machine-one that could perform rotations
as well as reflections-we could accomplish the same thing by performing just two
operations, as shown in the figure below.
A
c
C"
0
(I )
(2)
In this case, the grate is moved by applying the product R sR o,a . In the first solution,
we used R nR mR t. The individual transformations that make up the two products
are quite different, but the net effect is the same. So we can write:
238
THE ALGEBRA OF ISOMETRIES
Two transformations are said to be equal if they have the same effect on every point
in the plane. In other words, saying that T = S means that T(X) = S(X) for every
point X in the plane.
As the example shows, T = RsRo,a and S = RnRmR! does not mean that T
and S necessarily have the same description. The situation is similar to equality of
functions in trigonometry: if
f(x) = 1- 2sin 2 x
and
g(x) = cos2x,
then the functions f and g are equal (since we have pointwise equality), although
their descriptions are quite different.
Other examples of equal transformations that are defined differently are the halfturn
and a reflection in a point.
The halfturn about a point 0, denoted by Ho, is the transformation Ro,1soo. The
reflection in the point 0 is the transformation that takes each point P to the point P'
so that 0 is the midpoint of P P'. The point 0 is called the center of the halfturn or
the center of reflection. In the plane, since a reflection through a point is identical to
a halftum, there is no further need to talk about reflections through a point, and there
is really no need for a special symbol to denote it.
Closure
In Chapter 7, we mentioned that the composition of two isometries results in a
transformation that is also an isometry. In algebra, we would describe the situation
by saying that the set of isometries of the plane is closed under the operation of
composition.
More generally, if S is a set of elements and if o is a binary operation on S, we say
that S is closed under the operation o if for every pair of elements a and b in S the
product a obis also inS. For example, the set of positive integers is closed under
addition.
Associativity
If R, S, and T are three transformations, the product T S R means first apply R, then
S, and then T. For a point X, the notation TSR(X) means T (S (R(X))).
We can overrule this by using parentheses since the operations inside parentheses are
carried out first. The notation (T S) R is interpreted as follows: first, determine what
(T S) is. It will be some transformation, call it H, and H is usually different than
BASIC ALGEBRAIC PROPERTIES
239
either TorS. The notation (T S) R means H R; that is, first apply the transformation
R, then apply the transformation H.
For example, let us suppose that S and T are reflections about two different parallel
lines and R is a rotation. Then, as we will see later, T S is some translation H. So we
interpret (T S) R as meaning first do the rotation R, then follow it by the translation
H.
In a similar way, the notation T (SR) means that we should first determine what
SR is, namely, some different isometry L, and then take the product of L and T:
T (SR) = T L. Note, however, that the parentheses in this case could be omitted,
because in the absence of parentheses, the expression is evaluated from right to left.
The following theorem shows that (TS) Rand T (SR) are equal.
Theorem 8.1.1. (Associative Law)
The associative law holds for the product of transformations; that is, given three
transformations T, S, and R in the plane,
T (SR)
= (TS) R.
Proof. Let X be a point in the plane. We will evaluate T (SR) (X) and (TS) R(X).
The notation T (SR) (X) means "first evaluate SR(X), then evaluate T (SR(X))."
By definition, SR(X) = S (R(X)), so
T (SR) (X)= T (S (R(X))).
The notation (TS) R(X) tells us to evaluate TS (R(X)). Now, TS(Z) = T (S(Z))
for all Z in the plane. In particular, when Z = R(X), we get
TS(Z)
= T (S(Z)) = T (S (R(X))).
Thus,
(TS) R(X)
= T (S (R(X))).
Since (TS) Rand T (SR) have the same effect on every point X in the plane, we
conclude that they are equal transformations.
0
240
8.2
THE ALGEBRA OF ISOMETRIES
Groups of lsometries
In algebra, a set of elements g together with a binary operation · is called a group
and is denoted by (9, · ) if it possesses the following properties:
1. The set g is closed under the binary operation.
2. The associative law holds: (a · b) · c = a · (b · c) for all a, b, and c in g.
3. Qhasanidentityelement: thereissomee1ementeinQsuchthate·a
for every a in g.
= a·e =a
4. For every a in g, there is an inverse element a': there is an element a' in
such that a· a' =a'· a= e.
g
If it is also true that the commutative law holds (that is, if a · b = b · a for all a and b
in Q), then g is called an Abelian group.
The notation for the binary operation · is usually omitted, so that we write ab instead
of a· b.
We learned in Chapter 7 that every isometry has an inverse. This, along with the
results of the previous section, show that the family of isometries in the plane, together
with the composition operation, satisfy all four of the conditions listed above. We
can summarize this with the following theorem:
Theorem 8.2.1. The set of all isometries of the plane, together with the operation of
composition,forms a group.
Given an isometry T, we denote its inverse by T- 1 . Theorem 7.2.1 stated that
(RP,I.I)- 1 = RP,-1.1,
(R1)- 1 = R1,
(TAB)- 1 = TBA·
Example 8.2.2. What are the inverses ofGl,AB and Ho?
Solution. We have
(Gl,AB)- 1
= Gl,BA
and
(Ho)- 1
=
Ho.
D