Isometries
Definitions:
1) An isometry is a transformation that preserves distance. That is, if P and Q are any two points
and P’ and Q’ their images, then PQ = P’Q’.
2) A point (x,y) is invariant under a transformation F if it is a fixed point. That is,
F ( x, y )  ( x, y )  ( x, y ) .
3) A geometric property that is preserved under a transformation is called an invariant property
under the transformation. (Properties include distance between points, angle measures,
collinearity of points, parallel lines, etc)
4) Two figures are said to be congruent if one is the image of the other under some isometry.
5) If AB is a vector then a translation, TAB , is a transformation such that if P is any point in the
plane and TAB ( P)  P ' , then PP ' and AB are equivalent vectors. (same magnitude and parallel
to each other).
6) A rotation about a point O through an angle  , denoted RO, , is a transformation of the plane
such that RO, (O)  O and if P  O and RO, ( P)  P ' , then OP  OP ' and mPOP '   . O is
called the center of rotation.
7) A half-turn is a rotation of 180 degrees. (A half-turn is not a translation – half-turns turn things
up-side down)
8) A reflection, M , in a line is a transformation of the plane where if P  then M ( P)  P , and
if P 
 and M ( P)  P ' then
is the perpendicular bisector of PP ' .
9) A glide reflection is the composition of M and TAB where AB || .
10) A triangle ABC has clockwise (counterclockwise) orientation if we go around the triangle in
order A – B – C , the direction is clockwise (counterclockwise).
11) If a figures orientation is invariant under an isometry, then we call it a direct isometry. If it
reverses, then called indirect isometry.
Major Theorems about the product of two reflections:
1) M M n  RA,2 if  n  { A} , and the angle between the lines
and n , directed from n to , is
.
2) (Converse of (1)) For any rotation RA, , RA,  M M n where and n are any two lines
satisfying  n  { A} and the angle between the lines and n , directed from n to is  / 2 .
3) M M n  TAB if || n and AB  (and n) with || AB || twice the distance between
directed from n to .
4) (Converse of (3)) For any translation TAB , TAB = M M n where
satisfying || n , AB  , and the distance between n and
and n ,
and n are any two lines
(directed from n to
) is || AB || / 2 .
Example 1: Prove that a translation is an isometry. Let P, Q be
any two points in the plane and TAB ( P)  P ', TAB (Q)  Q ' . By
definition PP ' || AB || QQ ' and they all have the same magnitude.
Therefore, the quadrilateral PP’Q’Q has one pair of opposite sides
congruent and parallel, and so is a parallelogram. Since the opposite
sides are congruent, PQ = P’Q’. (Compare with the analytic proof
below).
Example 2: Show that angle measure is preserved
under any isometry. Given ABC and A’, B’, C’ the
images of A, B, C, respectively, under an isometry.
Since distance is preserved, we know that AB = A’B’,
AC=A’C’, and BD = B’C’. Therefore,
ABC  A ' B ' C ' by SSS. So, by corresponding parts,
ABC  A ' B ' C ' .
P'
B
Q'
P
A
Q
A'
A
B'
B
C
C'
Analytic Methods with Isometries
Trigonometry Preliminaries:
sin( x  y )  sin x cos y  cos x sin y
sin( x  y )  sin x cos y  cos x sin y
tan x  tan y
tan( x  y ) 
1  tan x tan y
tan x  tan y
tan( x  y ) 
1  tan x tan y
sin 2 x  2sin x cos x
cos( x  y )  cos x cos y  sin x sin y
cos( x  y )  cos x cos y  sin x sin y
cos 2 x  cos 2 x  sin 2 x  2cos 2 x  1  1  2sin 2 x
y  cos x is an even function so cos( x)  cos x
y  sin x is an odd function so sin( x)   sin x
Matrix Preliminaries
 a b   x   ax  by 
 c d   y    cx  dy 

  

a b   e
c d   g


f   ae  bg

h   ce  dg
a b
 ad  bc
c d
af  bh 
cf  dh 
a b
d e
g h
c
b
f g
e
i
c
a
h
f
d
c
a b
i
f
d e
Vector preliminaries
The magnitude or length of a vector <a,b> is r =
a, b  a 2  b 2
The polar form (when  and r are known) is
r cos , r sin 
4
r
2
b r sin 
The slope of the vector is 
 tan  .
a r cos
b
a
a
-5
5
Equations for Isometries
-2


Translation along vector <a,b>.

Rotation about the center O  (0,0) and through an angle of  degrees (pos or neg).
T[ a ,b ] : x   x  a ,
y  y  b
RO, : x  x cos  y sin , y  x sin  y cos

Reflection across the line y = mx; the angle between the positive x-axis and the line y = mx is  ,
so m  tan  .
M : x  x cos 2  y sin 2 , y   x sin 2  y cos 2
Example 3: A non-trivial translation has no fixed points. Ta,b ( x, y)  ( x, y)  ( x  a, y  b)  ( x, y) if
and only if x  a  x and y  b  y if and only if a = b = 0. So, only the trivial translation has fixed points.
Example 4: A translation is an isometry. Let A = ( x1 , y1 ) and B = ( x2 , y2 ) be any two points in the
plane. By definition of the translation, the images are A  ( x1, y1)  ( x1  a, y1  b) and
B  ( x2 , y2 )  ( x2  a, y2  b) . From the distance formula,
AB  ( x2  x1 )2  ( y2  y1 )2
 (( x2  a)  ( x1  a)) 2  (( y2  b)  ( y1  b)) 2
 ( x2  x1 ) 2  ( y2  y1 ) 2
 AB
Matrices and isometries
There are matrices that also define two of these transformations. Multiplication of matrices corresponds to
composition of functions. These are easily!!?? derived from the equations above.


 x  cos 
RO , :    
 y   sin 
 x  cos 2
M :  
 y   sin 2
 sin    x 
cos    y 
sin 2   x 
 cos 2   y 
The problem with 2X2 matrices is that there is no 2X2 matrix for a translation: Any 2X2 matrix
0 
0 
0 
times   is   , so it is impossible to translate the point   .
0 
0 
0 
To remedy the situation of not having matrices for all isometries, we need to alter the way we think about
points in the plane.
Solution: (Use 3X3 matrices) For each point ( x, y)  R2 identify it with the point ( x, y,1)  R3 . Notice
that this is just associating the (x,y) plane with the plane z = 1 in three-space.
3X3 Matrices for all motions of the plane



 x  1 0 a   x 
Translation Ta,b defined by  y   0 1 b   y 
 1  0 0 1   1 
 x  cos
Rotation about the origin defined by RO , :  y   sin 
 1   0
 sin 
cos
0
 x  cos 2
Reflection across y = mx definded by M :  y   sin 2
 1   0
sin 2
 cos 2
0
0  x 
0  y 
1   1 
0  x 
0   y 
1   1 
What if we want to rotate around a different point (h,k)? Follow the general point (x,y).
5
y
5
y
5
 (h,k)
y


x
x
x
(x,y)
-5
-5
5
(1) translate to origin
-5
-5
5
(2) Rotate about origin
-5
-5
5
(3) translate back
1) We first take the point (a,b) and translate it to the origin (along with all other points in the plane).
2) Then we rotate around the origin in this new plane.
3) Then we translate everything back to the original coordinate plane.

(3)
(2)
(1)
 x  1 0 h  cos   sin  0  1 0 h   x 
 y   0 1 k   sin  cos  0  0 1 k   y 
  


 
 1  0 0 1   0
0
1  0 0 1   1 
With matrices we have
.
cos   sin   h cos   k sin   h   x 
  sin  cos   h sin   k cos   k   y 
 0
  1 
0
1
The above matrix product defines a rotation about the point (h,k) through an angle of  degrees.
Example 5: Show that the product of two translations is a translation. What vector defines the product of
the two translations? I.e. How is it related to the vectors of the two translations?
1 0 a 
Since Ta,b defined by 0 1 b 
0 0 1 
1 0 c 
and Tc ,d  defined by 0 1 d  , the product (composition),
0 0 1 
1 0 a  1 0 c  1 0 a  c 
Ta,b Tc ,d  , is defined by the matrix product 0 1 b  0 1 d  = 0 1 b  d  which is the matrix
1 
0 0 1  0 0 1  0 0
for the translation Ta c,bd   Ta,bc,d  . The vector of the product is the sum of the original vectors.
Example 6: Show analytically that angle measure is preserved under a rotation. We will assume that we
know that angle measure is preserved under a translation (exercise 23 in the text). So, it is without loss of
generality that we form the angle at the origin with lines y = nx and y = mx, where n  tan  and
m  tan  (Note that we are also assuming that these are not vertical lines – this would be a special case).
So, the angle between these two lines is
y2= nx
WLOG    .
Every point on the line y = nx is of the form
(x, nx), and every point on y = mx is of the
form (x, mx). Using the equations for the
rotation of  degrees about the origin, the
image of (x, nx) is
y1 = mx
y1'
( x, y)  ( x cos  nx sin  , x sin   nx cos  )
 +
and the image of (x, mx) is
y2'
( x, y)  ( x cos  mx sin  , x sin   mx cos )
The slope of the image of y = nx is
 +


sin   n cos sin   tan cos tan   tan


 tan(   )
cos  n sin  cos  tan sin  1  tan tan 
Similarly, the image of y = mx has slope tan(    ) . So, the angle between the two image lines is
    (   )     , which is the same angle as that between the lines y = nx and y = mx.
Compositions of isometries. (Use the major theorems from page 1)
Theorem: Every motion of the plane is a composition of three or fewer reflections.
Proof: Clearly a reflection is one reflection. The two major theorems show that a translation and
a rotation are both compositions of two reflections. Since a glide reflection is the composition of
a reflection and a translation, it is the composition of three reflections.
Theorem: (Converse) Any composition of three or fewer reflections is a motion of the plane.
Proof: A single reflection is a reflection and again the theorems from page 1 show a composition
of two reflections is either a rotation or a translation. All that is left to consider is the
composition of three reflections. Let l , m, n be the three lines of reflection. The cases to consider
are:
Case 1: Two lines coincide.
Case 2: l || m || n
Case 3: l , m, n are concurrent.
Case 4: Two lines intersect at a point off of the third line.
Case 1: If two lines coincide, then the composition is a reflection.
Proof: If l  m then M l M m  M m M m  I , so the composition with M n is the reflection M n .
Case 2: If l || m || n then the composition is a reflection.
Proof: M n M m M l  M nTv where v is a vector perpendicular to the three lines, has
magnitude twice the distance between l and m and is directed from l to m (theorem 3 on page
1). By theorem 4 we can use any such two lines, so let p be a line parallel to n such that the
directed distance from p to n is the same as the directed distance from l to m . Then,
M m M l  Tv  M n M p and M n M m M l  M nTv  M n M n M p  IM p  M p .
Ml
l
MmM
m
Ml
l
MnMmM
l
n
MnM
m
l
MnMnMp
p
P
n
Case 3: Exercise
Case 4: If two lines intersect at a point of the third, then the composition is a glide reflection.
Proof: If l and m intersect at a point off of n , then M m M l is a rotation through an angle
twice that as the angle between l and m in the direction l to m . By theorem 2 we can write
M m M l  M m M l  where the angle between l  and m in the direction l  to m is the same as
the angle between l and m in the direction l to m and m || n . But then,
M n M m M l  M n M m M l   Tv M l  where v  m by theorem 1. Let w be the vector component of v in
the direction perpendicular to l  , then v  v  w  w and Tv  Tv  wTw and (v  w)  w ; so (v  w) || l  .
Since w  l  we can write Tw  M p M l  where p || l  and the distance between l  and p is half the
magnitude of w . We have, then
M n M m M l  M n M m M l 
 Tv M l 
 Tv  wTv M l 
 Tv  w M p M l  M l 
 Tv  w M p
By definition, since (v  w) || l  || p , this is a glide reflection.
v-w
D'
MnMm Ml=TvMl'
E'
w
v
l
Ml
n
m
Ml'
l'
E
F
D
MmMl =Mm'Ml'
m'
Exercises
No equations or matrices for #1, #2. See Examples 1 and 2.
1) Show that collinearity is invariant under an isometry. (3 points will suffice).
2) Prove that a rotation is an isometry.
Now you will use equations and matrices for #3 - #5. See examples 3 and 4
3) Show that a rotation about the origin is an isometry.
4) What are the invariant points under a reflection? Prove your answers analytically by using the
above equations. (So you can’t just say what they are from the definition)
5) Prove RO, RO,   RO,   using the above equations.
Now use matrices and equations again (and your trigonometry): See examples 5, 6
6) Use 3X3 matrices to show analytically that the product of two half-turns (rotations of 180
degrees) about two different centers is a translation. What is the vector of translation?
7) What is the product of a half-turn and a translation? Justify your answer analytically. Verify that
the set of all half-turns and all translations form a group of transformations. You do not need to
verify the associative property in detail.
8) Find the matrix representation (3X3)of a glide reflection with reflection line y 
1
x and
3
translation vector in the third quadrant (in standard position) with magnitude 10.
9) Prove analytically that the angle between two intersecting lines, y  nx and y  0 , is an invariant
under a reflection across the line y  mx .
Compositions of reflections: (Use the theorems on page 1 and the two theorems on page 6 to prove the
following.)
10) Prove Case 3: If the three lines are concurrent, then the composition of the three reflections is a
reflection.
11) Prove that the composition of two rotations about different centers is either a rotation or a
translation. Give conditions for when it is a rotation and when it is a translation.