TRANSFORMATIONS & ISOMETRIES
(Adopted from Geometry by Michael Hvidsten)
1.
Let f be a correspondence that assigns points in a plane to points in a plane (so
R 2 is both ‘source’ and ‘target’ for f.
2.
Example 1: f ( x, y )  ( x,1/ y )
3.
The set of allowable inputs of the correspondence is called the domain of the
correspondence. In Example 1, the domain of f, D( f )  {( x, y) | y  0}  R 2 .
4.
If to each point in the domain, there corresponds a unique output, we call the
correspondence a function. f in Example 1 is a function.
5.
The set of outputs of a function is called the range of the function. Observe that
the range of f in Example 1 is R( f )  {( x, y) | y  0}  R 2
6.
A function is said to be one-to-one if different inputs give different outputs, i.e.
(a, b)  (c, d )  f (a, b)  f (c, d )
or contrapositively
f ( a , b )  f ( c , d )  ( a , b )  ( c, d ) .
7.
The function f : X  Y is said to be onto, if for every y  Y , ,  x  X so that
f ( x)  y.
8.
So, the function f : R 2  R 2 is onto if R( f )  R 2 (target).
9.
The function f : R 2  R 2 which is both one-to-one and onto is called a
transformation of the plane.
10.
The function f : R 2  R 2 is called distance-preserving or a Euclidean
Isometry if for any pair of point P and Q, distance (P, Q) = distance ( f(P), f(Q)).
11.
Exercise 1: Is f ( x, y)  ( x, y 2 ) an isometry?
12.
i.
ii.
Theorem: Let f : R 2  R 2 be a Euclidean Isometry. Then
f is one-to-one
If f (P) = P’ and f (Q) = Q’, then f maps all points between P and Q to points
between P’ and Q’ ( i.e. f (PQ) = P’Q’ )
f maps lines to lines
f preserves angles
f is onto the plane
f preserves parallel lines
iii.
iv.
v.
vi.

So, an isometry is a length-preserving transformation

Define the inverse of a function.
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Assignment 4( due October 2, 2008)
1 Prove: Let f : R 2  R 2 be a Euclidean Isometry. Then f is onto the plane
2 Show that a function that is one-to-one and onto must have a unique inverse.
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13.
Explain the term ‘fixed-point’. (Fixed points of a transformation/isometry
are used to classify the transformation/isometry)
14.
Prove: Let f be an isometry. Let A and B be fixed points of f ( A  B ) . Then f
fixes every point on the line through A and B.
15.
Define the identity isometry (denoted by i ).
16.
Prove: Let f be an isometry with three non-collinear fixed-points. Then f = i.
17.
If two isometries agree on any three non-collinear points, then they agree
everywhere.
REFLECTION
18.
Define reflection
If
 m is a straight line.
 rm : R 2  R 2 is an isometry that fixes two points on m.
 rm  i
Then we call rm a reflection across the line of reflection m.
Note:
a) For each point A not on m, the unique point A'  rm ( A) is such that m is the
perpendicular bisector of A' A .
b) Each reflection is its own inverse and therefore an involution.
c) Let ABC  PQR . Then there is an isometry composed of at most three
reflections that maps ABC to PQR .
d) Every isometry can be written as the product of at most three reflections.
e) Let r , r be two reflections with lines of reflection l and m respectively. Let
m l
r (l )  l  . Then r  r  r  r
m l m
l
m
Outline of proof:
1. Let A and B be two distinct points on l, so r ( A), r ( B) are on l 
m
m
2. Show r ( A) & r ( B) are fixed points of r  r  r  r 
m l m
l
m
m
TRANSLATION
19.
Define translation
A translation is an isometry that is the composition of two reflections across two distinct
parallel lines or two reflections across the same line (called the identity or trivial
translation)





Prove: If   rn  rm is a non-trivial translation, such that for A  B , the points
A, B, ( A) &  ( B) form a quadrilateral, then the quadrilateral is a parallelogram.
Thus, for any point P, the segment P (P ) is of constant length and direction. The
vector from P to  ( P) is called the displacement vector of the translation.
If we consider points in R 2 as vectors from the origin, P  ( x, y ) , &
 ( P)  P  v  (v1 , v2 ) , we have the coordinate equation for translation given
by  ( x, y)  ( x, y)  (v1 , v2 ) . Conversely, any function of this type represents a
translation.
Prove: If   rn  rm is a non-trivial translation and k is a line perpendicular to m
and n at P and Q respectively, then the translation vector is given by the vector
2 PQ
Question: How many fixed-points does a non-trivial translation have?
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ROTATION
20.
Define rotation
A rotation is an isometry that is the composition of two reflections across two nonparallel lines.

Prove: Let two lines p and q intersect at O. Let P and Q be points on p and q
respectively. Let m be the bisector of POQ . Then
rm ( p)  q
i.
Outline of the proof:
 Let P and Q be chosen so that OP = OQ
 Let R be the intersection of PQ with m


Show m is the perpendicular bisector of PQ at R, thus rm ( P)  Q
Observe r (O )  O …
m
ii.
r  r  r  r (Use 18 (e) to prove this)
m p
q m

Prove: An isometry R  i is a rotation  R has exactly one fixed point.
Outline of )
A.
Let R  r  r where m and l intersect at O
m l
B.
Let R have another fixed point B  O
C.
B cannot be on both m and l
D.
Suppose not and come up with a contradiction.
Outline of )
A.
Let R (O )  O
B.
Let A another point, A  O
C.
Let l be the line through A and O
D.
let m be the bisector of AOR ( A)
E.
So r  R fixes A
m
r  R also fixes O
F.
m
r  R is either the identity or a reflection …
G.
m

(Triples of reflections about concurrent lines) Let l, m, n be three lines that
intersect at O. Then
a. r  r  r  r where p is a line through O
l m n
p
Outline:
Let f  r  r  r and let A be a point on, A  O
l m n
Let f ( A)  B . Then either A  B or A  B
If A  B , choose p = n, and if A  B , let p be the perpendicular bisector of
AOB
Claim: In either case, r  R fixes both A and O.
p




b. r  r  r  r  r  r
l m n
n m l

(Rotations transform points through a fixed angle) Let R be a rotation about a
fixed point O. Let A be any point, A  O . Let l be the line through A and O.
Then
a. There is a unique line m passing through O such that R  r  r
m l
b. If AOR ( A) is  degrees, then for any point P  O ,
POR (P)   degrees.
Outline for b)
 Choose B so that OB = OA = OR(B)
 Let w be the bisector of R( B)OA
 Then r ( A)  R( B)  r (r ( B))
w
m l
Note: The construction of a rotation about O of a specific angle  requires the choice
of two lines that meet O and make angle of  / 2
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

The coordinate form for a rotation about the origin by an angle  is given by
R ( x, y )  ( x cos   y sin  , x sin   y cos  )

A half-turn is a rotation about a point of  radians.
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Exercises:
Show that the lines invariant under a reflection r ( i.e. lines l such that
m
r (l )  l ) , where m is the line of reflection (also called the line of symmetry) for r ,
m
m
consists of the line m and all lines perpendicular to m.
Outline:
a) Since r fixes every point on m, l  m is invariant under r
m
m
b) If A is any point such that r ( A)  A , then m is the perpendicular bisector of
m
Ar ( A).
m
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1.
2.
Show that the composition of four reflections can be written as the composition of
two reflections.
Outline:
We have shown that a triple of reflections is equivalent to one reflection.
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Given the translation T  r  r with displacement vector v, show that
m l
1
T  r  r with displacement vector –v.
l m
Outline:
a) Determine (r  r )  (r  r ) & (r  r )  (r  r )
m l
l m
l m
m l
b) Let T  ( x, y )  v
c) Define S  ( x, y )  v and complete the proof
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3.
4.
Show that the composition of two translations is a translation. Find the
displacement vector for this composition.
Outline:
Use the coordinate equation of the translation
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5.
Show that composition of translations is commutative.
6.
Given a reflection across a line and a translation in the same direction as the line,
show that the composition of the reflection and the translation is commutative.
7.
Show that the inverse to a rotation about a point of  degrees is a rotation about
the same point of   degrees
8.
Show that the composition of two rotations centered about the same point is again
a rotation centered at that point.
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Assignment 5 ( due October 23, 2008)
1.
Suppose that two rotations R, R  centered at O have the same effect on a point
A  O . Show R  R  .
2.
Let A, B be distinct points. Let H , H be half-turns about A, B respectively.
A B
Show H  H is a translation in the direction of the vector from A to B .
B
A
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