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b)
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Math 466 - Homework Set 7 (16.12.2011)
1. If f is an isometry of the plane then show that the general equations
of f are
x
y
= ax by + c,
= (bx + ay + d)
with a2 + b2 = 1.
2. If f is a transformation from R2 to itself with equations of the form
given in q
Math 466 - Homework Set 6 (14.12.2011)
1. Consider the regular heptagon H in Figure 1. We know that the
symmetry group of H is the dihedral group D7 . Hence there are 7 reection
symmetries in lines 1 , 2 , , 7 and 7 rotation symmetries around the
origin t
Math 466 - Homework Set 5 (8.12.2011)
1. Let
be the line y = 0. Find a formula for the reection .
2. Let
be the line x = 0. Find a formula for the reection k .
3. Let k be the line y = mx where m = tan . Find a formula for the
reection k as follows :
(a)
Symmetries of an Equilateral Triangle
It is easier to represent the symmetries of an equilateral triangle using
permutation notation.
Two line form
Cyclic form
=
123
123
(1)(2)(3)
O,2/3 =
123
231
(1 2 3)
O,4/3 =
123
312
(1 3 2)
1 =
123
132
(2 3)
2 =
123
3
Matrix forms of Isometries
We know that
ab
cd
(x, y )
cos sin
sin cos
= (ax + cy, bx + dy ).
: Counterclockwise rotation about the origin through
.
1 0
01
: Reection about the y-axis.
1
0
0 1
: Reection about the x-axis.
01
10
: Reection about the line
Math 466 - Homework Set 4 (10.11.2011)
1. Let A = (0, 0), B = (5, 0), C = (0, 10), D = (4, 2), E = (1, 2) and
F = (12, 4). Assume that ABC DEF . Find equations of lines such
=
that the product of reections in these lines maps ABC to DEF .
2. Let , m and n
Math 466 Homework Set 3
(If any) find
.All lines of symmetry
.All points of symmetry
.All rotational symmetries
In each letter of English alphabet. If a letter has a point of symmetry can it have more than
one point of symmetry?
ABCDE
F
GHIJK
L
MNOPQ
RSTU
Math 466 - Homework Set 1 (1.10.2011)
1. Which of the mappings dened on the Cartesian plane by the equations below are transformations?
(a) f (x, y ) = (cos x, sin y ),
(b) g (x, y ) = (3y, x + 2).
2. Which of the transformations (if any) in exercise 1 ar
Chapter 5
Isometries II
Topics :
1. Even and Odd Isometries
2. Classification of Isometries
3. Equations for Isometries
e
e
e
e
e
Copyright c Claudiu C. Remsing, 2006.
All rights reserved.
65
66
M2.1 - Transformation Geometry
5.1
Even and Odd Isometries
A
Chapter 7
Similarities
Topics :
1. Classification of Similarities
2. Equations for Similarities
j
e
e
j
j
e
e
Copyright c Claudiu C. Remsing, 2006.
All rights reserved.
98
j
99
C.C. Remsing
7.1
Classication of Similarities
The image of a triangle as seen
Chapter 6
Symmetry
Topics :
1. Symmetry and Groups
2. The Cyclic and Dihedral Groups
3. Finite Symmetry Groups
m
e rrr
e
d
d
Copyright c Claudiu C. Remsing, 2006.
All rights reserved.
83
e rrr
e
e
e
84
M2.1 - Transformation Geometry
6.1
Symmetry and Group
Chapter 4
Isometries I
Topics :
1. Isometries as Product of Reflections
2. The Product of Two Reflections
3. Fixed Points and Involutions
e
e
e
e
e
e
Copyright c Claudiu C. Remsing, 2006.
All rights reserved.
49
50
M2.1 - Transformation Geometry
4.1
Isome
Chapter 2
Translations and Halfturns
Topics :
1. Translations
2. Halfturns
Copyright c Claudiu C. Remsing, 2006.
All rights reserved.
24
C.C. Remsing
2.1
25
Translations
Let E 2 be the Euclidean plane.
2.1.1 Definition.
A translation (or parallel displace
Chapter 3
Reections and Rotations
Topics :
1. Equations for a Reflection
2. Properties of a Reflection
3. Rotations
e
e
e
e
e
e
Copyright c Claudiu C. Remsing, 2006.
All rights reserved.
38
C.C. Remsing
3.1
39
Equations for a Reection
A reection will be d