Chapter 3
Reections and Rotations
Topics :
1. Equations for a Reflection
2. Properties of a Reflection
3. Rotations
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Copyright c Claudiu C. Remsing, 2006.
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C.C. Remsi
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
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
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 = ta
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
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
: Reec
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 reec
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
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.
Chapter 5
Isometries II
Topics :
1. Even and Odd Isometries
2. Classification of Isometries
3. Equations for Isometries
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Copyright c Claudiu C. Remsing, 2006.
All rights reserved.
65
66
M2.1
Chapter 7
Similarities
Topics :
1. Classification of Similarities
2. Equations for Similarities
j
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Copyright c Claudiu C. Remsing, 2006.
All rights reserved.
98
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C.C. Remsing
7.1
Classi
Chapter 6
Symmetry
Topics :
1. Symmetry and Groups
2. The Cyclic and Dihedral Groups
3. Finite Symmetry Groups
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Copyright c Claudiu C. Remsing, 2006.
All rights reserved.
83
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e
8
Chapter 4
Isometries I
Topics :
1. Isometries as Product of Reflections
2. The Product of Two Reflections
3. Fixed Points and Involutions
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Copyright c Claudiu C. Remsing, 2006.
All rights r
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
a)
b)
Ee E
X
X
X
a\
X
\,
E
e>
r)
c)
cfw_)
s)
h)
RRRRRRRRRRRR
eeeeeee
R fi R fr R N R 7 R fr R fl
&p&p&pep&p&p
T
T
f r-r I r-l I r-'l I r-.-l u
rr-U-t7-r7-17
T