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Lance Drager
Introduction
Orthogonal
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Classifying
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Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
Geometric Transformations and Wallpaper
Groups
Isometries of the Plane
Lance Drager
Department of Mathematics and Statistics
Texas Tech University
Lubbock, Texas
2010 Math Camp
Wallpaper
Groups
Outline
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
1 Introduction
2 Orthogonal Matrices
Orthogonal Matrices and Dot Product
Classifying Orthogonal Matrices
Exercises
3 Classifying Isometries
Translations
First Classification
Final Classification
Exercises
Wallpaper
Groups
Isometries
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
• A transformation T of the plane is an isometry if it is
one-to-one and onto and
dist(T (p), T (q)) = dist(p, q),
for all points p and q.
• If S and T are isometries, so is ST , where
(ST )(p) = S(T (p)).
• If T is an isometry, so is T −1 .
• Our goal is to find all the isometries.
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Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
• A matrix A is orthogonal if kAxk = kxk for all x ∈ R2 .
• An orthogonal matrix gives an isometry of the plane since
dist(Ap, Aq) = kAp − Aqk
= kA(p − q)k = kp − qk = dist(p, q)
• If A and B are orthogonal, so is AB.
• If A is orthogonal, it is invertible and A−1 is orthogonal.
• Rotation matrices are orthogonal.
y
q
kp − qk
p
x
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Lance Drager
Introduction
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Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
1 Introduction
2 Orthogonal Matrices
Orthogonal Matrices and Dot Product
Classifying Orthogonal Matrices
Exercises
3 Classifying Isometries
Translations
First Classification
Final Classification
Exercises
Wallpaper
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Dot Product 1
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
Theorem
A is orthogonal if and only if Ax · Ay = x · y for all x, y ∈ R2 .
Thus, an orthogonal matrix preserves angles.
Proof.
(=⇒)
kAxk2 = Ax · Ax = x · x = kxk2
(⇐=) Recall
2x · y = kxk2 + ky k2 − kx − y k2 .
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Proof Continued.
So, we have
2Ax · Ay = kAxk2 + kAy k2 − kAx − Ay k2
= kAxk2 + kAy k2 − kA(x − y )k2 (distributive law)
Classifying
Isometries
= kxk2 + ky k2 − kx − y k2
Translations
First
Classification
Final
Classification
Exercises
(A is orthogonal)
= 2x · y ,
and dividing by 2 gives the result.
Theorem
A matrix is orthogonal if and only if its columns are orthogonal
unit vectors.
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Introduction
Proof.
Orthogonal
Matrices
(=⇒)
Orthogonal
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Classifying
Orthogonal
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Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
kAei k = kei k = 1.
Ae1 · Ae2 = e1 · e2 = 0.
(⇐=) Let A = [u | v ] where u and v are orthogonal unit
vectors. Note Ax = x1 u + x2 v .
kAxk2 = Ax · Ax
= (x1 u + x2 v ) · (x1 u + x2 v )
= x12 u · u + 2x1 x2 u · v + x22 v · v
= x12 (1) + 2x1 x2 (0) + x22 (1)
= x12 + x22 = kxk2
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Orthogonal
Matrices and
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Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
1 Introduction
2 Orthogonal Matrices
Orthogonal Matrices and Dot Product
Classifying Orthogonal Matrices
Exercises
3 Classifying Isometries
Translations
First Classification
Final Classification
Exercises
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Classifying Orthogonal Matrices 1
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
• If A is orthogonal, Col1 (A) = (cos(θ), sin(θ)) for some θ.
So the possibilities are
cos(θ) − sin(θ)
A=
= R(θ),
sin(θ)
cos(θ)
cos(θ)
sin(θ)
A=
= S(θ).
sin(θ) − cos(θ)
• In the first case A = R(θ) is a rotation.
• What about S(θ)?
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Classifying Orthogonal Matrices 1
Introduction
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Final
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• If A is orthogonal, Col1 (A) = (cos(θ), sin(θ)) for some θ.
So the possibilities are
cos(θ) − sin(θ)
A=
= R(θ),
sin(θ)
cos(θ)
cos(θ)
sin(θ)
A=
= S(θ).
sin(θ) − cos(θ)
• In the first case A = R(θ) is a rotation.
• What about S(θ)?
• There are orthogonal unit vectors u and v so that
S(θ)u = u and S(θ)v = −v .
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First
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Final
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• In fact, let u = (cos(θ/2), sin(θ/2)) and
v = (− sin(θ/2), cos(θ/2)).
•
S(θ)u =
cos(θ)
sin(θ)
sin(θ) − cos(θ)
cos(θ/2)
sin(θ/2)
cos(θ) cos(θ/2) + sin(θ) sin(θ/2)
=
sin(θ) cos(θ/2) − cos(θ) sin(θ/2)
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First
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Final
Classification
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• In fact, let u = (cos(θ/2), sin(θ/2)) and
v = (− sin(θ/2), cos(θ/2)).
•
S(θ)u =
cos(θ)
sin(θ)
sin(θ) − cos(θ)
cos(θ/2)
sin(θ/2)
cos(θ) cos(θ/2) + sin(θ) sin(θ/2)
=
sin(θ) cos(θ/2) − cos(θ) sin(θ/2)
cos(θ − θ/2)
cos(θ/2)
=
=
= u.
sin(θ − θ/2)
sin(θ/2)
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Translations
First
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•
S(θ)v =
cos(θ)
sin(θ)
sin(θ) − cos(θ)
− sin(θ/2)
cos(θ/2)
− cos(θ) sin(θ/2) + sin(θ) cos(θ/2)
=
− sin(θ) sin(θ/2) − cos(θ) cos(θ/2)
sin(θ − θ/2)
=
− cos(θ − θ/2)
sin(θ/2)
− sin(θ/2)
=
=−
= −v .
− cos(θ/2)
cos(θ/2)
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Classifying Orthogonal Matrices 4
• S(θ)u = u and S(θ)v = −v . If x = tu + sv , then
S(θ)x = tu − sv .
y
x
sv
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
M
tu
v
u
−sv
x
S(θ)x
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Classifying Orthogonal Matrices 4
• S(θ)u = u and S(θ)v = −v . If x = tu + sv , then
S(θ)x = tu − sv .
y
x
sv
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
M
tu
v
u
−sv
x
S(θ)x
• S(θ) is a reflection with its mirror line at an angle of θ/2.
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Matrices
Exercises
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Translations
First
Classification
Final
Classification
Exercises
1 Introduction
2 Orthogonal Matrices
Orthogonal Matrices and Dot Product
Classifying Orthogonal Matrices
Exercises
3 Classifying Isometries
Translations
First Classification
Final Classification
Exercises
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Orthogonal Matix Exercises 1
Lance Drager
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Exercises
• S(θ)2 = I , so S(θ)−1 = S(θ).
• Let A be an orthogonal matrix. Then A is a rotation (or
the identity) if and only if det(A) = 1 and A is a reflection
if and only if det(A) = −1.
a c
• If A =
define the transpose of A by
b d
a b
AT =
. Show that A is orthogonal if and only if
c d
A−1 = AT .
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2
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Exercises Continued
• Use the addition laws for sine and cosine to verify the
important identities
R(θ)S(φ) = S(θ + φ),
S(φ)R(θ) = S(φ − θ),
S(θ)S(φ) = R(θ − φ)
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Introduction
Orthogonal
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Matrices and
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Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
1 Introduction
2 Orthogonal Matrices
Orthogonal Matrices and Dot Product
Classifying Orthogonal Matrices
Exercises
3 Classifying Isometries
Translations
First Classification
Final Classification
Exercises
Wallpaper
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Translations
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
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Translations
First
Classification
Final
Classification
Exercises
• For v ∈ R2 , define Tv : R2 → R2 by Tv (x) = x + v . We
say Tv is translation by v.
−1
• Tu Tv = Tu+v and Tv
= T−v .
• Translations are isometries
dist(Tv (x), Tv (y )) = kTv (x) − Tv (y )k
= k(x + v ) − (y + v )k
= kx + v − y − v k
= kx − y k
= dist(x, y ).
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Outline
Lance Drager
Introduction
Orthogonal
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Matrices and
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Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
1 Introduction
2 Orthogonal Matrices
Orthogonal Matrices and Dot Product
Classifying Orthogonal Matrices
Exercises
3 Classifying Isometries
Translations
First Classification
Final Classification
Exercises
Wallpaper
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Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Three Distances Determine a Point
• Let p1 , p2 and p3 be noncolinear points. A point x is
uniquely determined by the three numbers r1 = dist(x, p1 ),
r2 = dist(x, p2 ) and r3 = dist(x, p3 ).
Classifying
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Translations
First
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p1
p2
p3
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• If an isometry T fixes three noncolinear points p1 , p2 , p3
then T is the identity.
dist(x, pi ) = dist(T (x), T (pi )) = dist(T (x), pi ),
i = 1, 2, 3,
=⇒ x = T (x).
Theorem
Every isometry T can be written as T (x) = Ax + v where A is
an orthogonal matrix, i.e., T is multiplication by an orthogonal
matrix followed by a translation.
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Translations
First
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Final
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Exercises
Proof.
1
The points 0, e1 , e2 are noncolinear.
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First
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Proof.
1
The points 0, e1 , e2 are noncolinear.
2
Let u = −T (0). Then Tu T (0) = 0.
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Final
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Proof.
1
The points 0, e1 , e2 are noncolinear.
2
Let u = −T (0). Then Tu T (0) = 0.
3
dist(Tu T (e1 ), 0) = 1, so we can find a rotation matrix R
so that RTu T (e1 ) = e1 .
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Classifying
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Translations
First
Classification
Final
Classification
Exercises
Proof.
1
The points 0, e1 , e2 are noncolinear.
2
Let u = −T (0). Then Tu T (0) = 0.
3
dist(Tu T (e1 ), 0) = 1, so we can find a rotation matrix R
so that RTu T (e1 ) = e1 .
4
RTu T (e2 ) = ±e2 .
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Proof of First Classification
Lance Drager
Introduction
Orthogonal
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Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
Proof.
1
The points 0, e1 , e2 are noncolinear.
2
Let u = −T (0). Then Tu T (0) = 0.
3
dist(Tu T (e1 ), 0) = 1, so we can find a rotation matrix R
so that RTu T (e1 ) = e1 .
4
RTu T (e2 ) = ±e2 .
5
If RTu T (e2 ) = −e2 , let S = S(0) be reflection through
the x-axis, otherwise let S = I .
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Proof of First Classification
Lance Drager
Introduction
Orthogonal
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Classifying
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Translations
First
Classification
Final
Classification
Exercises
Proof.
1
The points 0, e1 , e2 are noncolinear.
2
Let u = −T (0). Then Tu T (0) = 0.
3
dist(Tu T (e1 ), 0) = 1, so we can find a rotation matrix R
so that RTu T (e1 ) = e1 .
4
RTu T (e2 ) = ±e2 .
5
If RTu T (e2 ) = −e2 , let S = S(0) be reflection through
the x-axis, otherwise let S = I .
6
SRTu (0) = 0, SRTu T (e1 ) = e2 , and SRTu T (e2 ) = e2
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Translations
First
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Final
Classification
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Proof.
1
The points 0, e1 , e2 are noncolinear.
2
Let u = −T (0). Then Tu T (0) = 0.
3
dist(Tu T (e1 ), 0) = 1, so we can find a rotation matrix R
so that RTu T (e1 ) = e1 .
4
RTu T (e2 ) = ±e2 .
5
If RTu T (e2 ) = −e2 , let S = S(0) be reflection through
the x-axis, otherwise let S = I .
6
SRTu (0) = 0, SRTu T (e1 ) = e2 , and SRTu T (e2 ) = e2
7
SRTu T = I , so T = T−u R −1 S −1
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Translations
First
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Final
Classification
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Proof.
1
The points 0, e1 , e2 are noncolinear.
2
Let u = −T (0). Then Tu T (0) = 0.
3
dist(Tu T (e1 ), 0) = 1, so we can find a rotation matrix R
so that RTu T (e1 ) = e1 .
4
RTu T (e2 ) = ±e2 .
5
If RTu T (e2 ) = −e2 , let S = S(0) be reflection through
the x-axis, otherwise let S = I .
6
SRTu (0) = 0, SRTu T (e1 ) = e2 , and SRTu T (e2 ) = e2
7
SRTu T = I , so T = T−u R −1 S −1
8
Let A = R −1 S −1 and v = −u. Then T (x) = Ax + v
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Translations
First
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1 Introduction
2 Orthogonal Matrices
Orthogonal Matrices and Dot Product
Classifying Orthogonal Matrices
Exercises
3 Classifying Isometries
Translations
First Classification
Final Classification
Exercises
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Notation and Algebra
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Classifying
Orthogonal
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• If T (x) = Ax + v write T = (A | v ).
• Calculate the product (composition) in this notation
(A | u)(B | v )x = (A | u)(Bx + v )
= A(Bx + v ) + u
Classifying
Isometries
= ABx + Av + u
Translations
First
Classification
Final
Classification
Exercises
= (AB | u + Av )x.
• (A | u)(B | v ) = (AB | u + Av )
• The identity transformation is (I | 0). Translation by v is
(I | v ).
• (A | u)−1 = (A−1 | − A−1 u)
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• Consider (R | v ) where R = R(θ) 6= I is a rotation.
• Look for a fixed point p, i.e., (R | v )p = p
Rp + v = p
⇐⇒ p − Rp = v
⇐⇒ (I − R)p = v .
• If (I − R) is invertible, there is a unique solution p.
• (I − R) is invertible because
(I − R)x = 0 ⇐⇒ x = Rx ⇐⇒ x = 0. (Use the Big
Theorem.)
• There is a unique fixed point p, and we can write
(R | v ) = (R | p − Rp).
• (R | p − Rp)x = Rx + p − Rp = p + R(x − p).
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• y = (R | p − Rp)x = p + R(x − p) says that this isometry
is rotation though angle θ around the point p,which is
called the center of rotation or rotocenter.
y
y
R(x − p)
Translations
First
Classification
Final
Classification
Exercises
θ
x−p
x
p
x
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• Consider (S | w ) where S = S(θ) is a reflection matrix.
• Let u and v be the vectors with Su = u and Sv = −v .
We can write w = αu + βv , so (S | w ) = (S | αu + βv ).
• First Case: α = 0. So we have (S | βv ).
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• Consider (S | w ) where S = S(θ) is a reflection matrix.
• Let u and v be the vectors with Su = u and Sv = −v .
We can write w = αu + βv , so (S | w ) = (S | αu + βv ).
• First Case: α = 0. So we have (S | βv ).
• The point p = βv /2 is fixed.
(S | βv )p = S(βv /2) + βv = −βv /2 + βv = βv /2 = p.
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• Consider (S | w ) where S = S(θ) is a reflection matrix.
• Let u and v be the vectors with Su = u and Sv = −v .
We can write w = αu + βv , so (S | w ) = (S | αu + βv ).
• First Case: α = 0. So we have (S | βv ).
• The point p = βv /2 is fixed.
(S | βv )p = S(βv /2) + βv = −βv /2 + βv = βv /2 = p.
• The fixed points are exactly x = p + tu.
(S | 2p)(p + tu) = Sp + tSu + 2p = −p + tu + 2p = p + tu.
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• The line through p parallel to u is parametrized by
x = p + tu, for t ∈ R.
y
tu
p
Translations
First
Classification
Final
Classification
Exercises
x
v
u
x
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• We can write a point x as x = p + tu + sv . Then
y = (S | 2p)x = p + tu − sv .
y
x
sv
tu + sv
M
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
tu
p
−sv
tu − sv
v
y
u
x
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Translations
First
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Classification
Exercises
• (S | βv ) = (S | 2p) is reflection through the mirror line M
given by y = p + tu.
• Second Case: α 6= 0. We have (S | αu + βv ).
(S | αu + βv ) = (I | αu)(S | βv )
This is a reflection followed by a translation parallel to the
mirror line. This is called a glide refection or just a glide.
The mirror line of the reflection is called the glide line.
• A glide has no fixed points.
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First
Classification
Final
Classification
Exercises
Glide Reflection Picture
• A glide with a horizontal glide line, and the translation
vector shown in blue.
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Matrices and
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Orthogonal
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Classifying
Isometries
Translations
First
Classification
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Classification
Exercises
Theorem
Every isometry of the plane falls into on of the following five
mutually exclusive classes.
1
The identity.
2
A translation (not the identity).
3
A rotation about some point (not the identity);
4
A reflection through some mirror line.
5
A glide along some glide line.
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Outline
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Introduction
Orthogonal
Matrices
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Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
1 Introduction
2 Orthogonal Matrices
Orthogonal Matrices and Dot Product
Classifying Orthogonal Matrices
Exercises
3 Classifying Isometries
Translations
First Classification
Final Classification
Exercises
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Exercises
1
Consider the cases for the product T1 T2 of two isometries
T1 and T2 . Describe what happens geometrically (e.g.,
rotation angle, location of the mirror line, etc.). In what
cases do the isometries commute, i.e., when does
T1 T2 = T2 T1
2
Project: For a rotation (R | v ) give a geometric description
of how to find the rotocenter p = (I − R)−1 v .
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
3
As part of the first exercise, given two rotations (with
possibly different rotocenters) show that the composition
is a rotation or a translation.
4
Project: Given two rotations with different rotocenters
give a geometric description of how to find the rotocenter
of the composition.