Geometric Transformations and Wallpaper Groups Isometries of the Plane Lance Drager

advertisement
Wallpaper
Groups
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
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.
Wallpaper
Groups
Orthogonal Matrices
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
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
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
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 .
Wallpaper
Groups
Dot Product 2
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
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.
Wallpaper
Groups
Dot Product 3
Lance Drager
Introduction
Proof.
Orthogonal
Matrices
(=⇒)
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
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
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
Lance Drager
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)?
Wallpaper
Groups
Lance Drager
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)?
• There are orthogonal unit vectors u and v so that
S(θ)u = u and S(θ)v = −v .
Wallpaper
Groups
Classifying Orthogonal Matrices 2
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
• 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)
Wallpaper
Groups
Classifying Orthogonal Matrices 2
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
• 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)
Wallpaper
Groups
Classifying Orthogonal Matrices 3
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
•
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)
Wallpaper
Groups
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
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
Wallpaper
Groups
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
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.
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
Orthogonal Matix Exercises 1
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
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 .
Wallpaper
Groups
Lance Drager
Introduction
Exercises on Orthogonal Matrices
2
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
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)
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
Translations
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
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 ).
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
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
Isometries
Translations
First
Classification
Final
Classification
Exercises
p1
p2
p3
Wallpaper
Groups
First Classification
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
• 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.
Wallpaper
Groups
Proof of First Classification
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
Proof.
1
The points 0, e1 , e2 are noncolinear.
Wallpaper
Groups
Proof of First Classification
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
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.
Wallpaper
Groups
Proof of First Classification
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
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 .
Wallpaper
Groups
Proof of First Classification
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
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 .
Wallpaper
Groups
Proof of First Classification
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
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 .
Wallpaper
Groups
Proof of First Classification
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
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 .
6
SRTu (0) = 0, SRTu T (e1 ) = e2 , and SRTu T (e2 ) = e2
Wallpaper
Groups
Proof of First Classification
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
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 .
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
Wallpaper
Groups
Proof of First Classification
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
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 .
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
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
Notation and Algebra
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
• 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)
Wallpaper
Groups
Rotations
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
• 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).
Wallpaper
Groups
Picture of Rotation
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
• 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
Wallpaper
Groups
Lance Drager
Isometries with a Reflection Matrix
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
• 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 ).
Wallpaper
Groups
Lance Drager
Isometries with a Reflection Matrix
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
• 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.
Wallpaper
Groups
Lance Drager
Isometries with a Reflection Matrix
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
• 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.
Wallpaper
Groups
The Line of Fixed Points
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
• 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
Wallpaper
Groups
Reflection Picture
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
• 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
Wallpaper
Groups
Reflections and Glides
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
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.
Wallpaper
Groups
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
Classification
Exercises
Glide Reflection Picture
• A glide with a horizontal glide line, and the translation
vector shown in blue.
Wallpaper
Groups
Classification Theorem
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
Classifying
Isometries
Translations
First
Classification
Final
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.
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
Exercises on Isometries
Lance Drager
Introduction
Orthogonal
Matrices
Orthogonal
Matrices and
Dot Product
Classifying
Orthogonal
Matrices
Exercises
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.
Download