Introduction Orthogonal Geometric Transformations and...

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WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

OrthogonalMatrices andDot Product

ClassifyingOrthogonalMatrices

Exercises

ClassifyingIsometries

Translations

FirstClassification

FinalClassification

Exercises

Geometric Transformations and WallpaperGroups

Isometries of the Plane

Lance Drager

Department of Mathematics and StatisticsTexas Tech University

Lubbock, Texas

2010 Math Camp

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Outline

1 Introduction

2 Orthogonal MatricesOrthogonal Matrices and Dot ProductClassifying Orthogonal MatricesExercises

3 Classifying IsometriesTranslationsFirst ClassificationFinal ClassificationExercises

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Isometries

• A transformation T of the plane is an isometry if it isone-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.

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Orthogonal Matrices

• A matrix A is orthogonal if ‖Ax‖ = ‖x‖ for all x ∈ R2.

• An orthogonal matrix gives an isometry of the plane since

dist(Ap,Aq) = ‖Ap − Aq‖= ‖A(p − q)‖ = ‖p − q‖ = 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.

‖p− q‖

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Outline

1 Introduction

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Dot Product 1

TheoremA is orthogonal if and only if Ax · Ay = x · y for all x , y ∈ R2.Thus, an orthogonal matrix preserves angles.

Proof.(=⇒)

‖Ax‖2 = Ax · Ax = x · x = ‖x‖2

(⇐=) Recall

2x · y = ‖x‖2 + ‖y‖2 − ‖x − y‖2.

WallpaperGroups

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Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Dot Product 2

Proof Continued.So, we have

2Ax · Ay = ‖Ax‖2 + ‖Ay‖2 − ‖Ax − Ay‖2

= ‖Ax‖2 + ‖Ay‖2 − ‖A(x − y)‖2 (distributive law)

= ‖x‖2 + ‖y‖2 − ‖x − y‖2 (A is orthogonal)

= 2x · y ,

and dividing by 2 gives the result.

TheoremA matrix is orthogonal if and only if its columns are orthogonalunit vectors.

WallpaperGroups

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Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Dot Product 3

Proof.(=⇒)

‖Aei‖ = ‖ei‖ = 1.

Ae1 · Ae2 = e1 · e2 = 0.

(⇐=) Let A = [u | v ] where u and v are orthogonal unitvectors. Note Ax = x1u + x2v .

‖Ax‖2 = Ax · Ax

= (x1u + x2v) · (x1u + x2v)

= x21 u · u + 2x1x2u · v + x2

2 v · v= x2

1 (1) + 2x1x2(0) + x22 (1)

= x21 + x2

2 = ‖x‖2

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Outline

1 Introduction

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Classifying Orthogonal Matrices 1

• If A is orthogonal, Col1(A) = (cos(θ), sin(θ)) for some θ.So the possibilities are

[cos(θ) − sin(θ)sin(θ) cos(θ)

]= R(θ),

[cos(θ) sin(θ)sin(θ) − cos(θ)

]= S(θ).

• In the first case A = R(θ) is a rotation.

• What about S(θ)?

• There are orthogonal unit vectors u and v so thatS(θ)u = u and S(θ)v = −v .

WallpaperGroups

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Exercises

• If A is orthogonal, Col1(A) = (cos(θ), sin(θ)) for some θ.So the possibilities are

[cos(θ) − sin(θ)sin(θ) cos(θ)

]= R(θ),

]= S(θ).

• In the first case A = R(θ) is a rotation.

• What about S(θ)?

• There are orthogonal unit vectors u and v so thatS(θ)u = u and S(θ)v = −v .

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Exercises

• In fact, let u = (cos(θ/2), sin(θ/2)) andv = (− sin(θ/2), cos(θ/2)).

S(θ)u =

] [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)

WallpaperGroups

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Exercises

• In fact, let u = (cos(θ/2), sin(θ/2)) andv = (− sin(θ/2), cos(θ/2)).

S(θ)u =

] [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)sin(θ − θ/2)

[cos(θ/2)sin(θ/2)

WallpaperGroups

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Exercises

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FirstClassification

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Exercises

S(θ)v =

] [− 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)

− cos(θ/2)

]= −

[− sin(θ/2)

cos(θ/2)

]= −v .

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Exercises

• S(θ)u = u and S(θ)v = −v . If x = tu + sv , thenS(θ)x = tu − sv .

S(θ)x

• S(θ) is a reflection with its mirror line at an angle of θ/2.

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Exercises

• S(θ)u = u and S(θ)v = −v . If x = tu + sv , thenS(θ)x = tu − sv .

S(θ)x

• S(θ) is a reflection with its mirror line at an angle of θ/2.

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Outline

1 Introduction

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Orthogonal Matix Exercises 1

Exercises

• S(θ)2 = I , so S(θ)−1 = S(θ).

• Let A be an orthogonal matrix. Then A is a rotation (orthe identity) if and only if det(A) = 1 and A is a reflectionif and only if det(A) = −1.

• If A =

[a cb d

]define the transpose of A by

[a bc d

]. Show that A is orthogonal if and only if

A−1 = AT .

WallpaperGroups

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OrthogonalMatrices

Exercises

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Exercises

Exercises on Orthogonal Matrices2

Exercises Continued

• Use the addition laws for sine and cosine to verify theimportant identities

R(θ)S(φ) = S(θ + φ),

S(φ)R(θ) = S(φ− θ),

S(θ)S(φ) = R(θ − φ)

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Outline

1 Introduction

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Translations

• For v ∈ R2, define Tv : R2 → R2 by Tv (x) = x + v . Wesay Tv is translation by v.

• TuTv = Tu+v and T−1v = T−v .

• Translations are isometries

dist(Tv (x),Tv (y)) = ‖Tv (x)− Tv (y)‖= ‖(x + v)− (y + v)‖= ‖x + v − y − v‖= ‖x − y‖= dist(x , y).

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Outline

1 Introduction

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Three Distances Determine a Point

• Let p1, p2 and p3 be noncolinear points. A point x isuniquely determined by the three numbers r1 = dist(x , p1),r2 = dist(x , p2) and r3 = dist(x , p3).

WallpaperGroups

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OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

First Classification

• 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).

TheoremEvery isometry T can be written as T (x) = Ax + v where A isan orthogonal matrix, i.e., T is multiplication by an orthogonalmatrix followed by a translation.

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Proof of First Classification

Proof.

1 The points 0, e1, e2 are noncolinear.

2 Let u = −T (0). Then TuT (0) = 0.

3 dist(TuT (e1), 0) = 1, so we can find a rotation matrix Rso that RTuT (e1) = e1.

4 RTuT (e2) = ±e2.

5 If RTuT (e2) = −e2, let S = S(0) be reflection throughthe x-axis, otherwise let S = I .

6 SRTu(0) = 0, SRTuT (e1) = e2, and SRTuT (e2) = e2

7 SRTuT = I , so T = T−uR−1S−1

8 Let A = R−1S−1 and v = −u. Then T (x) = Ax + v

WallpaperGroups

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Exercises

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FirstClassification

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Exercises

Proof.

2 Let u = −T (0). Then TuT (0) = 0.

4 RTuT (e2) = ±e2.

WallpaperGroups

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OrthogonalMatrices

Exercises

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FirstClassification

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Exercises

Proof.

2 Let u = −T (0). Then TuT (0) = 0.

4 RTuT (e2) = ±e2.

WallpaperGroups

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OrthogonalMatrices

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Exercises

Proof.

2 Let u = −T (0). Then TuT (0) = 0.

4 RTuT (e2) = ±e2.

WallpaperGroups

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Exercises

Proof.

2 Let u = −T (0). Then TuT (0) = 0.

4 RTuT (e2) = ±e2.

WallpaperGroups

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OrthogonalMatrices

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Exercises

Proof.

2 Let u = −T (0). Then TuT (0) = 0.

4 RTuT (e2) = ±e2.

WallpaperGroups

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OrthogonalMatrices

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FirstClassification

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Exercises

Proof.

2 Let u = −T (0). Then TuT (0) = 0.

4 RTuT (e2) = ±e2.

WallpaperGroups

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Introduction

OrthogonalMatrices

Exercises

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FirstClassification

FinalClassification

Exercises

Proof.

2 Let u = −T (0). Then TuT (0) = 0.

4 RTuT (e2) = ±e2.

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Outline

1 Introduction

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Notation and Algebra

• 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

= ABx + Av + u

= (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−1u)

WallpaperGroups

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Exercises

Rotations

• 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 BigTheorem.)

• 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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Picture of Rotation

• y = (R | p − Rp)x = p + R(x − p) says that this isometryis rotation though angle θ around the point p,which iscalled the center of rotation or rotocenter.

xx− pθ

R(x− p)

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Exercises

Isometries with a Reflection Matrix

• 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.

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

WallpaperGroups

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Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

The Line of Fixed Points

• The line through p parallel to u is parametrized byx = p + tu, for t ∈ R.

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Reflection Picture

• We can write a point x as x = p + tu + sv . Theny = (S | 2p)x = p + tu − sv .

tu + sv

tu− sv

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Reflections and Glides

• (S |βv) = (S | 2p) is reflection through the mirror line Mgiven 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 themirror 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.

WallpaperGroups

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Exercises

Glide Reflection Picture

• A glide with a horizontal glide line, and the translationvector shown in blue.

WallpaperGroups

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Exercises

Classification Theorem

TheoremEvery isometry of the plane falls into on of the following fivemutually 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.

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Outline

1 Introduction

WallpaperGroups

Lance Drager

Introduction

OrthogonalMatrices

Exercises

Translations

FirstClassification

FinalClassification

Exercises

Exercises on Isometries

Exercises

1 Consider the cases for the product T1T2 of two isometriesT1 and T2. Describe what happens geometrically (e.g.,rotation angle, location of the mirror line, etc.). In whatcases do the isometries commute, i.e., when doesT1T2 = T2T1

2 Project: For a rotation (R | v) give a geometric descriptionof how to find the rotocenter p = (I − R)−1v .

3 As part of the first exercise, given two rotations (withpossibly different rotocenters) show that the compositionis a rotation or a translation.

4 Project: Given two rotations with different rotocentersgive a geometric description of how to find the rotocenterof the composition.

Introduction Orthogonal Geometric Transformations and...

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