modern geometry · 2018. 10. 2. · motions in the plane re ections any subset of the plane is...

Motions in the Plane

MODERN GEOMETRYMotions in the Plane

Ederlina Ganatuin-Nocon

SPECIAL Term, AY2014-2015

Ederlina Ganatuin-Nocon MODERN GEOMETRY


Reflections

Any subset of the plane is called a figure.

Let ` be a line passing through a point P and havingnormal unit vector N . Two points X and X ′ aresymmetrical about ` if the midpoint of the segmentXX ′ is the foot F of the perpendicular from X to `.

X and X ′ are symmetrical about ` if we have

F =1

2(X +X ′).



Reflections

This means that

F =1

2(X +X ′) = X − 〈X − P,N〉N,

1

2X ′ =

1

2X − 〈X − P,N〉N,

X ′ = X − 2〈X − P,N〉N.



Reflections

Definition

For a line ` the reflection in ` is the mapping Ω` of E2 to E2

defined byΩ`X = X − 2〈X − P,N〉N,

where N is a unit normal to ` and P is any point of `.



Reflections

If F is a figure such that Ω`F = F , then we say that F issymmetric about `. The line ` is called a line of symmetryor axis of symmetry of F .



Reflections

Theorem

i. d(Ω`X,Ω`Y ) = d(X,Y ) for all X,Y ∈ E2

ii. Ω`Ω`X = X for all X ∈ E2.

iii. Ω` : E2 → E2 is a bijection.



Reflections

Theorem

Ω`X = X if and only if X ∈ `.



Congruence and Isometries

If F is a figure and Ω` is any reflection, then Ω`F is called themirror image of F in the line `.




The figure and its mirror image are of the same size and shape.If Ωm is a second reflection, then ΩmΩ`F is again the same sizeand shape.




Definition

A mapping T of E2 onto E2 is said to be an isometry if for anyX,Y ∈ E2,

d(TX, TY ) = d(X,Y ).




Definition

Two figures F1 and F2 are congruent if there exists anisometry T such that TF1 = F2.




Remark

1 Every reflection is an isometry.

2 Not every isometry is a reflection.




Every isometry T is a bijection of E2 onto E2.If TX = TY , then 0 = d(TX, TY ) = d(X,Y ) andtherefore, X = Y . This tells us that T is one-to-one.

The inverse map T−1 exists and it is also an isometry since

d(T−1X,T−1Y ) = d(TT−1X,TT−1Y ) = d(X,Y ).

If T and S are isometries, then so is TS.

d(TSX, TSY ) = d(SX, SY ) = d(X,Y ).




Theorem

i. If T and S are isometries, so is TS (TS = T S).

ii. If T is an isometry, then so is T−1.

iii. The identity map I of E2 is an isometry.

In other words, the set of all isometries is a group called theisometry group of E2. It is denoted by I (E2).



Symmetry Groups

Let F be a figure in E2. Then the set

S (F ) = T ∈ I (E2)| TF = F

is a subgroup of I (E2) called the symmetry group of F .The size of the symmetry group of F is a measure of the degreeof symmetry of the figure.



Symmetry Groups

An equilateral triangle has a symmetry group of size 6.

An isosceles triangle has a symmetry group of size 2.

The circle has an infinite symmetry group generated byreflections in all diameters.



Translations

Suppose m and n are parallel lines. We choose P arbitrarily onm and choose Q to be the foot of the perpendicular from P ton. Then if N is a unit normal to m (and hence to n), we get

ΩmΩnx = Ωnx− 2〈Ωnx− P,N〉N= x− 2〈x−Q,N〉N − 2〈x− P,N〉N +

4〈x−Q,N〉〈N,N〉N= x+ 2〈P −Q,N〉N= x+ 2(P −Q)



Translations

Definition

Let ` be any line, and let m and n be perpendicular to `. Thetransformation ΩmΩn is called a translation along `. Ifm 6= n, the translation is said to nontrivial.



Translations

Remark

When two lines in E2 are perpendicular to `, then they areparallel.

When two lines are parallel, there is a line (in fact,infinitely many lines) that is perpendicular to both.

In the Euclidean plane, a translation does not determine aunique line. Although it determines a parallel family.



Translations

Theorem

Let T be a translation along `. If `′ is any line parallel to ` thenT is also a translation along `.



Translations

Theorem

Let T be a nontrivial translation along `. Then `′ has adirection vector v such that

Tx = x+ v (1)

for all x ∈ E2. Conversely, if v is a nonzero vector and ` is anyline with direction vector v, then the transformation (1) is atranslation along `.



Translations

Definition

The set of all lines perpendicular to a given line ` in E2 is calleda pencil of parallels. The line ` is called a commonperpendicular for the pencil.

We note that taking any line m in E2 together with all linesparallel to m would be an equivalent construction of pencil ofparallels.



Translations

Theorem

Let TRANS(`) denote all translations along `. Then TRANS(`)is an abelian group isomorphic to the additive group of realnumbers.



TranslationsThree Reflections Theorem

Theorem

Let α, β, and γ be three lines of a pencil P with commonperpendicular `. Then there is a unique fourth line δ of thispencil such that

ΩαΩβΩγ = Ωδ.




Let P be the pencil of all lines that are perpendicular to a line`. We denote by REF(P) the group generated by all reflectionsof the form Ωm where m ∈P. In other words, REF(P) is thesmallest subgroup of I (E2) containing all such Ωm.This implies that TRANS(`) is a subgroup of REF(P).




Let α, β, and γ be three lines in the pencil P corresponding tothe numbers a, b and c. Then

ΩαΩβΩγ = Ωα T2(b−c)ΩαΩβΩγx = Ωα(x+ 2(b− c)N)

= Ωα(x+ µN), where µ = 2(b− c)= x+ µN − 2〈x+ µN − P − aN,N〉N= x− 2〈x− P,N〉N + (2a− µ)N

= x− 2〈x− P,N〉N + 2(a− b+ c)N

= x− 2〈x− (P + (a− b+ c)N), N〉N




In the equation

ΩαΩβΩγx = x− 2〈x− (P + (a− b+ c)N), N〉N

we see that the right side is the formula for reflection in the lineδ ∈P passing through the point P + dN where d = a− b+ c.



TranslationsRepresentation Theorem for Translations

Theorem

Let T = ΩαΩβ be any member of TRANS(`). If m and n arearbitrary lines perpendicular to `, there exist lines m′ and n′

such thatT = ΩmΩm′ = Ωn′Ωn.



Translations

Corollary

Every element of REFP is either a translation along ` or areflection in a line of P.

Ωb TµΩa T2(a−b) Ωa−µ/2Tλ Ωb+λ/2 Tλ+µ

Ωa is short for Ωα where α = P + aN + [N⊥].



Translations

Let v be any vector in E2. Define τv as the translation

τvx = x+ v.

(If v = 0, the τv = I, the trivial translation.)

We use the notation T (E2) to denote the set of all translationsin E2.



Translations

Theorem

The set T (E2) is an abelian subgroup of I (E2).

Corollary

The group T (E2) is isomorphic to the group R2 with vectoraddition.



Rotations

Let ` = P + [v] be a line with unit direction vector v. There is aunique real number θ ∈ (−π, π] such that

v = (cos θ, sin θ).

The unit normal v⊥ can be written as

N = (− sin θ, cos θ).

We now try to express Ω` in terms of θ. We note that

Ω`x = x− 2〈x− P,N〉NΩ`x− P = x− P − 2〈x− P,N〉N



Rotations

Let `0 be the line through 0 parallel to ` then

Ω`0x = x− 2〈x,N〉N.

Hence, Ω`x− P = x− P − 2〈x− P,N〉N implies

Ω`x− P = Ω`0(x− P )

Ω`x = Ω`0(x− P ) + P

Therefore,Ω` = τPΩ`0τ−P .



Rotations

Now we deal with Ω`0 . Note that for any x

〈x,N〉 = −x1 sin θ + x2 cos θ.

Writing our vectors as column vectors, we get

Ω`0

[x1x2

]=

[x1x2

]− 2(−x1 sin θ + x2 cos θ)

[− sin θ

cos θ

]=

[(1− 2 sin2 θ)x1 + (2 sin θ cos θ)x2(2 sin θ cos θ)x1 + (1− 2 cos2 θ)x2

]=

[cos 2θ sin 2θsin 2θ − cos 2θ

] [x1x2

]



Rotations

We see that Ω`0 is linear. From hereon we use the notation

ref θ =


]We now investigate the matrix algebra of these reflections.Consider another line m through P and the associated line m0.Then if (cosφ, sinφ) is a direction vector of m, we have

ref θ · ref φ =


] [cos 2φ sin 2φsin 2φ − cos 2φ

]=

[cos 2(θ − φ) − sin 2(θ − φ)sin 2(θ − φ) cos 2(θ − φ)

]



Rotations

We the notation

rotα =

[cosα − sinαsinα cosα

]so that

ref θ · ref φ = rot 2(θ − φ).

Because this linear mapping takes the standard unit basisvector ε1 to v = (cos θ, sin θ) and takes ε2 tov⊥ = (− sin θ, cos θ), it is reasonable to think of rot θ as arotation by θ radians in the positive sense.



Rotations

Definition

If α and β are lines passing through a point P , the isometryΩαΩβ is called a rotation about P . The special case α = β isallowed so that the identity is (by definition) a rotation aboutP no matter what P is. If a rotation is not the identity, we referto it as nontrivial rotation. If α ⊥ β, the rotationΩαΩβ = HP is called a half-turn about P .



Rotations

Theorem

The set of all rotations about the origin is an abelian groupcalled SO(2).

The symbol SO(2) stands for the special orthogonal group ofE2.



Rotations

Theorem

i. ref θ rotφ = ref

(θ −

φ

2

)

ii. rot θ ref φ = ref

(φ+

θ

2

)iii. ref θ ref φ ref ψ = ref(θ − φ+ ψ)



Rotations

Theorem

The set of all rotations about the origin and reflections in linesthrough the origin is a group called the orthogonal group and isdenoted by O(2). SO(2) is a subgroup of index 2 in O(2).



Rotations

The following is the group multiplication table of O(2).

ref φ rotβ

ref θ rot 2(θ − φ) ref

(θ −

β

2

)

rotα ref

(φ+

α

2

)rot (α+ β)



Rotations

Theorem

Let P be the pencil of all lines through a point P . ThenREF (P) ∼= O(2) and ROT (P ) ∼= SO(2).



Rotations

Theorem

Let α, β and γ be three lines through P . Then there is a uniqueline δ through P such that

ΩαΩβΩγ = Ωδ



Rotations

Theorem

Let T = ΩαΩβ be any member of ROT (P ), and let ` be any linethrough P . Then there exist unique lines m and m′ through Psuch that

T = Ω`Ωm = Ωm′Ω`.



Glide Reflections

A glide reflection is a reflection followed by a translationalong the mirror. If ` = P + [v], the glide reflection defined by `and v is given by

τvΩ`x = x− 2〈x− P,N〉N + v,

where N is the unit vector v⊥/|v|.



Glide Reflections

Note that

Ω`τvx = x+ v − 2〈x+ v − P,N〉N= x+ v − 2〈x− P,N〉N

since 〈v,N〉 = 0. Thus, the reflection and translation that makeup the glide reflection commute.

Because τv = I is a possibility, each reflection is also glidereflection. But this type of glide reflection is said to be trivial.



Glide Reflections

Theorem

Let α, β and γ be three distinct lines that are not concurrent andnot all parallel. Then ΩαΩβΩγ is a nontrivial glide reflection.



Glide Reflections

Outline of Proof:

Assume that α meets β atP . Let ` be the linethrough P perpendicularγ.

Let F be the point ofintersection of ` and γ.

Using the representationtheorem for rotations,there exists a line mthrough P such thatΩαΩβ = ΩmΩ` andΩαΩβΩγ = ΩmΩ`Ωγ .



Glide Reflections

Let n be the line throughF perpendicular to m andlet n′ be the line throughF perpendicular to n.

Observe thatΩ`Ωγ = Ωn′Ωn = HF .

Then

ΩαΩβΩγ = ΩmΩn′Ωn.



Glide Reflections

Note that ΩmΩn′ is atranslation along n.

Because F does not lie onm, then n′ and m aredistinct.

This shows that ΩαΩβΩγ

is a nontrivial glidereflection.



Glide Reflections

If α does not meet β but instead, β meets γ, apply thesame argument to ΩγΩβΩα = τvΩ`, thenΩαΩβΩγ = (τvΩ`)

−1 = Ω−1` τ−1v = Ω`τ−v which is also anontrivial glide reflection.



Glide Reflections

Theorem

Let T be a glide reflection, and let Ωα be any reflection. ThenΩαT is a translation or rotation.



Glide Reflections

Outline of Proof:

Let ` be the axis of the glide reflection T .

CASE 1: ` intersects αLet P be the point of intersection. By the representationtheorem for translations, we may write T = Ω`ΩaΩb, wherea passes through P , and both a and b are perpendicular to`. Then

ΩαT = ΩαΩ`ΩaΩb.



Glide Reflections

But now α, ` and a all pass through P . By the threereflections theorem there is a line c through P such that

ΩαT = ΩcΩb.

Thus, ΩαT is either a translation or a rotation.



Glide Reflections

CASE 2: ` ‖ αThen

ΩαT = ΩαΩ`ΩaΩb = ΩαΩaΩ`Ωb.

Note that b ⊥ ` and α ⊥ a so that ΩαΩa and Ω`Ωb aredistinct half-turns.

But the product of two half-turns is a translation.



Motions

Definition

An isometry that is the product of a finite number of reflectionsis called a motion.



Motions

Theorem

Every motion is the product of two or three suitably chosenreflections.

Corollary

The group of motions consists of all translations, rotations,reflections, and glide reflections.



Motions

Theorem

Every isometry of E2 is a motion.



Motions

Outline of Proof:

CASE 1: T (0) = 0. In this case, we show that rot θ or ref θ forsome value of θ. In other words, T ∈ O(2). (The next lemma isto be used in this case.)



Motions

CASE 2: T (P ) = P for some point P . Then τ−PTτ−P is anisometry leaving 0 fixed and is hence, a member of O(2) byCase 1. Thus, T = τP (rot θ)τ−P or T = τP (ref θ)τ−P . In eithercase, T is a motion.



Motions

CASE 3: T has no fixed points. P = T (0). Then τ−P T leaves0 fixed and is, hence either rot θ or ref θ. In any case,T = τP rot θ or T = τP ref θ, so that T is a motion.



Motions

The following lemma shows that an isometry leaving the originfixed has a particularly nice algebraic form - it must be linear.

Lemma

If T is an isometry with T (0) = 0, then

(i) 〈Tx, Ty〉 = 〈x, y〉(ii) T = rot θ of T = ref θ for some θ.



Motions

Proof:

For (i), we have

〈x, y〉 =1

2(|x|2 + |y|2 − |x− y|2),

〈Tx, Ty〉 =1

2(|Tx|2 + |Ty|2 − |Tx− Ty|2).

Now,

|Tx| = d(0, Tx) = d(T (0), Tx) = d(0, x) = |x|.

Similarly, |Ty| = |y|. Also

|Tx− Ty| = d(Tx, Ty) = d(x, y) = |x− y|.



Motions

For (ii), let ε1 = (1, 0) and ε2 = (0, 1). If x = x1ε1 + x2ε2, thenTε1, T ε2 is an orthonormal basis for E2.

Hence,Tx = 〈Tx, Tε1〉Tε1 + 〈Tx, Tε2〉Tε2.

Using the result of (i) we have

Tx = 〈x, ε1〉Tε1 + 〈x, ε2〉Tε2 = x1Tε1 + x2Tε2.

Now, Tε1 is a unit vector. Writing

Tε1 = λ1ε1 + λ2ε2

we see that by the Cauchy-Schwartz Inequality,

|λ1| = |〈Tε1, ε1〉| ≤ |Tε1||ε1| = 1.



Motions

Similarly,|λ2| ≤ 1 and |Tε1|2 = λ21 + λ22

so that λ21 + λ22 = 1. This implies there is a unique θ ∈ (−π, π]such that λ1 = cos θ and λ2 = sin θ. Now,

〈Tε1, T ε2〉 = 〈ε1, ε2〉 = 0

and therefore,Tε2 = ±(Tε1)

⊥.

Therefore,Tε2 = ±((− sin θ)ε1 + (cos θ)ε2).



Motions

In matrix form, we have that either

Tx =

[cos θ − sin θsin θ cos θ

] [x1x2

]= rot θ · x

or

Tx =

[cos θ sin θsin θ − cos θ

] [x1x2

]= ref(θ/2) · x

for all ∈ E2.



Fixed Points and Fixed Lines of Isometries

Theorem

(i) A nontrivial translation has no fixed points.

(ii) A nontrivial rotation has exactly one fixed point, thecenter of rotation.

(iii) A reflection has a line of fixed points, the axis of reflection.

(iv) A nontrivial glide reflection has no fixed points.

(v) The identity has a plane of fixed points.




Corollary

The fixed point set of an isometry must be one of the following:

(i) a point (rotation)

(ii) a line (reflection)

(iii) the empty set (translation or glide reflection)

(iv) the whole plane E2 (the identity).




Theorem

(i) A nontrivial translation along a line ` has a pencil ofparallels as its fixed lines. This pencil consists of all linesparallel to `.

(ii) A half-turn centered at C has the pencil of all linesthrough C as its set of fixed lines. A nontrivial rotationthat is not a half turn has no fixed lines.

(iii) A reflection Ωm has the line m and its pencil of commonperpendiculars as its fixed lines

(iv) A nontrivial glide reflection has exactly one fixed line - itsaxis.

(v) The identity leaves all lines fixed.


modern geometry · 2018. 10. 2. · motions in the plane re ections any subset of the plane is...

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