Transformational Plane Geometry - Problems with Solutions | MATH 355, Study notes of Mathematics

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Transformational Plane Geometry

Solutions to Selected Exercises

Ronald N. Umble

July 6, 2007

4 CHAPTER 1. ISOMETRIES

• β is not a collineation.
• γ is not a collineation.
• δ is a collineation: a 2 X + b 3 Y + c = 0.
•  is a collineation: X + Y − 3 = 0.
• η is a collineation: b 3 X + aY + (− 2 a + c) = 0.
• ρ is not a collineation.
• σ is a collineation: −aX − bY + c = 0.
• τ is a collineation: aX + bY + (− 2 a + 3b + c) = 0

12. −X + 10Y + 2 = 0
13. α([xy ]) = [ (^) yx 3 ]

1.2 Reflections

2. L or F
3. 97 cm
4. σl([xy ]) = 15 [ (^) −^34 xx−−^43 yy+12+24 ] σl([− 35 ]) = [− 73 ]

Equation of l Point P σl(P ) Equation of l Point P σl(P ) X = 0 [xy ] [−yx ] Y = − 3 [xy ] [−^6 y− x] Y = 0 [ (^) −xy ] [xy ] X = −^32 [^53 ] [− 38 ] Y = X [^32 ] [^23 ] Y = −X [^03 ] [− 03 ] Y = X [xy ] [ yx ] Y = −X [^03 ] [− 03 ] X = 2 [− 32 ] [^63 ] Y = 2X [^05 ] [^43 ] Y = − 3 [− −^41 ] [− −^45 ] 2 Y = 3X + 5 [xy ] 131 [− 125 xx+12− 8 yy+20−^30 ]

12. a) X + Y = 0 b) X + Y − 10 = 0

1.3. TRANSLATIONS 5

13. • σl([^00 ]) = [ (^) −^42 ]
    • σl([ (^) −^13 ]) = [ (^) −^13 ]
    • σl([− 12 ]) = [ (^) −^63 ]
    • σl([^24 ]) = [^62 ]
14. • σm([^00 ]) = 15 [− 126 ]
    • σm([ (^) −^41 ]) = 15 [ 312 ]
    • σm([− 53 ]) = [ (^) −^13 ]
    • σm([^36 ]) = 15 [^276 ]
15. 18X − Y + 44 = 0

1.3 Translations

2. a) [ (^) −^61 ] b) x′^ = x + 6, y′^ = y − 1 c) τ ([^00 ]) = [ (^) −^61 ], τ ([ (^) −^37 ]) = [ (^) −^98 ], τ ([− −^52 ]) = [ (^) −^13 ] d) τ ([− 16 ]) = [^00 ]
3. a) [^34 ] b) x′^ = x + 3, y′^ = y + 4 c) τ ([^00 ]) = [^34 ], τ ([^12 ]) = [^46 ], τ ([− −^34 ]) = [^00 ] d) τ ([− −^34 ]) = [^00 ]
4. a) [− 67 ] b) x′^ = x − 7, y′^ = y + 6

1.5. HALFTURNS 7

4. glide reflection
5. glide reflection
6. translation or halfturn
7. translation or halfturn
8. translation or halfturn

4. 12 [ (^) −^38 ]
5. a) x′^ = 4 − x, y′^ = 6 − y b) ϕP ([^12 ]) = [^34 ], ϕP ([− 52 ]) = [^61 ] c) 5X − Y − 21 = 0
6. a) x′^ = − 6 − x, y′^ = 4 − y b) ϕP ([^12 ]) = [− 27 ], ϕP ([− 52 ]) = [− −^41 ] c) 5X − Y + 27 = 0
7. x′′^ = − 10 − x′^ = − 10 − (6 − x) = x − 16, y′′^ = 14 − y′^ = 14 − (− 4 − y) = y + 18 This is a translation with vector [− 1816 ].
8. x′′^ = 2c − x′^ = 2c − (2a − x) = x + 2(c − a), y′′^ = 2d − y′^ = 2d − (2b − y) = y + 2(d − b) The translation vector is 2[cd−−ab ].
9. a) x′′^ = − 6 − x, y′′^ = 10 − y b) C = P = [− 53 ].

8 CHAPTER 1. ISOMETRIES

1.6 Properties of Translations and Halfturns

1. All real numbers a = b 6 = 0.
2. Right-side-up. The transformation can be thought of as the composi- tion of two halfturns and is therefore a translation.

1.7 General Rotations

1. ρO, 30 ([^36 ]) = 12 [^3

√ 3 − 6 3+6√ 3 ]

2. ρQ, 45 ([^36 ]) = 12 [ 5

√ 2 − 6 7 √2+10 ]

3. a) (√ 3 − 32 )X + (1 + 3 √ 2 3 )Y + 4 = 0 b) −√ 22 X + 5 √ 2 2 Y + 13 − 14 √2 = 0
4. a) x′^ = x − (x + y − 2) = −y + 2, y′^ = y − (x + y − 2) = −x + 2, x′′^ = x′^ = −y + 2, y′′^ = y′^ − 2(y′^ − 3) = 6 − y′^ = 6 − (−x + 2) = x + 4

⇒

[ (^) x′′ y′′

]

[ 0 − 1

] [ (^) x y

]

[ 2

]

This can be rewritten as [ (^) x′′ y′′

]

[ 0 − 1

] [ (^) x + 1 y − 3

]

[ − 1

]

or

[ (^) x′′ y′′

]

[ (^) cos 90◦ (^) − sin 90◦ sin 90◦^ cos 90◦

] [ (^) x − (−1) y − 3

]

[ − 1

]

b) The equation in part (a) represents a 90◦^ rotation about the point C = [− 31 ]. Lines l and m intersect at [− 31 ] with a directed angle from l to m of 45◦.

10 CHAPTER 2. CONGRUENCE

2. a) x′^ = 4 − x, y′^ = 4 − y b) C = [^22 ], θ = 180◦
3. Any two lines l and m intersecting at the origin with directed angle from l to m of 45◦, e.g. l : Y = 0, m : Y = X
4. Any two lines l and m intersecting at C = [^34 ] with directed angle from l to m of 30◦, e.g. l : Y − 4 = 0, m : X − √ 3 Y + 4√ 3 − 3 = 0

2.4 The Fundamental Theorem

1. 4 DEF = σl( 4 ABC) where l is given by 2 X − 4 Y − 5 = 0.
2. a) 4 DEF = (σm ◦ σl)( 4 ABC) where l and m are parallel lines producing a translation with vector [ (^) −^66 ], e.g. l : X − Y = 0, m : X − Y − 6 = 0 b) 4 DEF = (σm ◦ σl)( 4 ABC) where l and m are lines intersecting at [^15 ] with directed angle from l to m of 45◦, e.g. l : Y − 5 = 0, m : X − Y + 4 = 0 c) 4 DEF = (σm ◦ σl)( 4 ABC) where l and m produce a halfturn about [ (^) −^51 ], e.g. l : X − 5 = 0, m : Y + 1 = 0

2.4. THE FUNDAMENTAL THEOREM 11

d) 4 DEF = (σn ◦ σm ◦ σl)( 4 ABC) where l, m, and n produce a glide reflection with axis X + Y + 3 = 0 and glide vector [ (^) −^66 ], e.g. l : X + Y + 3 = 0, m : X − Y = 0, n : X − Y − 6 = 0 e) 4 DEF = σl( 4 ABC) where l is given by X + Y − 2 = 0.

Chapter 3

Classification of Isometries

3.1 The Angle Addition Theorem, part I

1. a) l : X − Y = 0, m : Y = 0, n : X + Y − 2 = 0 b) D = [^11 ] c) E = [− −^11 ]
2. a) l : Y = 0, m : X = 0, n : X + √ 3 Y − √3 = 0 b) D = [

√ 3 0 ],^ Θ = 300◦ c) E = 14 [

√ 3 3 ],^ Φ = 180◦

3. a) l : X − 4 = 0, m : X + Y − 4 = 0, n : (√ 3 − 2)X + Y − 4 = 0 b) C = [ (^12) −^44 √ 3 ], Θ = 210◦ c) D = [^4

√ 3 − 8 0 ],^ Φ = 210◦ 13

14 CHAPTER 3. CLASSIFICATION OF ISOMETRIES

3.2 Parity

1. l : X − Y = 0, m : X − Y + 2 = 0 σm ◦ σl is a translation.
2. l : X − Y + 1 = 0, m : X + Y = 0 σm ◦ σl is a rotation (halfturn).
3. l : X = 0, m : X = 0 σm ◦ σl is the identity.
4. l : X + Y = 0, m : X + Y − 2 = 0 σm ◦ σl is a translation.
5. l : X + 3Y = 0, m : X + 3Y − 5 = 0 σm ◦ σl is a translation.

3.3 The Geometry of Conjugation

1. The line b is the image of line a under a halfturn about point B: b = ϕB (a)
2. The point B is the reflection of point A in line b: B = σb(A)

3.5 The Classification Theorem

1. Halfturns and translations are dilatations (see sect. 1.6). Rotations other than halfturns are clearly not dilatations. The image of line a

16 CHAPTER 3. CLASSIFICATION OF ISOMETRIES

Chapter 4

Symmetry

4.2 Groups of Symmetries

1. ι = ρ^0120 ρ 240 = ρ^2120 σm = ρ 120 ◦ σl σn = ρ^2120 ◦ σl So K = {ρ 120 , σl} generates D 3

ιn^ = ι ∀n ∈ Z ρ^0120 = ι, ρ^1120 = ρ 120 , ρ^2120 = ρ 240 , ρ^3120 = ι, · · · ρ^0240 = ι, ρ^1240 = ρ 240 , ρ^2240 = ρ 120 , ρ^3240 = ι, · · · σ^0 l = ι, σ^1 l = σl, σ^2 l = ι, · · · σ^0 m = ι, σ m^1 = σm, σ m^2 = ι, · · · σ^0 n = ι, σ^1 n = σn, σ n^2 = ι, · · · Thus, no single element of D 3 generates the entire group. 17

4.4 The Frieze Groups

• 4.4. THE FRIEZE GROUPS
  • 4. Top row (left to right): D 2 , D 5 , C
    • Middle row (left to right): C 3 , not a rosette (D∞), D
    • Bottome row (left to right): C 6 , D
  • 5. Top row (left to right): C 6 , D 8 , C
    • Middle row (left to right): D 8 , C 14 , C
    • Bottome row (left to right): C 8 , D 6 , C
  • 2. F
  • 4. a) F
       • b) F
         • c) F
       • d) F
         • e) F
  • 5. top row left: F 4 , top row right: F
    • second row left: F 5 , second row right: F
    • third row left: F 1 , third row right: F
    • fourth row left: F 5 , fourth row right: F
    • fifth row left: F 4 , fifth row right: F
    • sixth row left: F 7 , sixth row right: F
    • bottom row left: F 6 , bottom row right: F
  • 6. a) F
       • b) F
       • d) F c) not a frieze (no minimum length translation)

20 CHAPTER 4. SYMMETRY

e) F 3 f) F 2 g) F 6 h) F 7 i) F 2 j) F 2 k) F 6 l) F 3 m) F 1 n) F 4 o) F 1 p) F 4 q) F 2 r) F 5

4.5 The Wallpaper Groups

1. pmm
2. left: p 4 m, right: pm
3. a) p 4 b) pg c) p 2 d) pmg e) pg f) p 4 m