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Transformation Operations in Algebra: Rotation, Translation & Inversion

Explore properties of rotation, translation, screw operations, and glides in algebra. Learn about intersecting and parallel mirrors, inversion, and screw axes. Discover the mathematics behind these transformations.

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Transformation Operations in Algebra: Rotation, Translation & Inversion

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  1. Algebra of operations Thm 1: Rotation about A thru angle  followed by translation T to axis = rotation thru same angle  about B on bisector of AA' at T/2 cot (/2) from AA'

  2. Algebra of operations a a -1 a a Thm 1: Rotation about A thru angle afollowed by translation T to axis = rotation thru same angleaabout B on bisector of AA' at T/2 cot (a/2) from AA' A T = B B A T = 1

  3. Algebra of operations a a -1 a a -1 -1 a a Thm 1: Rotation about A thru angle  followed by translation T to axis = rotation thru same angle  about B on bisector of AA' at T/2 cot (/2) from AA' A T = B B A T = 1 B A = T = T'

  4. Algebra of operations Thm 1: Rotation about A thru angle  followed by translation T to axis = rotation thru same angle  about B on bisector of AA' at T/2 cot (/2) from AA' A T = B B A T = 1 B A = T = T' a a B -1 a a -1 -1 a a A A'

  5. Algebra of operations -1 -1 Meaning of A B A: The operation B as transformed by A 2 B 3 (2B3) A (1B'4) 1 B' 4 1 A 2 2 B 3 3 A 4 A 2 -1 1 A B' A B 3 4 A

  6. Algebra of operations a,T a a Thm 2: Rotation about A thru angle  followed by translation T to axis = screw. Rotation & translation operations permutable A T = T A = A

  7. Algebra of operations a,T a a a a a,T a Thm 2: Rotation about A thru angle  followed by translation T to axis = screw. Rotation & translation operations permutable A T = T A = A Then: T = T + T A T = A T T = B T = B

  8. Algebra of operations a,T a a a a a,T a Thm 2: Rotation about A thru angle  followed by translation T to axis = screw. Rotation & translation operations permutable A T = T A = A Then: T = T + T A T = A T T = B T = B B screw axis A A'

  9. Algebra of operations 2m 1 2 2 1 Intersecting mirrors: m m = A Parallel mirrors: m T = m

  10. Algebra of operations Inversion: i T = A m T = A m = i p 1, 1 2 p 2,

  11. Algebra of operations p 1, 1 p 2, 2 t Inversion: i T = A m T = A m = i Glides: Define: m = m t

  12. Algebra of operations Inversion: i T = A m T = A m = i Glides: Define: m = m t p 1, 1 p 2 2, Glide symbols t a c axial diagonal diamond 1/8 b n d

  13. Algebra of operations • Inversion: • i T = A m T • = A m = i • Glides: • Define: m = m t • m T = m T T • m T = m 1 1, p p 2, 2 t 1 1 2 2, t

  14. Algebra of operations • Inversion: • i T = A m T • = A m = i • Glides: • Define: m = m t • m T = m T T • m T = m 1 1, p p 2, 2 t 1 1 2 2, t

  15. Algebra of operations • Glides: • m T = m T T • = m T T • = m T • = m t 1 1, t t 1 t 2 2, t +

  16. Algebra of operations • Glides: • m T = m T T • = m T T • = m T • = m t 1 1, t t 1 t 2 2, t +

  17. Algebra of operations • 2 screw & glide: • A m = A t m • = A m t' 1 t t p p, t t p

  18. Algebra of operations • 2 screw & glide: • A m = A t m • = A m t' • = i t' • = i t = i 1 t t p, t p t p t 1 t -1 2 1

  19. Algebra of operations • 2 screw & glide: • A m = A t m • = A m t' • = i t' • = i t = i 1 t t p, t p t p t 1 -1 t 2 1 -1 t i 1 2 t i 1 1 t

  20. Algebra of operations • 2 screw & glide: • A m = A t m • = A m t' • = i t' • = i t = i 1 t t p p, t t p t 1 -1 t 2 1 -1 t i 1 2 t i 1 1 t

  21. Algebra of operations • 2 screw & glide: • A m = A t m • = A m t' • = i t' • = i t = i 1 t t p p, t t p t 1 -1 t 2 1 -1 t i 1 2 t i 1 1 t

  22. Algebra of operations • 2 screw & glide: • A m = A t m • = A m t' • = i t' • = i t = i 1 t t p p, t t p t 1 -1 t 2 1 -1 t i 1 2 t i 1 1 t

  23. Algebra of operations • 2 screw & glide: • A m = A t m • = A m t' • = i t' • = i t = i 1 t p t p, t t p t 1 -1 t 2 1 -1 t i 1 2 t i 1 1 t

  24. Algebra of operations 2m p p • Two 2-fold axes at angle  apart: • A B = C' (from Euler calc.) C 2m 2m m 2 2

  25. Algebra of operations • In general: • A B = C' • If screw axes: • A B = t A B t = t C' t • t C t = C' g a b a g a,t b 2 2 a,t 1 1 1 2 -1 g g t t 2 2 t t 1 1 C" C + m 2m C' B A

  26. Algebra of operations • In general: • A B = C' • If screw axes: • A B = t A B t = t C' t • t C t = C' • t C' t = C t t • = C" g b a a,t g a,t b 2 1 2 a 1 1 2 -1 g g 1 g g 2 2 1 C t t 2 2 t t 1 1 C" + m 2m C' B A

  27. Algebra of operations • In general: • A B = C' • If screw axes: • A B = t A B t = t C' t • t C t = C' • t C' t = C t t • = C" b a g a,t a,t a b 2 g 1 1 2 1 2 -1 g g g 2 g 2 1 1 C t t 2 2 Not a screw axis - no t II C t t 1 1 C" + m 2m C' B A

  28. Algebra of operations t t 2 2 t t 1 1 • If screw axes: • A B = t A B t = t C' t • t C t = C' • t C' t = C t t • = C" g 2 a,t a,t b a 1 2 1 2 1 -1 g g g 1 g 1 2 2 C" If A, B do not intersect, but are separated in direction along C, C" has screw axis translation component C + m 2m C' B A

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