1 / 28

Algebra of Operations: Theorems for Transformations

Learn about the properties of rotations, translations, and transformations in algebraic operations. Understand how rotations followed by translations can be simplified. Delve into the concepts of screw motions and glides in mathematical operations.

youngrichard
Download Presentation

Algebra of Operations: Theorems for Transformations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related