1 Algebra of operations Thm 1: Rotation about A thru angle followed by translation T to axis =...
-
Upload
tobias-mccormick -
Category
Documents
-
view
221 -
download
0
description
Transcript of 1 Algebra of operations Thm 1: Rotation about A thru angle followed by translation T to axis =...
![Page 1: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/1.jpg)
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' atT/2 cot (/2) from AA'
![Page 2: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/2.jpg)
2
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' atT/2 cot (/2) from AA'
A T = B
B A T = 1
![Page 3: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/3.jpg)
3
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' atT/2 cot (/2) from AA'
A T = B
B A T = 1
B A = T = T'
![Page 4: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/4.jpg)
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' atT/2 cot (/2) from AA'
A T = B
B A T = 1
B A = T = T'
A A'
B
![Page 5: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/5.jpg)
5
Algebra of operations
Meaning of A B A:
The operation B as transformed by A
2 B 3
(2B3) A (1B'4)
1 B' 4
1 A 22 B 33 A 4
A
B B'
2
34
1
A
A
A
![Page 6: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/6.jpg)
6
Algebra of operations
Thm 2:
Rotation about A thru angle followed by translation T to axis= screw. Rotation & translation operations permutable
A T = T A = A
![Page 7: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/7.jpg)
7
Algebra of operations
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
![Page 8: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/8.jpg)
8
Algebra of operations
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
A A'
Bscrew axis
![Page 9: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/9.jpg)
9
Algebra of operations
Intersecting mirrors:
m m = A
Parallel mirrors:
m T = m
![Page 10: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/10.jpg)
10
Algebra of operations
Inversion:
i T = A m T
= A m = i
![Page 11: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/11.jpg)
11
Algebra of operations
Inversion:
i T = A m T
= A m = i
Glides:
Define: m = m t
![Page 12: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/12.jpg)
12
Algebra of operations
Inversion:
i T = A m T
= A m = i
Glides:
Define: m = m t
Glide symbols
axial diagonal diamond
1/8
ca
b
dn
![Page 13: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/13.jpg)
13
Algebra of operations
Inversion:
i T = A m T
= A m = i
Glides:
Define: m = m
m T = m T T
m T = m
![Page 14: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/14.jpg)
14
Algebra of operations
Inversion:
i T = A m T
= A m = i
Glides:
Define: m = m
m T = m T T
m T = m
![Page 15: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/15.jpg)
15
Algebra of operations
Glides:
m T = m T T
= m T T
= m T
= m
![Page 16: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/16.jpg)
16
Algebra of operations
Glides:
m T = m T T
= m T T
= m T
= m
![Page 17: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/17.jpg)
17
Algebra of operations
2 screw & glide:
A m = A t m
= A m t'
t
![Page 18: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/18.jpg)
18
Algebra of operations
2 screw & glide:
A m = A t m
= A m t'
= i t'
= i t = i
t
![Page 19: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/19.jpg)
19
Algebra of operations
2 screw & glide:
A m = A t m
= A m t'
= i t'
= i t = i
t
t1
t1-1
i1
i2
![Page 20: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/20.jpg)
20
Algebra of operations
2 screw & glide:
A m = A t m
= A m t'
= i t'
= i t = i
t
t1
t1-1
i1
i2
![Page 21: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/21.jpg)
21
Algebra of operations
2 screw & glide:
A m = A t m
= A m t'
= i t'
= i t = i
t
t1
t1-1
i1
i2
![Page 22: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/22.jpg)
22
Algebra of operations
2 screw & glide:
A m = A t m
= A m t'
= i t'
= i t = i
t
t1
t1-1
i1
i2
![Page 23: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/23.jpg)
23
Algebra of operations
2 screw & glide:
A m = A t m
= A m t'
= i t'
= i t = i
t
t1
t1-1
i1
i2
![Page 24: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/24.jpg)
24
Algebra of operations
Two 2-fold axes at angle apart:
A B = C' (from Euler calc.)
22
C
![Page 25: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/25.jpg)
25
Algebra of operations
C
In general:
A B = C'
If screw axes:
A B = t A B t = t C' t
t C t = C'
t
t
C'
C"
BA
t2 t1
t1 t2 +
![Page 26: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/26.jpg)
26
Algebra of operations
C'
C"
BA
t2 t1
t1 t2 +
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"
t
C
t
![Page 27: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/27.jpg)
27
Algebra of operations
C'
C"
BA
t2
t1 t2 +
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"
t
C
t
t1
Not a screw axis - no t II C
![Page 28: 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'](https://reader036.fdocuments.us/reader036/viewer/2022062413/5a4d1b477f8b9ab0599a3ad2/html5/thumbnails/28.jpg)
28
Algebra of operations
If screw axes:
A B = t A B t = t C' t
t C t = C'
t C' t = C t t
= C"
t
C
t
C'
C"
BA
t2 t1
t1 t2 +
If A, B do not intersect, but areseparated in direction along C,C" has screw axis translation component