1 Algebra of operations m 1: tation about A thru angle followed by translation T to axis rotation thru same angle about B on bisector of AA' at 2 cot (/2) from AA'
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'
Jan 18, 2018
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' at T/2 cot ( /2) from AA' A T = B B A T = 1 B A = T = T'
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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' atT/2 cot (/2) from AA'
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
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'
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
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
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
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
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
9
Algebra of operations
Intersecting mirrors:
m m = A
Parallel mirrors:
m T = m
10
Algebra of operations
Inversion:
i T = A m T
= A m = i
11
Algebra of operations
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
Glide symbols
axial diagonal diamond
1/8
ca
b
dn
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
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
15
Algebra of operations
Glides:
m T = m T T
= m T T
= m T
= m
16
Algebra of operations
Glides:
m T = m T T
= m T T
= m T
= m
17
Algebra of operations
2 screw & glide:
A m = A t m
= A m t'
t
18
Algebra of operations
2 screw & glide:
A m = A t m
= A m t'
= i t'
= i t = i
t
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
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
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
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
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
24
Algebra of operations
Two 2-fold axes at angle apart:
A B = C' (from Euler calc.)
22
C
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 +
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
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
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
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