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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'
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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' at T/2 cot ( /2) from AA ' A T = B B A T = 1
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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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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
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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 2 2 B 3 3 A 4 A B B' 2 3 4 1 A A A
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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
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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
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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' B screw axis
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9 Algebra of operations Intersecting mirrors: m m = A Parallel mirrors: m T = m
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10 Algebra of operations Inversion: i T = A m T = A m = i
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11 Algebra of operations Inversion: i T = A m T = A m = i Glides: Define: m = m t
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12 Algebra of operations Inversion: i T = A m T = A m = i Glides: Define: m = m t Glide symbols axialdiagonaldiamond 1/8 ca b dn
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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
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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
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15 Algebra of operations Glides: m T = m T T = m T T = m T = m
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16 Algebra of operations Glides: m T = m T T = m T T = m T = m
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17 Algebra of operations 2 screw & glide: A m = A t m = A m t' t
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18 Algebra of operations 2 screw & glide: A m = A t m = A m t' = i t' = i t = i t
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19 Algebra of operations 2 screw & glide: A m = A t m = A m t' = i t' = i t = i t t 1 t 1 i 1 i 2
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20 Algebra of operations 2 screw & glide: A m = A t m = A m t' = i t' = i t = i t t 1 t 1 i 1 i 2
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21 Algebra of operations 2 screw & glide: A m = A t m = A m t' = i t' = i t = i t t 1 t 1 i 1 i 2
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22 Algebra of operations 2 screw & glide: A m = A t m = A m t' = i t' = i t = i t t 1 t 1 i 1 i 2
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23 Algebra of operations 2 screw & glide: A m = A t m = A m t' = i t' = i t = i t t 1 t 1 i 1 i 2
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24 Algebra of operations Two 2-fold axes at angle apart: A B = C' (from Euler calc.) 2 2 C
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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 C' C" B A t 2 t 1 t 1 t 2 +
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26 Algebra of operations C' C" B A t 2 t 1 t 1 t 2 + 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
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27 Algebra of operations C' C" B A t 2 t 1 t 2 + 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 1 Not a screw axis - no t II C
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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" B A t 2 t 1 t 1 t 2 + If A, B do not intersect, but are separated in direction along C, C" has screw axis translation component
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