1 euler construction spherical c 1 c 2 c 3 formed by pts where axes intersect sphere cos w = cos u...
TRANSCRIPT
1
Euler constructionEuler construction
Spherical C1C2C3 formed bypts where axes intersect sphere
cos w = cos u cos v + sin u sin v cos W
u = 180 - Uv = 180 - Vw = 180 -W
cos w =
Spherical C1C2C3 formed bypts where axes intersect sphere
cos w = cos u cos v + sin u sin v cos W
u = 180 - Uv = 180 - Vw = 180 -W
cos w = cos W + cos U cos Vcos W + cos U cos V
sin U sin Vsin U sin V
C1C1
C2C2
C3C3
V = b /2V = b /2
W = g /2W = g /2
vv
ww
uu
U = a /2U = a /2
2
Euler constructionEuler construction
cos w =
1
2
3
4
6
cos w =
1
2
3
4
6
cos W + cos U cos Vcos W + cos U cos V
sin U sin Vsin U sin V
, , or , , or U, V, or W cos U, V, W sin U, V, WU, V, or W cos U, V, W sin U, V, W
360 180 -1 0
180 90 0 1
181 60 1/2 3/2
90 45 1/ 2 1/ 2
60 30 3/2 1/2
360 180 -1 0
180 90 0 1
181 60 1/2 3/2
90 45 1/ 2 1/ 2
60 30 3/2 1/2
3
Euler constructionEuler construction
cos w =cos w =cos W + cos U cos Vcos W + cos U cos V
sin U sin Vsin U sin V
222 0 90222 0 90
cos w wcos w w
4
Euler constructionEuler construction
cos w =cos w =cos W + cos U cos Vcos W + cos U cos V
sin U sin Vsin U sin V
222 0 90
223 1/2 60
222 0 90
223 1/2 60
cos w wcos w w
5
Euler constructionEuler construction
cos w =cos w =cos W + cos U cos Vcos W + cos U cos V
sin U sin Vsin U sin V
222 0 90
223 1/2 60
224 1/ 2 45
222 0 90
223 1/2 60
224 1/ 2 45
cos w wcos w w
6
Euler constructionEuler construction
cos w =cos w =cos W + cos U cos Vcos W + cos U cos V
sin U sin Vsin U sin V
222 0 90
223 1/2 60
224 1/ 2 45
226 3/2 30
222 0 90
223 1/2 60
224 1/ 2 45
226 3/2 30
cos w wcos w w
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Point groups so farPoint groups so far
Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6
8
Point groups so farPoint groups so far
Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6
What about combining proper & improper along same axis?
P I I
Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6
What about combining proper & improper along same axis?
P I I
9
Point groups so farPoint groups so far
Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6
What about combining proper & improper along same axis?
P I I
1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m
Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6
What about combining proper & improper along same axis?
P I I
1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m
10
Point groups so farPoint groups so far
Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6
What about combining proper & improper along same axis?
P I I
1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m
Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6
What about combining proper & improper along same axis?
P I I
1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m
11
Point groups so farPoint groups so far
Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6
What about combining proper & improper along same axis?
P I I
1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m
Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6
What about combining proper & improper along same axis?
P I I
1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m
12
Point groups so farPoint groups so far
Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6
What about combining proper & improper along same axis?
P I I
1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m
Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6
What about combining proper & improper along same axis?
P I I
1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m
13
Point groups so farPoint groups so far
Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6
What about combining proper & improper along same axis?
P I I
1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m
13 monaxial groups …….. 19 to go
Monoaxial groups - 1, 2, 3, 4, 6, i, m, 3, 4, 6
What about combining proper & improper along same axis?
P I I
1/1 = 1 2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m
13 monaxial groups …….. 19 to go
14
Point groups so farPoint groups so far
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
222 = 2mm
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
222 = 2mm
15
Point groups so farPoint groups so far
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
222 = 2mm
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
222 = 2mm
16
Point groups so farPoint groups so far
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
222 = 2mm
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
222 = 2mm
17
Point groups so farPoint groups so far
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
223 = 3mm
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
223 = 3mm
18
Point groups so farPoint groups so far
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
223 = 3mm
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
223 = 3mm
19
Point groups so farPoint groups so far
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
223 = 3mm
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
223 = 3mm
20
Point groups so farPoint groups so far
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
224 = 4mm
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
224 = 4mm
21
Point groups so farPoint groups so far
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
224 = 42m
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
224 = 42m
22
Point groups so farPoint groups so far
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
224 = 42m
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
224 = 42m
23
Point groups so farPoint groups so far
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
226 = 6mm
226 = 62m
13 to go
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
226 = 6mm
226 = 62m
13 to go
24
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
226 = 6mm
226 = 62m
332 = 3 = 2/m 3 = 332
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
226 = 6mm
226 = 62m
332 = 3 = 2/m 3 = 332
More point groupsMore point groups
3 2 3 2
3 2 3 2
25
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
226 = 6mm
226 = 62m
332 = 3 = 2/m 3 = 332
432 = 43m
432 = 432 = 3
Polyaxial groups - 222, 223, 224, 226, 233, 234 …… 13 to go
How? Combine P & I - remember P I I & I I P
226 = 6mm
226 = 62m
332 = 3 = 2/m 3 = 332
432 = 43m
432 = 432 = 3
More point groupsMore point groups
3 2 3 2
3 2 3 2
4 2 4 2 m mm m
26
More point groupsMore point groups
2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m
2 2 2 2 2 2 3 2 2 3 2 2 2
4 2 2 4 2 2 6 2 2 6 2 2
2/2 = 2/m 3/3 = 3 4/4 = 4/m 6/6 = 6/m
2 2 2 2 2 2 3 2 2 3 2 2 2
4 2 2 4 2 2 6 2 2 6 2 2
==2 2 22 2 2 mmmmmm
==2 2 22 2 2 mmmmmm mm
33==
==4 2 24 2 2 mmmmmm
==6 2 26 2 2 mmmmmm
27
Triclinic 1 1
Monoclinic 2 m 2/m
Orthorhmbic 222 2mm
Tetragonal 4 4 4/m 422 4mm 42m
Cubic 23 432 43m 3
Trigonal 3 3 32 3m 3
Hexagonal 6 6 6/m 622 6mm 62m
Triclinic 1 1
Monoclinic 2 m 2/m
Orthorhmbic 222 2mm
Tetragonal 4 4 4/m 422 4mm 42m
Cubic 23 432 43m 3
Trigonal 3 3 32 3m 3
Hexagonal 6 6 6/m 622 6mm 62m
SummarySummary
mmmmmm2 2 22 2 2
4 2 24 2 2mmmmmm
mm22
334 2 4 2 m mm m
mm22
6 2 26 2 2mmmmmm