3-D Rotational Matrix
48 3-D Rotational Matrices |
Remarks | ||||
1 | x2(α)y2(β)z2(γ) | cosαcosγ +sinαsinβsinγ | -sinαcosβ | -cosαsinγ+sinαsinβcosγ | 01=(10)T |
sinαcosγ-cosαsinβsinγ | cosαcosβ | -sinαsinγ-cosαsinβcosγ | |||
cosβsinγ | sinβ | cosβcosγ | |||
2 | x1(α)y1(β)z1(γ) | cosαcosγ -sinαsinβsinγ | sinαcosβ | cosαsinγ+sinαsinβcosγ | 02=(09)T |
-sinαcosγ-cosαsinβsinγ | cosαcosβ | -sinαsinγ+cosαsinβcosγ | |||
-cosβsinγ | -sinβ | cosβcosγ | |||
3 | y2(β)z2(γ)x2(α) | cosαcosγ | -sinαcosγ | -sinγ | 03=(08)T |
-cosαsinβsinγ+sinαcosβ | sinαsinβsinγ+cosαcosβ | -sinβcosγ | |||
cosαcosβsinγ+sinαsinβ | -sinαcosβsinγ+cosαsinβ | cosβcosγ | |||
4 | y1z1x1 | cosαcosγ | sinαcosγ | sinγ | 04=(07)T |
-cosαsinβsinγ-sinαcosβ | -sinαsinβsinγ+cosαcosβ | sinβcosγ | |||
-cosαcosβsinγ+sinαsinβ | -sinαcosβsinγ-cosαsinβ | cosβcosγ | |||
5 | z2x2y2 | cosαcosγ | -sinαcosβcosγ-sinβsinγ | sinαsinβcosγ-cosβsinγ | 05=(12)T |
sinα | cosαcosβ | -cosαsinβ | |||
cosαsinγ | -sinαcosβsinγ+sinβcosγ | sinαsinβsinγ+cosβcosγ | |||
6 | z1x1y1 | cosαcosγ | sinαcosβcosγ-sinβsinγ | sinαsinβcosγ+cosβsinγ | 06=(11)T |
-sinα | cosαcosβ | cosαsinβ | |||
-cosαsinγ | -sinαcosβsinγ-sinβcosγ | -sinαsinβsinγ+cosβcosγ | |||
7 | x2z2y2 | cosαcosγ | -sinαcosβ-cosαsinβsinγ | sinαsinβ-cosαcosβsinγ | x2z2y2=(y1z1x1)T |
sinαcosγ | cosαcosβ-sinαsinβsinγ | -cosαsinβ-sinαcosβsinγ | 07=(04)T | ||
sinγ | sinβcosγ | cosβcosγ
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8 | x1z1y1 | cosαcosγ | sinαcosβ-cosαsinβsinγ | sinαsinβ+cosαcosβsinγ | x1z1y1=(y2z2x2)T |
-sinαcosγ | cosαcosβ+sinαsinβsinγ | cosαsinβ-sinαcosβsinγ | 08=(03)T | ||
-sinγ | -sinβcosγ | cosβcosγ | |||
9 | z2y2x2 | cosαcosγ-sinαsinβsinγ | -sinαcosγ-cosαsinβsinγ | -cosβsinγ | z2y2x2=(x1y1z1)T |
sinαcosβ | cosαcosβ | -sinβ | 09=(02)T | ||
cosαsinγ+sinαsinβcosγ | -sinαsinγ+cosαsinβcosγ | cosβcosγ | |||
10 | z1y1x1 | cosαcosγ+sinαsinβsinγ | sinαcosγ-cosαsinβsinγ | cosβsinγ | z1y1x1=(x2y2z2)T |
-sinαcosβ | cosαcosβ | sinβ | 10=(01)T | ||
-cosαsinγ+sinαsinβcosγ | -sinαsinγ-cosαsinβcosγ | cosβcosγ | |||
11 | y2x2z2 | cosαcosγ | -sinα | -cosαsinγ | y2x2z2=(z1x1y1)T |
sinαcosβcosγ-sinβsinγ | cosαcosβ | -sinαcosβsinγ-sinβcosγ | 11=(06)T | ||
sinαsinβcosγ+cosβsinγ | cosαsinβ | -sinαsinβsinγ+cosβcosγ | |||
12 | y1x1z1 | cosαcosγ | sinα | cosαsinγ | y1x1z1=(z2x2y2)T |
-sinαcosβcosγ-sinβsinγ | cosαcosβ | -sinαcosβsinγ+sinβcosγ | 12=(05)T | ||
sinαsinβcosγ-cosβsinγ | -cosαsinβ | sinαsinβsinγ+cosβcosγ | |||
13 | x1(α)y1(β)z2(γ) | cosαcosγ+sinαsinβsinγ | sinαcosβ | -cosαsinγ+sinαsinβcosγ | 13=(16)T |
-sinαcosγ+cosαsinβsinγ | cosαcosβ | sinαsinγ+cosαsinβcosγ | |||
cosβsinγ | -sinβ | cosβcosγ | |||
14 | x2(α)y2(β)z1(γ) | cosαcosγ-sinαsinβsinγ | -sinαcosβ | cosαsinγ+sinαsinβcosγ | 14=(15)T |
sinαcosγ+cosαsinβsinγ | cosαcosβ | sinαsinγ-cosαsinβcosγ | |||
-cosβsinγ | sinβ | cosβcosγ | |||
15 | z2y1x1 | cosαcosγ-sinαsinβsinγ | sinαcosγ+cosαsinβsinγ | -cosβsinγ | z2y1x1=(x2y2z1)T |
-sinαcosβ | cosαcosβ | sinβ | 15=(14)T | ||
cosαsinγ+sinαsinβcosγ | sinαsinγ-cosαsinβcosγ | cosβcosγ | |||
16 | z1y2x2 | cosαcosγ+sinαsinβsinγ | -sinαcosγ+cosαsinβsinγ | cosβsinγ | z1y2x2=(x1y1z2)T |
sinαcosβ | cosαcosβ | -sinβ | 16=(13)T | ||
-cosαsinγ+sinαsinβcosγ | sinαsinγ+cosαsinβcosγ | cosβcosγ | |||
17 | z2x1y1 | cosαcosγ | sinαcosβcosγ+sinβsinγ | -cosβsinγ+sinαsinβcosγ | 17=(20)T |
-sinα | cosαcosβ | cosαsinβ | |||
cosαsinγ | sinαcosβsinγ-sinβcosγ | sinαsinβsinγ+cosβcosγ | |||
18 | z1x2y2 | cosαcosγ | -sinαcosβcosγ+sinβsinγ | cosβsinγ+sinαsinβcosγ | 18=(19)T |
sinα | cosαcosβ | -cosαsinβ | |||
-cosαsinγ | sinαcosβsinγ+sinβcosγ | -sinαsinβsinγ+cosβcosγ | |||
19 | y1x1z2 | cosαcosγ | sinα | -cosαsinγ | y1x1z2=(z1x2y2)T |
-sinαcosβcosγ+sinβsinγ | cosαcosβ | sinαcosβsinγ+sinβcosγ | 19=(18)T | ||
cosβsinγ+sinαsinβcosγ | -cosαsinβ | -sinαsinβsinγ+cosβcosγ | |||
20. | y2x2z1 | cosαcosγ | -sinα | cosαsinγ | y2x2z1=(z2x1y1)T |
sinαcosβcosγ+sinβsinγ | cosαcosβ | sinαcosβsinγ-sinβcosγ | 20=(17)T | ||
-cosβsinγ+sinαsinβcosγ | cosαsinβ | sinαsinβsinγ+cosβcosγ | |||
21 | z1x1y2 | cosαcosγ | sinαcosβcosγ+sinβsinγ | cosβsinγ-sinαsinβcosγ | 21=(23)T |
-sinα | cosαcosβ | -cosαsinβ | |||
-cosαsinγ | sinβcosγ-sinαcosβsinγ | sinαsinβsinγ+cosβcosγ | |||
22 | z2x2y1 | cosαcosγ | -sinαcosβcosγ+sinβsinγ | -sinαsinβcosγ-cosβsinγ | 22=(24)T |
sinα | cosαcosβ | cosαsinβ | |||
cosαsinγ | -sinβcosγ-sinαcosβsinγ | -sinαsinβsinγ+cosβcosγ | |||
23 | y1x2z2 | cosαcosγ | -sinα | -cosαsinγ | y1x2z2=(z1x1y2)T |
sinαcosβcosγ+sinβsinγ | cosαcosβ | sinβcosγ-sinαcosβsinγ | 23=(21)T | ||
cosβsinγ-sinαsinβcosγ | -cosαsinβ | sinαsinβsinγ+cosβcosγ | |||
24 | y2x1z1 | cosαcosγ | sinα | cosαsinγ | y2x1z1=(z2x2y1)T |
-sinαcosβcosγ+sinβsinγ | cosαcosβ | -sinβcosγ-sinαcosβsinγ | 24=(22)T | ||
-sinαsinβcosγ-cosβsinγ | cosαsinβ | -sinαsinβsinγ+cosβcosγ | |||
25 | x1(α)y2(β)z2(γ) | cosαcosγ-sinαsinβsinγ | sinαcosβ | -cosαsinγ-sinαsinβcosγ | 25=(26)T |
-sinαcosγ-cosαsinβsinγ | cosαcosβ | sinαsinγ-cosαsinβcosγ | |||
cosβsinγ | sinβ | cosβcosγ | |||
26 | z1y1x2 | cosαcosγ-sinαsinβsinγ | -sinαcosγ-cosαsinβsinγ | cosβsinγ | z1y1x2=(x1y2z2)T |
sinαcosβ | cosαcosβ | sinβ | 26=(25)T | ||
-cosαsinγ-sinαsinβcosγ | sinαsinγ-cosαsinβcosγ | cosβcosγ | |||
27 | z2y2x1 | cosαcosγ+sinαsinβsinγ | sinαcosγ-cosαsinβsinγ | -cosβsinγ | 27=(28)T |
-sinαcosβ | cosαcosβ | -sinβ | |||
cosαsinγ-sinαsinβcosγ | sinαsinγ+cosαsinβcosγ | cosβcosγ | |||
28 | x2y1z1 | cosαcosγ+sinαsinβsinγ | -sinαcosβ | cosαsinγ-sinαsinβcosγ | x2y1z1=(z2y2x1)T |
sinαcosγ-cosαsinβsinγ | cosαcosβ | sinαsinγ+cosαsinβcosγ | 28=(27)T | ||
-cosβsinγ | -sinβ | cosβcosγ | |||
29 | y1z1x2 | cosαcosγ | -sinαcosγ | sinγ | 29=(30)T |
sinαcosβ-cosαsinβsinγ | cosαcosβ+sinαsinβsinγ | sinβcosγ | |||
-cosαcosβsinγ-sinαsinβ | sinαcosβsinγ-cosαsinβ | cosβcosγ | |||
30 | x1z2y2 | cosαcosγ | sinαcosβ-cosαsinβsinγ | -cosαcosβsinγ-sinαsinβ | x2z2y2=(y1z1x2)T |
-sinαcosγ | cosαcosβ+sinαsinβsinγ | sinαcosβsinγ-cosαsinβ | 30=(29)T | ||
sinγ | sinβcosγ | cosβcosγ | |||
31 | x2z1y1 | cosαcosγ | -sinαcosβ-cosαsinβsinγ | cosαcosβsinγ-sinαsinβ | 31=(32)T |
sinαcosγ | cosαcosβ-sinαsinβsinγ | sinαcosβsinγ+cosαsinβ | |||
-sinγ | -sinβcosγ | cosβcosγ | |||
32 | y2z2x1 | cosαcosγ | sinαcosγ | -sinγ | y2z2x1=(x2z1y1)T |
-sinαcosβ-cosαsinβsinγ | cosαcosβ-sinαsinβsinγ | -sinβcosγ | 32=(31)T | ||
cosαcosβsinγ-sinαsinβ | sinαcosβsinγ+cosαsinβ | cosβcosγ | |||
33 | x1z1y2 | cosαcosγ | sinαcosβ+cosαsinβsinγ | cosαcosβsinγ-sinαsinβ | 33=(34)T |
-sinαcosγ | cosαcosβ-sinαsinβsinγ | -sinαcosβsinγ-cosαsinβ | |||
-sinγ | sinβcosγ | cosβcosγ | |||
34 | y1z2x2 | cosαcosγ | -sinαcosγ | -sinγ | y1z2x2=(x1z1y2)T |
sinαcosβ+cosαsinβsinγ | cosαcosβ-sinαsinβsinγ | sinβcosγ | 34=(33)T | ||
cosαcosβsinγ-sinαsinβ | -sinαcosβsinγ-cosαsinβ | cosβcosγ | |||
35 | y2z1x1 | cosαcosγ | sinαcosγ | sinγ | 35=(36)T |
cosαsinβsinγ-sinαcosβ | sinαsinβsinγ+cosαcosβ | -sinβcosγ | |||
-cosαcosβsinγ-sinαsinβ | -sinαcosβsinγ+cosαsinβ | cosβcosγ | |||
36 | x2z2y1 | cosαcosγ | cosαsinβsinγ-sinαcosβ | -cosαcosβsinγ-sinαsinβ | x2z2y1=(y2z1x1)T |
sinαcosγ | sinαsinβsinγ+cosαcosβ | -sinαcosβsinγ+cosαsinβ | 36=(35)T | ||
sinγ | -sinβcosγ | cosβcosγ | |||
37 | x1(α)y2(β)z1(γ) | cosαcosγ+sinαsinβsinγ | sinαcosβ | cosαsinγ-sinαsinβcosγ | 37=(38)T |
-sinαcosγ+cosαsinβsinγ | cosαcosβ | -sinαsinγ-cosαsinβcosγ | |||
-cosβsinγ | sinβ | cosβcosγ | |||
38 | z2y1x2 | cosαcosγ+sinαsinβsinγ | -sinαcosγ+cosαsinβsinγ | -cosβsinγ | z2y1x2=(x1y2z1)T |
sinαcosβ | cosαcosβ | sinβ | 38=(37)T | ||
cosαsinγ-sinαsinβcosγ | -sinαsinγ-cosαsinβcosγ | cosβcosγ | |||
39 | x2(α)y1(β)z2(γ) | cosαcosγ-sinαsinβsinγ | -sinαcosβ | -cosαsinγ-sinαsinβcosγ | 39=(40)T |
sinαcosγ+cosαsinβsinγ | cosαcosβ | -sinαsinγ+cosαsinβcosγ | |||
cosβsinγ | -sinβ | cosβcosγ | |||
40 | z1y2x1 | cosαcosγ-sinαsinβsinγ | sinαcosγ+cosαsinβsinγ | cosβsinγ | z1y2x1=(x2y1z2)T |
-sinαcosβ | cosαcosβ | -sinβ | 40=(39)T | ||
-cosαsinγ-sinαsinβcosγ | -sinαsinγ+cosαsinβcosγ | cosβcosγ | |||
41 | y1x2z1 | cosαcosγ | -sinα | cosαsinγ | 41=(42)T |
sinαcosβcosγ-sinβsinγ | cosαcosβ | sinαcosβsinγ+snβcosγ | |||
-sinαsinβcosγ-cosβsinγ | -cosαsinβ | -sinαsinβsinγ+cosβcosγ | |||
42 | z2x1y2 | cosαcosγ | sinαcosβcosγ-sinβsinγ | -sinαsinβcosγ-cosβsinγ | z2x1y2=(y1x2z1)T |
-sinα | cosαcosβ | -cosαsinβ | 42=(41)T | ||
cosαsinγ | sinαcosβsinγ+sinβcosγ | -sinαsinβsinγ+cosβcosγ | |||
43 | z1x2y1 | cosαcosγ | -sinαcosβcosγ-sinβsinγ | -sinαsinβcosγ+cosβsinγ | 43=(44)T |
sinα | cosαcosβ | cosαsinβ | |||
-cosαsinγ | sinαcosβsinγ-sinβcosγ | sinαsinβsinγ+cosβcosγ | |||
44 | y2x1z2 | cosαcosγ | sinα | -cosαsinγ | y2x1z2=(z12x2y1)T |
-sinαcosβcosγ-sinβsinγ | cosαcosβ | sinαcosβsinγ-sinβcosγ | 44=(43)T | ||
-sinαsinβcosγ+cosβsinγ | cosαsinβ | sinαsinβsinγ+cosβcosγ | |||
45 | y1z2x1 | cosαcosγ | sinαcosγ | -sinγ | 45=(46)T |
cosαsinβsinγ-sinαcosβ | sinαsinβsinγ+cosαcosβ | sinβcosγ | |||
cosαcosβsinγ+sinαsinβ | sinαcosβsinγ-cosαsinβ | cosβcosγ | |||
46 | x2z1y2 | cosαcosγ | cosαsinβsinγ-sinαcosβ | cosαcosβsinγ+sinαsinβ | x2z1y2=(y1z2x1)T |
sinαcosγ | sinαsinβsinγ+cosαcosβ | sinαcosβsinγ-cosαsinβ | 46=(45)T | ||
-sinγ | sinβcosγ | cosβcosγ | |||
47 | x1z2y1 | cosαcosγ | sinαcosβ+cosαsinβsinγ | -cosαcosβsinγ+sinαsinβ | 47=(48)T |
-sinαcosγ | -sinαsinβsinγ+cosαcosβ | sinαcosβsinγ+cosαsinβ | |||
sinγ | -sinβcosγ | cosβcosγ | |||
48 | y2z1x2 | cosαcosγ | -sinαcosγ | sinγ | y2z1x2=(x1z2y1)T |
sinαcosβ+cosαsinβsinγ | -sinαsinβsinγ+cosαcosβ | -sinβcosγ | 48=(47)T | ||
-cosαcosβsinγ+sinαsinβ | sinαcosβsinγ+cosαsinβ | cosβcosγ | |||
interchangeSin&Cosof matrices | |||||
1a | x'2y'2z'2 | sinαsinγ +cosαcosβcosγ | -cosαsinβ | -sinαcosγ+cosαcosβsinγ | 01a=(10a)T |
cosαsinγ-sinαcosβcosγ | sinαsinβ | -cosαcosγ-sinαcosβsinγ | |||
sinβcosγ | cosβ | sinβsinγ | |||
2a | x'1y'1z'1 | sinαsinγ -cosαcosβcosγ | cosαsinβ | sinαcosγ+cosαcosβsinγ | 02a=(09a)T |
-cosαsinγ-sinαcosβcosγ | sinαsinβ | -cosαcosγ+sinαcosβsinγ | |||
-sinβcosγ | -cosβ | sinβsinγ | |||
3a | y'2z'2x'2 | sinαsinγ | -cosαsinγ | -cosγ | 03a=(08a)T |
-sinαcosβcosγ+cosαsinβ | cosαcosβcosγ+sinαsinβ | -cosβsinγ | |||
sinαsinβcosγ+cosαcosβ | -cosαsinβcosγ+sinαcosβ | sinβsinγ | |||
4a | y'1z'1x'1 | sinαsinγ | cosαsinγ | cosγ | 04a=(07a)T |
-sinαcosβcosγ-cosαsinβ | -cosαcosβcosγ+sinαsinβ | cosβsinγ | |||
-sinαsinβcosγ+cosαcosβ | -cosαsinβcosγ-sinαcosβ | sinβsinγ | |||
5a | z'2x'2y'2 | sinαsinγ | -cosαsinβsinγ-cosβcosγ | cosαcosβsinγ-sinβcosγ | 05a=(12a)T |
cosα | sinαsinβ | -sinαcosβ | |||
sinαcosγ | -cosαsinβcosγ+cosβsinγ | cosαcosβcosγ+sinβsinγ | |||
6a | z'1x'1y'1 | sinαsinγ | cosαsinβsinγ-cosβcosγ | cosαcosβsinγ+sinβcosγ | 06a=(11a)T |
-cosα | sinαsinβ | sinαcosβ | |||
-sinαcosγ | -cosαsinβcosγ-cosβsinγ | -cosαcosβcosγ+sinβsinγ | |||
7a | x'2z'2y'2 | sinαsinγ | -cosαsinβ-sinαcosβcosγ | cosαcosβ-sinαsinβcosγ | x'2z'2y'2=(y'1z'1x'1)T |
cosαsinγ | sinαsinβ-cosαcosβcosγ | -sinαcosβ-cosαsinβcosγ | 07a=(04a)T | ||
cosγ | cosβsinγ | sinβsinγ
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8a | x'1z'1y'1 | sinαsinγ | cosαsinβ-sinαcosβcosγ | cosαcosβ+sinαsinβcosγ | x'1z'1y'1=(y'2z'2x'2)T |
-cosαsinγ | sinαsinβ+cosαcosβcosγ | sinαcosβ-cosαsinβcosγ | 08a=(03a)T | ||
-cosγ | -cosβsinγ | sinβsinγ | |||
9a | z'2y'2x'2 | sinαsinγ-cosαcosβcosγ | -cosαsinγ-sinαcosβcosγ | -sinβcosγ | z'2y'2x'2=(x'1y'1z'1)T |
cosαsinβ | sinαsinβ | -cosβ | 09a=(02a)T | ||
sinαcosγ+cosαcosβsinγ | -cosαcosγ+sinαcosβsinγ | sinβsinγ | |||
10a | z'1y'1x'1 | sinαsinγ+cosαcosβcosγ | cosαsinγ-sinαcosβcosγ | sinβcosγ | z'1y'1x'1=(x'2y'2z'2)T |
-cosαsinβ | sinαsinβ | cosβ | 10a=(01a)T | ||
-sinαcosγ+cosαcosβsinγ | -cosαcosγ-sinαcosβsinγ | sinβsinγ | |||
11a | y'2x'2z'2 | sinαsinγ | -cosα | -sinαcosγ | y'2x'2z'2=(z'1x'1y'1)T |
cosαsinβsinγ-cosβcosγ | sinαsinβ | -cosαsinβcosγ-cosβsinγ | 11a=(06a)T | ||
cosαcosβsinγ+sinβcosγ | sinαcosβ | -cosαcosβcosγ+sinβsinγ | |||
12a | y'1x'1z'1 | sinαsinγ | cosα | sinαcosγ | y'1x'1z1= (z'2x'2y'2)T |
-cosαsinβsinγ-cosβcosγ | sinαsinβ | -cosαsinβcosγ+cosβsinγ | 12a=(05a)T | ||
cosαcosβsinγ-sinβcosγ | -sinαcosβ | cosαcosβcosγ+sinβsinγ | |||
13a | x'1y'1z'2 | sinαsinγ+cosαcosβcosγ | cosαsinβ | -sinαcosγ+cosαcosβsinγ | 13a=(16a)T |
-cosαsinγ+sinαcosβcosγ | sinαsinβ | cosαcosγ+sinαcosβsinγ | |||
sinβcosγ | -cosβ | sinβsinγ | |||
14a | x'2y'2z'1 | sinαsinγ-cosαcosβcosγ | -cosαsinβ | sinαcosγ+cosαcosβsinγ | 14a=(15a)T |
cosαsinγ+sinαcosβcosγ | sinαsinβ | cosαcosγ-sinαcosβsinγ | |||
-sinβcosγ | cosβ | sinβsinγ | |||
15a | z'2y'1x'1 | sinαsinγ-cosαcosβcosγ | cosαsinγ+sinαcosβcosγ | -sinβcosγ | 15a=(14a)T |
-cosαsinβ | sinαsinβ | cosβ | |||
sinαcosγ+cosαcosβsinγ | cosαcosγ-sinαcosβsinγ | sinβsinγ | |||
16a | z'1y'2x'2 | sinαsinγ+cosαcosβcosγ | -cosαsinγ+sinαcosβcosγ | sinβcosγ | 16a=(13a)T |
cosαsinβ | sinαsinβ | -cosβ | |||
-sinαcosγ+cosαcosβsinγ | cosαcosγ+sinαcosβsinγ | sinβsinγ | |||
17a | z'2x'1y'1 | sinαsinγ | cosαsinβsinγ+cosβcosγ | -sinβcosγ+cosαcosβsinγ | 17a=(20a)T |
-cosα | sinαsinβ | sinαcosβ | |||
sinαcosγ | cosαsinβcosγ-cosβsinγ | cosαcosβcosγ+sinβsinγ | |||
18a | z'1x'2y'2 | sinαsinγ | -cosαsinβsinγ+cosβcosγ | sinβcosγ+cosαcosβsinγ | 18a=(19a)T |
cosα | sinαsinβ | -sinαcosβ | |||
-sinαcosγ | cosαsinβcosγ+cosβsinγ | -cosαcosβcosγ+sinβsinγ | |||
19a | y'1x'1z'2 | sinαsinγ | cosα | -sinαcosγ | y'1x'1z'2=(z'1x'2y'2)T |
-cosαsinβsinγ+cosβcosγ | sinαsinβ | cosαsinβcosγ+cosβsinγ | 19a=(18a)T | ||
sinβcosγ+cosαcosβsinγ | -sinαcosβ | -cosαcosβcosγ+sinβsinγ | |||
20a | y'2x'2z'1 | sinαsinγ | -cosα | sinαcosγ | 20a=(17a)T |
cosαsinβsinγ+cosβcosγ | sinαsinβ | cosαsinβcosγ-cosβsinγ | |||
-sinβcosγ+cosαcosβsinγ | sinαcosβ | cosαcosβcosγ+sinβsinγ | |||
21a | z'1x'1y'2 | sinαsinγ | cosαsinβsinγ+cosβcosγ | sinβcosγ-cosαcosβsinγ | 21a=(23a)T |
-cosα | sinαsinβ | -sinαcosβ | |||
-sinαcosγ | cosβsinγ-cosαsinβcosγ | cosαcosβcosγ+sinβsinγ | |||
22a | z'2x'2y'1 | sinαsinγ | -cosαsinβsinγ+cosβcosγ | -cosαcosβsinγ-sinβcosγ | 22a=(24a)T |
cosα | sinαsinβ | sinαcosβ | |||
sinαcosγ | -cosβsinγ-cosαsinβcosγ | -cosαcosβcosγ+sinβsinγ | |||
23a | y'1x'2z'2 | sinαsinγ | -cosα | -sinαcosγ | 23a=(21a)T |
cosαsinβsinγ+cosβcosγ | sinαsinβ | cosβsinγ-cosαsinβcosγ | |||
sinβcosγ-cosαcosβsinγ | -sinαcosβ | cosαcosβcosγ+sinβsinγ | |||
24a | y'2x'1z'1 | sinαsinγ | cosα | sinαcosγ | 24a=(22a)T |
-cosαsinβsinγ+cosβcosγ | sinαsinβ | -cosβsinγ-cosαsinβcosγ | |||
-cosαcosβsinγ-sinβcosγ | sinαcosβ | -cosαcosβcosγ+sinβsinγ | |||
25a | x'1y'2z'2 | sinαsinγ-cosαcosβcosγ | cosαsinβ | -sinαcosγ-cosαcosβsinγ | 25a=(26a)T |
-cosαsinγ-sinαcosβcosγ | sinαsinβ | cosαcosγ-sinαcosβsinγ | |||
sinβcosγ | cosβ | sinβsinγ | |||
26a | z'1y'1x'2 | sinαsinγ-cosαcosβcosγ | -cosαsinγ-sinαcosβcosγ | sinβcosγ | 26a=(25a)T |
cosαsinβ | sinαsinβ | cosβ | |||
-sinαcosγ-cosαcosβsinγ | cosαcosγ-sinαcosβsinγ | sinβsinγ | |||
27a | z'2y'2x'1 | sinαsinγ+cosαcosβcosγ | cosαsinγ-sinαcosβcosγ | -sinβcosγ | 27a=(28a)T |
-cosαsinβ | sinαsinβ | -cosβ | |||
sinαcosγ-cosαcosβsinγ | cosαcosγ+sinαcosβsinγ | sinβsinγ | |||
28a | x'2y'1z'1 | sinαsinγ+cosαcosβcosγ | -cosαsinβ | sinαcosγ-cosαcosβsinγ | 28a=(27a)T |
cosαsinγ-sinαcosβcosγ | sinαsinβ | cosαcosγ+sinαcosβsinγ | |||
-sinβcosγ | -cosβ | sinβsinγ | |||
29a | y'1z'1x'2 | sinαsinγ | -cosαsinγ | cosγ | 29a=(30a)T |
cosαsinβ-sinαcosβcosγ | sinαsinβ+cosαcosβcosγ | cosβsinγ | |||
-sinαsinβcosγ-cosαcosβ | cosαsinβcosγ-sinαcosβ | sinβsinγ | |||
30a | x'1z'2y'2 | sinαsinγ | cosαsinβ-sinαcosβcosγ | -sinαsinβcosγ-cosαcosβ | 30a=(29a)T |
-cosαsinγ | sinαsinβ+cosαcosβcosγ | cosαsinβcosγ-sinαcosβ | |||
cosγ | cosβsinγ | sinβsinγ | |||
31a | x2z1y1 | sinαsinγ | -cosαsinβ-sinαcosβcosγ | sinαsinβcosγ-cosαcosβ | 31a=(32a)T |
cosαsinγ | sinαsinβ-cosαcosβcosγ | cosαsinβcosγ+sinαcosβ | |||
-cosγ | -cosβsinγ | sinβsinγ | |||
32a | y'2z'2x'1 | sinαsinγ | cosαsinγ | -cosγ | 32a=(31a)T |
-cosαsinβ-sinαcosβcosγ | sinαsinβ-cosαcosβcosγ | -cosβsinγ | |||
sinαsinβcosγ-cosαcosβ | cosαsinβcosγ+sinαcosβ | sinβsinγ | |||
33a | x'1z'1y'2 | sinαsinγ | cosαsinβ+sinαcosβcosγ | sinαsinβcosγ-cosαcosβ | 33a=(34a)T |
-cosαsinγ | sinαsinβ-cosαcosβcosγ | -cosαsinβcosγ-sinαcosβ | |||
-cosγ | cosβsinγ | sinβsinγ | |||
34a | y'1z'2x'2 | sinαsinγ | -cosαsinγ | -cosγ | 34a=(33a)T |
cosαsinβ+sinαcosβcosγ | sinαsinβ-cosαcosβcosγ | cosβsinγ | |||
sinαsinβcosγ-cosαcosβ | -cosαsinβcosγ-sinαcosβ | sinβsinγ | |||
35a | y'2z'1x'1 | sinαsinγ | cosαsinγ | cosγ | 35a=(36a)T |
sinαcosβcosγ-cosαsinβ | cosαcosβcosγ+sinαsinβ | -cosβsinγ | |||
-sinαsinβcosγ-cosαcosβ | -cosαsinβcosγ+sinαcosβ | sinβsinγ | |||
36a | x'2z'2y'1 | sinαsinγ | sinαcosβcosγ-cosαsinβ | -sinαsinβcosγ-cosαcosβ | 36a=(35a)T |
cosαsinγ | cosαcosβcosγ+sinαsinβ | -cosαsinβcosγ+sinαcosβ | |||
cosγ | -cosβsinγ | sinβsinγ | |||
37a | x'1y'2z'1 | sinαsinγ+cosαcosβcosγ | cosαsinβ | sinαcosγ-cosαcosβsinγ | 37a=(38a)T |
-cosαsinγ+sinαcosβcosγ | sinαsinβ | -cosαcosγ-sinαcosβsinγ | |||
-sinβcosγ | cosβ | sinβsinγ | |||
38a | z'2y'1x'2 | sinαsinγ+cosαcosβcosγ | -cosαsinγ+sinαcosβcosγ | -sinβcosγ | 38a=(37a)T |
cosαsinβ | sinαsinβ | cosβ | |||
sinαcosγ-cosαcosβsinγ | -cosαcosγ-sinαcosβsinγ | sinβsinγ | |||
39a | x'2y'1z'2 | sinαsinγ-cosαcosβcosγ | -cosαsinβ | -sinαcosγ-cosαcosβsinγ | 39a=(40a)T |
cosαsinγ+sinαcosβcosγ | sinαsinβ | -cosαcosγ+sinαcosβsinγ | |||
sinβcosγ | -cosβ | sinβsinγ | |||
40a | z'1y'2x'1 | sinαsinγ-cosαcosβcosγ | cosαsinγ+sinαcosβcosγ | sinβcosγ | 40a=(39a)T |
-cosαsinβ | sinαsinβ | -cosβ | |||
-sinαcosγ-cosαcosβsinγ | -cosαcosγ+sinαcosβsinγ | sinβsinγ | |||
41a | y'1x'2z'1 | sinαsinγ | -cosα | sinαcosγ | 41a=(42a)T |
cosαsinβsinγ-cosβcosγ | sinαsinβ | cosαsinβcosγ+cosβsinγ | |||
-cosαcosβsinγ-sinβcosγ | -sinαcosβ | -cosαcosβcosγ+sinβsinγ | |||
42a | z'2x'1y'2 | sinαsinγ | cosαsinβsinγ-cosβcosγ | -cosαcosβsinγ-sinβcosγ | 42a=(41a)T |
-cosα | sinαsinβ | -sinαcosβ | |||
sinαcosγ | cosαsinβcosγ+cosβsinγ | -cosαcosβcosγ+sinβsinγ | |||
43a | z'1x'2y'1 | sinαsinγ | -cosβcosγ-cosαsinβsinγ | sinβcosγ-cosαcosβsinγ | 43a=(44a)T |
cosα | sinαsinβ | sinαcosβ | |||
-sinαcosγ | -cosβsinγ+cosαsinβcosγ | sinβsinγ+cosαcosβcosγ | |||
44a | y'2x'1z'2 | sinαsinγ | cosα | -sinαcosγ | 44a=(43a)T |
-cosβcosγ-cosαsinβsinγ | sinαsinβ | -cosβsinγ+cosαsinβcosγ | |||
sinβcosγ-cosαcosβsinγ | sinαcosβ | sinβsinγ+cosαcosβcosγ | |||
45a | y'1z'2x'1 | sinαsinγ | cosαsinγ | -cosγ | 45a=(46a)T |
sinαcosβcosγ-cosαsinβ | sinαsinβ+cosαcosβcosγ | cosβsinγ | |||
sinαsinβcosγ+cosαcosβ | -sinαcosβ+cosαsinβcosγ | sinβsinγ | |||
46a | x'2z'1y'2 | sinαsinγ | -cosαsinβ+sinαcosβcosγ | cosαcosβ+sinαsinβcosγ | 46a=(45a)T |
cosαsinγ | sinαsinβ+cosαcosβcosγ | -sinαcosβ+cosαsinβcosγ | |||
-cosγ | cosβsinγ | sinβsinγ | |||
47a | x'1z'2y'1 | sinαsinγ | cosαsinβ+sinαcosβcosγ | cosαcosβ-sinαsinβcosγ | 47a=(48a)T |
-cosαsinγ | sinαsinβ-cosαcosβcosγ | sinαcosβ+cosαsinβcosγ | |||
cosγ | -cosβsinγ | sinβsinγ | |||
48a | y'2z'1x'2 | sinαsinγ | -cosαsinγ | cosγ | 48a=(47a)T |
cosαsinβ+sinαcosβcosγ | sinαsinβ-cosαcosβcosγ | -cosβsinγ | |||
cosαcosβ-sinαsinβcosγ | sinαcosβ+cosαsinβcosγ | sinβsinγ | |||
EULERANGLES | |||||
1u | x2(α)y2(β)x2(γ) | cosαcosγ-sinαcosβsinγ | -cosαsinγ-sinαcosβcosγ | sinαsinβ | 1u=(10u)T |
sinαcosγ+cosαcosβsinγ | - sinαsinγ+cosαcosβcosγ | -cosαsinβ | |||
sinβsinγ | sinβcosγ | cosβ | |||
2u | x1(α)y1(β)x1(γ) | cosαcosγ-sinαcosβsinγ | cosαsinγ+sinαcosβcosγ | sinαsinβ | 2u=(9u)T |
-sinαcosγ-cosαcosβsinγ | -sinαsinγ+cosαcosβcosγ | cosαsinβ | |||
sinβsinγ | -sinβcosγ | cosβ | |||
3u | x2(α)y1(β)x2(γ) | cosαcosγ-sinαcosβsinγ | -cosαsinγ-sinαcosβcosγ | -sinαsinβ | 3u=(8u)T |
sinαcosγ+cosαcosβsinγ | - sinαsinγ+cosαcosβcosγ | cosαsinβ | |||
-sinβsinγ | -sinβcosγ | cosβ | |||
4u | x1(α)y2(β)x1(γ) | cosαcosγ-sinαcosβsinγ | cosαsinγ+sinαcosβcosγ | -sinαsinβ | |
-sinαcosγ-cosαcosβsinγ | - sinαsinγ+cosαcosβcosγ | -cosαsinβ | |||
-sinβsinγ | sinβcosγ | cosβ | |||
5u | y2(α)x2(β)y2(γ) | cosβ | -sinβcosγ | sinβsinγ | |
cosαsinβ | -sinαsinγ+cosαcosβcosγ | -sinαcosγ-cosαcosβsinγ | |||
sinαsinβ | cosαsinγ+sinαcosβcosγ | cosαcosγ-sinαcosβsinγ | |||
6u | y1(α)x1(β)y1(γ) | cosβ | sinβcosγ | sinβsinγ | |
-cosαsinβ | -sinαsinγ +cosαcosβcosγ | sinαcosγ+cosαcosβsinγ | |||
sinαsinβ | -cosαsinγ-sinαcosβcosγ | cosαcosγ-sinαcosβsinγ | |||
7u | y1(α)x2(β)y1(γ) | cosβ | -sinβcosγ | -sinβsinγ | |
cosαsinβ | -sinαsinγ +cosαcosβcosγ | sinαcosγ+cosαcosβsinγ | |||
-sinαsinβ | -cosαsinγ-sinαcosβcosγ | cosαcosγ-sinαcosβsinγ | |||
8u | cosαcosγ-sinαcosβsinγ | sinαcosγ+cosαcosβsinγ | -sinβsinγ | 8u=(3u)T | |
-cosαsinγ-sinαcosβcosγ | - sinαsinγ+cosαcosβcosγ | -sinβcosγ | |||
-sinαsinβ | cosαsinβ | cosβ | |||
9u | cosαcosγ-sinαcosβsinγ | -sinαcosγ-cosαcosβsinγ | sinβsinγ | 9u=(2u)T | |
cosαsinγ+sinαcosβcosγ | -sinαsinγ+cosαcosβcosγ | -sinβcosγ | |||
sinαsinβ | cosαsinβ | cosβ | |||
10u | cosαcosγ-sinαcosβsinγ | sinαcosγ+cosαcosβsinγ | sinβsinγ | 10u=(1u)T | |
-cosαsinγ-sinαcosβcosγ | - sinαsinγ+cosαcosβcosγ | sinβcosγ | |||
sinαsinβ | -cosαsinβ | cosβ | |||
y2(α)x1(β)y2(γ) | cosβ | sinβcosγ | -sinβsinγ | ||
-cosαsinβ | -sinαsinγ +cosαcosβcosγ | -sinαcosγ-cosαcosβsinγ | |||
-sinαsinβ | cosαsinγ+sinαcosβcosγ | cosαcosγ-sinαcosβsinγ | |||
x1 | x2 | ||||
cosα | sinα | 0 | cosα | -sinα | 0 |
-sinα | cosα | 0 | sinα | cosα | 0 |
0 | 0 | 1 | 0 | 0 | 1 |
x'1 | x'2 | ||||
sinα | cosα | 0 | sinα | -cosα | 0 |
-cosα | sinα | 0 | cosα | sinα | 0 |
0 | 0 | 1 | 0 | 0 | 1 |
y1 | y2 | ||||
1 | 0 | 0 | 1 | 0 | 0 |
0 | cosβ | sinβ | 0 | cosβ | -sinβ |
0 | -sinβ | cosβ | 0 | sinβ | cosβ |
y'1 | y'2 | ||||
1 | 0 | 0 | 1 | 0 | 0 |
0 | sinβ | cosβ | 0 | sinβ | -cosβ |
0 | -cosβ | sinβ | 0 | cosβ | sinβ |
z1 | z2 | ||||
cosγ | 0 | sinγ | cosγ | 0 | -sinγ |
0 | 1 | 0 | 0 | 1 | 0 |
-sinγ | 0 | cosγ | sinγ | 0 | cosγ |
z'1 | z'2 | ||||
sinγ | 0 | cosγ | sinγ | 0 | -cosγ |
0 | 1 | 0 | 0 | 1 | 0 |
-cosγ | 0 | sinγ | cosγ | 0 | sinγ |
Signature | + | - | no. | matrix no. | |
9 | 0 | nil | |||
8 | 1 | nil | |||
7 | 2 | 16 + 16 |
(4,7)(14,15)(17,20)(25,26) (29,30)(43,44)(45,46)(47,48) (4a,7a)(14a,15a)(17a,20a)(25a,26a) (29a,30a)(43a,44a)(45a,46a)(47a,48a) |
||
6 | 3 | 16 + 16 | (1,10)(6,11)(13,16)(18,19)(22,24)(27,28)(37,38)(41,42)(1a,10a)(6a,11a)(13a,16a)(18a,19a)(22a,24a)(27a,28a)(37a,38a)(41a,42a) | ||
5 | 4 | 16 + 16 | (2,9)(3,8)(5,12)(21,23)(31,32)(33,34)(35,36)(39,40)(2a,9a)(3a,8a)(5a,12a)(21a,23a)(31a,32a)(33a,34a)(35a,36a)(39a,40a) | ||
TOTAL | 48 + 48 | ||||
* Above are case of Rotational Matrices
in 3-D Euclidian Space (R3) which a) preserve origin. b) maintain
Euclidian distance isometry c) preserve orientation i.e. handedness
in space. * R=Rz(α)*Ry(γ)*Rx(β) is an intrinsic rotation where α,γ,β are called Tait-Bryan Angles about z,y,x axis respectively. Intrinsic rotations occur about the axis of a co-ordinate system attached to a moving body(XYZ). Structure of rotation is X-Y-Z, Y-Z-X, Z-X-Y, X-Z-Y, Z-Y-X, Y-X-Z * R=Rz(β)*Ry(γ)*Rx(α) is an extrinsic rotation where α,γ,β are called Proper Euler Angles about x,y,z axis respectively. Extrinsic rotations occur about the axis of a co-ordinate system attached to a fixed frame of reference. The Structure of rotation is x-y-x, x-z-x, y-z-y, y-x-y,z-x-z,z-y-z. Two frames of reference are connected by Euler Angle provided they possess the same handedness in 3-D space.
In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a rotation group. The theorem is named after Leonhard Euler, who proved it in 1775 by means of spherical geometry. The axis of rotation is known as an Euler axis, typically represented by a unit vector ê. Its product by the rotation angle is known as an axis-angle. The extension of the theorem to kinematics yields the concept of instant axis of rotation, a line of fixed points. In linear algebra terms, the theorem states that, in 3D space, any two Cartesian coordinate systems with a common origin are related by a rotation about some fixed axis. This also means that the product of two rotation matrices is again a rotation matrix and that for a non-identity rotation matrix one eigenvalue is 1 and the other two are both complex, or both equal to −1. The eigenvector corresponding to this eigenvalue is the axis of rotation connecting the two systems. * The trace of the real 3-D rotation matrix is 1+2cosθ. Trace is invariant under orthogonal matrix similarity transformation. Since the orthogonal matrices are normal matrices and any normal matrix can be diagonalized by a unitary transformation, orthogonal matrices can be diagonalized through a unitary transformation. * Then, any orthogonal matrix is either a rotation or an improper rotation. A general orthogonal matrix has only one real eigen value, either +1 or −1. When it is +1 the matrix is a rotation. When −1, the matrix is an improper rotation. * An m × m matrix A has m orthogonal eigenvectors if and only if A is normal, that is, if A†A = AA†.[b] This result is equivalent to stating that normal matrices can be brought to diagonal form by a unitary similarity transformation: * These are matrices which form SO(3) group under matrix multiplication where determinant is +1. These are sub-groups of O(3) groups of matrices whose determinant can be +1 or -1.These are sub-groups of GL(3). matrices. * These are non-abelian group unlike SO(2) which is abelian. * The group is a manifold and therefore is a Lie Group under the same composition. * It has dimension 3- 1 co-ordinate for the point and 2 co-ordinates for the axis. *If α =180 degree, β =γ= 0 degree, rotation is put at x1,y1,z1, then (x,y) co-ordinates are reflected to (-x.-y) while z co-ordinate remains unchanged. If in addition, z1=0, and c33=-1, then the coordinate becomes (-x.-y.-z). But c33=-1 is not a rotation matrix element but reflection matrix. |
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Norm and Distance in Vectors : * The distance is a 2-vector function d(x,y) while the norm is a 1-vector function ||v|| . However, we frequently use the norm to calculate the distance by means of difference of 2 vectors, ||y - x || . A norm always induces a distance , but the reverse is not true. A trivial distance has no equivalent norm as d(x,x)=0 and d(x,y)=1 when x is not equal to y. |
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