1 Combining rotation axes Should be obvious that: C 1 C 2 = C 3 C1C1 C2C2 C1C1 C2C2 C3C3 C3C3.

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Presentation transcript:

1 Combining rotation axes Should be obvious that: C 1 C 2 = C 3 C1C1 C2C2 C1C1 C2C2 C3C3 C3C3

2 Combining rotation axes But not obvious what types of rotations can intersect at what angles C1C1 C2C2 C1C1 C2C2 C3C3 C3C3

3 Combining rotation axes But not obvious what types of rotations can intersect at what angles Use Euler construction

4 Spherical C 1 C 2 C 3 formed by pts where axes intersect sphere Euler construction C1C1 C2C2 C3C3 C3’C3’ C1’C1’

5 C1C1 C2C2 C3C3 C3’C3’ C1’C1’ Spherical C 1 C 2 C 3 formed by pts where axes intersect sphere C 1 takes C 3 to C 3 ’ C 2 takes C 3 ’ back to C 3 but C 1 C 3 ’

6 Euler construction C1C1 C2C2 C3C3 C3’C3’ C1’C1’ Spherical C 1 C 2 C 3 formed by pts where axes intersect sphere C 1 takes C 3 to C 3 ’ C 2 takes C 3 ’ back to C 3 but C 1 C 3 ’   

7 Euler construction Spherical C 1 C 2 C 3 formed by pts where axes intersect sphere cos w = cos u cos v + sin u sin v cos W u = U v = V w = 180 -W C1C1 C2C2 C3C3 V =  W =  U =  v w u

8 Euler construction Spherical C 1 C 2 C 3 formed by pts where axes intersect sphere cos w = cos u cos v + sin u sin v cos W u = U v = V w = 180 -W cos w = C1C1 C2C2 C3C3 V =  W =  U =  v w u cos W + cos U cos V sin U sin V

9 Euler construction cos w = cos W + cos U cos V sin U sin V  or  U, V, or W cos U, V, W sin U, V, W

10 Euler construction cos w = cos W + cos U cos V sin U sin V For 22X, find w (X = 2, 3, 4, 6)