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

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

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

4. 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. 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. 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. 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 = 180 - U v = 180 - V w = 180 -W C1C1 C2C2 C3C3 V =  W =  U =  v w u

8. 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 = 180 - U v = 180 - 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. 9 Euler construction cos w = 1 2 3 4 6 cos W + cos U cos V sin U sin V  or  U, V, or W cos U, V, W sin U, V, W

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