The coordinates after two 3d rotations.?

Question

Picture the unit circle. Now imagine the line from the origin to point (1,0). Now lets add a dimension of depth to this point making it (1, 0, 0). Suppose i rotate this point 45 degrees on the z axis to get (sqrt2/2,sqrt2/2,0). Now suppose i rotate it again but on the Y axis 45 degrees (and in all honesty i'm not even sure where exactly it would end up) and yet again 45 degrees on the x axis. Would the point now be at (sqrt3/3,sqrt3/3,sqrt3/3)? And h**l if you actually know how to calculate this without making a freaking unit sphere in google sketchup do share.

Answer 1

After the second rotation, it would be at (imagine just rotating (sqrt(2)/2, 0, 0) first) at

(1/2, sqrt(2)/2, 1/2)

After the third rotation, it would be at

(1/2, 1/2 + sqrt(2)/4, 1/2 - sqrt(2)/4).

This is a bit more difficult to see, but follows but simply doing the rotation to the last two coordinates.

Another thing to be careful of: the *order* in which you do the rotations makes a difference! If you had done the x axis rotation first, then the y axis, and then the z axis, the 1/2 would have been in the z position.

If you know how to do rotations in two dimensions, then rotations around the axes in three dimensions is fairly easy: just do the 2D rotation to the coordinates you are not rotating about. For more complicated rotations, it is best to do a matrix multiplication.