More of a thought experiment here, knowledge for knowledges sake.

Let's say you can create infinite points that rotate smoothly and at the same speed as each other through a full revolution - let's for simplicity say 1 revolution a second.

Let's also say you make these rotating points children of other rotating points that is - if you have one point as the child of a "root" point, then the combined rotation would be 2 rotations per second.

Now, let's say you can rotate the orientation of those points through any alignment in 3D space, so if you rotate your child point through 180 degrees on the axis perpendicular to rotation, the net rotation would be 0 revolutions per second as the two rotations cancel each other out.

Say also you can offset each child rotation point, so that:

Root point A rotates at point 0,0,0 with alignment 0,0,0 at 1 revolution per second. Point B is a child of root point A, and has offset of 0,0,1 and is aligned to 0,180,0 (so rotating in the opposite direction to point A, netting a global rotation of point B to be 0 revolutions per second, however the point moves about a circular orbit around point A.) Point C is a child of point B, and is given offset 0,0,0 to point B and is aligned to 0,0,0 offset of point B, so that the net rotation is now -1 revolutions per second, still following the circular orbit of 1r around global point 0,0,0. Finally, Point D is a child of point C, and is given offset 0,0,1 to point C and is aligned 0,180,0 offset to point C. (That is, the net rotation of point C is 0 revolutions per second and point C will move in a linear fashion along the z axis, from point 0,0,-2 to 0,0,2, with the speed of a sine wave.)

I understand this is hard to visualise, but perhaps pen and paper will help you.

It is possible, through creative rotation of these point/child relationships to create swivelling points, that is points that rotate patially through a complete revolution, then swing back around and complete the reverse side of the rotation, and finally completing the rotation (be it still in a sine wave shape).

My question, is this: What is the structure and ratio of the aligment of these point/child relationships so to have a net result of a point that can swivel through x degrees accurately without any net deviation into any other axis? (That is, for example, a point that rotates only on the z axis from 0 to 90 degrees and back again, for example with no net rotation on the y or x axes.)

Remember: The constraints are that all points must rotate at the same speed. There is no limit on the amount of child points any one point can have, and there is no limit on the angle, or positional offset that any of these points can have.

Is it possible to rotate to an exact angle every time, and is it possible to completely eliminate ALL out of plane fluctuations, given these constraints?

If you need clarification on any point, please do ask, I am very curious to see the results of this discussion.

Graphical Representation

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    $\begingroup$ I am unable to visualise what you mean. Why don't you add a drawing? $\endgroup$ – copper.hat Jul 12 '14 at 1:21
  • $\begingroup$ Done, please see above. $\endgroup$ – D. Brumby Jul 13 '14 at 4:52

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