equation for the path of a point on a spinning gyroscope

Factual Questions
1

i’m ten years out of college so this is not any sort of homework assignment. i was watching the movie Contact the other day in which there is a giant gyroscopic device with three wheels thus. a normal gyroscope’s job is i guess to keep the interior stable, but in this device, the outermost ring is fixed, and each interior ring is fixed to a motor such each on it own would only spin at a constant speed around one axis.

so i’m wondering, how would you graph the path that a point on the inner most ring takes as it traces along the surface of an imaginary sphere whose radius is the same as the innermost ring? how would you graph it if there were only two rings spinning? one ring?

i have any would like to use the graphiing program grapher for max OS X. i suspect the equations would be something similar to what that page shows for a torus knot, although i surmise that the torus is using a cylindrical equation and what i need here is some kind of spherical equation. grapher gives two basic equation set ups besides cylindrical, one is a cartesian matrix of x,y,z - and the other a sherical matrix of r,?,?. but what equations do i use? i imagine r is just a constant since it’s only moving along the surface of an imaginary sphere. but i’m not sure what to put for ? and ?, and for t.

2

What happens if you graph r=1 (or any constant),t,2*t, in a sherical matrix.

3

I’m used to doing these calculations in cartesian coordinates; I’m sure someone will chime in with an easier method of doing it in spherical coordinates. Until then…

You can think of the transformation of a point [x; y; z] on the center ring to a point in the air as the concatenation of three rotation matrices. The first and third are rotations about the y axis and the second is a rotation about the x axis. Call the three angles i, j, and k. The transformation is then:
[ cos(i), 0, sin(i); 0, 1, 0; -sin(i), 0, cos(i) ] * [ 1, 0, 0; 0, cos(j), sin(j); 0, -sin(j), cos(j) ] * [ cos(k), 0, sin(k); 0, 1, 0; -sin(k), 0, cos(k) ] * [x; y; z]

You then just need to calculate the values of i, j, and k over time, and pick a starting location x, y, z, and multiply it through.