Combination of rotations


This is the sphere of rotations, on which a rotation over a degrees in a plane is represented as any directed arc of a/2 degrees of the great circle in which that planar direction cuts the sphere.

We can use this to show how rotations combine. Given two rotations over a and b degrees in different planes, we can either to the a rotation first and then the b rotation, or vice versa. The results are different.

To combine two rotations, determine one of the intersection points O of their great circles. Then find two points on each of the great circles so that you can move along the blue a-circle to O with an arc of length a/2, and then from O along the green b-circle with an arc of length b/2. Connect the points with a new great circle arc in red: this is the plane of the combined notation; the length of the red arc is (half) the total rotation angle.


Performing this recipe in the reverse order (first along green, then along blue) gives a different great circle, so a different rotation -- though the rotation angle is the same.

This addition of arcs along the sphere can be represented algebraically as the multiplication of quaternions or rotors in geometric algebra.


UNFORTUNATELY CINDERELLA DOES NOT APPEAR TO SAVE THE SPHERICAL VIEW THIS WAS DESIGNED FOR


Play with it yourself: dragging the red points changes the rotation angles (arc lengths relative to O) as well as the rotation planes.

Created with Cinderella by Leo Dorst 20010716.