Objects with Constrained Rotation.

[Hornbein]

TL;DR Summary

Here in 3D one may either have an object constrained to rotate only in a single plane or have a freely rotating object that can rotate in any combination of the three planes. But not exactly two planes. True?

Here in 3D one may either have an object constrained to rotate only in a single plane or have a freely rotating object that can rotate in any combination of the three planes. It seems to me that you can't have an object constrained to rotate in only combinations of two planes. You can't have something free to rotate in axy + byz but not in axy + byz + cxz. But I don't know how to prove that so maybe this isn't true. Mechanical only : no electronics. Little help?

Note: those planes are the bivectors of geometric algebra. This uses planes of rotation, which are the complements of axises of rotation. But I believe reader who prefer other notation will get the idea.

 

[Hornbein]

Aha now I see it. Have a sphere with a rod through the center protruding from the north and south poles. Have a ring that's larger than the sphere with a slot on the inner edge. Put the ends of the rod in the slot. If the ring is in the yz plane then the rotational plane of the sphere can have any combination of the xy and xz planes but no element of the yz plane.

 

[jack action]

Doesn't this work as you expect?

I also found an "ellipsoid" joint:

Or the "saddle" joint:

These are the kinds of joints you find in your articulations, like your wrists and fingers (you can look up condyloid joint):

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