I am talking about axles in machines, on which we can put gears, belts and other devices that help us convert the motion.
So this structure has to have (at least) the following properties:
- distance - works like a line, i can 'put' more stuff on it and be able to 'differentiate'(not in a mathematical sense) between them, essentially a $d(x_1, x_2)$ function has to be defined on it
- has finite or infinite length
- if i were to put a gear on it then the direction and the angular velocity of the rotation of the gear would be equivalent
- anticommutativity of its rotations(2 exist in 3D)
- the 'speed' of two parallel axes can be compared by a gear placed on each axle, so that they touch
- the touching point doesn't move at all, but the gears of a touching point have opposite rotation and same angular velocity
- touching gears need not be of same sizes
Of course if you liked to, you could generalize this to any curve.
Initially i thought of the cross-product and normals to a bi-vector. But those don't fulfil all requirements.
Is there a specific object that describes the motions and properties of a physical axle? (even only as a crude model)
Screw theory seems like what you want, or is at least highly relevant. I do not know how to apply it to your problem specifically without more details.