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I'm still confused as to why being able to keep Sao Paulo fixed prevents us from representing a rotation from Pittsburgh to Sau Paulo. Can we not rotate vertically, then horizontally, to get there? Is this just saying that to form a 3d rotation, we need 3 degrees of freedom and therefore 3 numbers?


I don't quite get where the 3rd degree of freedom comes from. Is it the direction of rotation?


Can we define a rotation by how the axis of rotation changes over time? I suppose this would be very difficult to do for abnormal rotations though.


Is there a general way to determine the degrees of freedom of a model statistically?


I think it makes sense that we need three degrees of freedom, but the example with Sao Paulo seemed to make it more confusing. Can you provide another example?