Prove Two/Three Euler Angles Needed for Orientation Change

[TheFerruccio]

Hi, I have a simple question.

Given two orientations of a body in space, how can it be proven that at least two (or three?) Euler angles are needed to get from one orientation to the other? I am designing a mechanism and someone suggested moving it in a way that would require two hinges. I know that it is intuitively true, but I would like to prove it with math.

 

[D H]

You only need *one* rotation to get from any orientation to another.

Rotations in three space form a mathematical group, a rather important one. It's called SO(3), short for the three dimensional special orthogonal group.

 

[TheFerruccio]

So you can get from any orientation to any other orientation using a single axis of rotation?

I am not sure how orthogonal groups work. That is beyond what I know. Is that what I need to know in order to understand this?

*edit* I found this: http://en.wikipedia.org/wiki/Rotation_group_SO(3)#Axis_of_rotation

I am guessing that is what you mean. So, basically, every rotation between any two orientations has an axis of rotation? That's hard to wrap my head around. I am imagining specific examples where that doesn't seem to be the case, but, I guess it must be the case.

 

[D H]

It's Euler's rotation theorem. (Wiki article: http://en.wikipedia.org/wiki/Euler's_rotation_theorem) Rotate an object two, three, or more different ways, and there's always some axis about which you could have rotated the object and get from the original orientation to the final orientation in one single rotation.

That said, just because a single axis rotation must exist does not necessarily mean that a physical device can rotate about that axis. For example, robotic arms need multiple joints because each joint has a limited number of degrees of freedom. Getting the arm into some desired orientation can sometimes be tricky.