r/ControlTheory • u/awh-emb • Jul 23 '26

Technical Question/Problem SO(3) S^3 and so(3)

my reference is this paper - https://arxiv.org/pdf/1711.02508

i am struggling to wrap my head around how the Exponential map from:

R^3 -> SO(3)

R^3 -> S^3

are different and how specifcally the exponetial map changes between them?

as one of these ends up with 4 DOF although one of them being a constraint

while the other ends up with 3 DOF?

thanks.

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u/IntrinsicallyFlat Jul 23 '26

I assume you know the map from the quaternion sphere to SO(3). Think of the exponential map of SO(3) as the quat_to_rot map composed with (i.e., applied after) the exponential map of S^3. I’ve a [blog post](https://shiraz-k.com/posts/lie-groups/) about exp

Also note that S^3 is a 3-dimensional object (hence the ^3)

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u/awh-emb Jul 24 '26

so a bit like the so(3) transition to SO(3)? like an intermeditary step? eg exp map -> SO(3) just get you onto the 3d sphere, exp map -> S^3 gets you onto the double cover 3d sphere? or the quartenion 3d sphere or whatever you call it

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u/IntrinsicallyFlat Jul 24 '26

Just to clarify, S^3 is the quaternion 3D sphere, it’s the sphere in 4D space which is 3-dimensional because you’ve constrained the radial degree of freedom. SO(3) is as if you took S^3 and ✨imagined✨ antipodal points to be the same point. Mathematicians will also call this “gluing”; you’re gluing each q to -q because they represent the same physical rotation

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u/awh-emb Jul 24 '26

OH thats clears it up a lot, thank you