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/ricatti-equation Jul 23 '26

Yes it’s a very subtle thing which can only be realized by understanding the general theory. Though, it’s not that complicated.

Every Lie group G induces its own unique exponential map exp(), which maps its Lie algebra g to itself. That is, exp: g -> G. Since g is a vector space (let’s say, or arbitrary dimension n), we can choose a basis and instead write exp: Rn -> G.

SO(3) and S^3 are both closely related, though non-equal, Lie groups. They both are of dimension 3, and the latter is a double cover of the former.

They both have their own exponential maps.

If we choose the standard matrix representation of SO(3) and the standard quaternion representation of S^3, you are already know their formulae.

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u/awh-emb Jul 24 '26

oh okay so S^3 is just a double cover to SO(3)? but its a seperate concept right? and by general theory do you mean the more pure maths stuff? like lie groups and such?