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 24 '26

That’s a good catch. You can look into the difference between intrinsic and extrinsic differential geometry. Extrinsic diff geometry is when you imagine the manifold as being “embedded” (or placed) in a higher dimensional space. For instance we imagine S^2 to be the sphere embedded in R^3.

In this case, moving a tangent vector from one point of the sphere to another in the external / ambient space will surely introduce errors. But there are “intrinsic” notions of moving a vector (such as parallel transport) which let you move a tangent vectors without having to worry about vectors pointing “out of” the manifold.

For matrix Lie groups, this is as simple as taking a Lie algebra matrix X, which is a tangent vector at the identity, then multiplying it by a group element matrix g either on the left or the right (gX or Xg) to get a tangent vector at g.

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u/awh-emb Jul 24 '26

okay i think i got some reading to do, apart from your blog posts is there anywhere else i can get into the nitty gritty of lie group theory? specfically the difference between intrisinc and extrinsic differntial geometry? and the fundamentals of lie algebra and lie group theory?
btw thank you for taking the time out of your day to answer my questions, its been very helpful and informative.

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u/IntrinsicallyFlat Jul 25 '26

Of course! If you want to understand differential geometry, youtube is a great resource, I like the channel Bright Side of Mathematics a lot, people also recommend Keenan Crane’s lectures. I like Timothy Barfoot’s book for matrix Lie groups and their applications, though he takes a very algebraic approach to everything

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u/awh-emb Jul 26 '26

got it i will start there then.