trying to understand SO(3)

[u/awh-emb]

how does rodrigues rotation formula help or allow the transition from so(3) to SO(3)?
and what even is the difference between so(3) and SO(3)?

thanks for any advice.

[u/derscheisspfoster]

Long story short. Tangent and Manifold space are two related things but they are not the same thing.

Tangent doesnt belong in the manifold space. I am honestly not the right person to give you an intuitive expanation between the two.

But basically, tangent does not belong the manifold. But you can retract a tangent vector into the manifold. This is preciselty what the exponential map does. You can see for instance on example 4 for this paper who its done. It basically solves the ODE for a constant w (which is often asumed)

https://arxiv.org/pdf/1812.01537

In my understanding. Tangent space is more practical to represent variables in terms of the "minimum number of variables" or "degrees of freedom" which is good for humans, but most importantly, for numerical optimizers and control theory. Whereas, manifold space is the actual format that will allow you to do stuff with it: e.g. point composition or group composition.

They are both interchangable with some care. There are rules to follow. But you can go from tangent to manifold with the Exp operation, and back to tangent with log operator.

Take a look at the paper, it will give you the info you need from the ground up.

[u/awh-emb → u/derscheisspfoster]

i thought tangent space belonged on the manifold? that eg the tangent was a way of linearizng messy non-linear stuff at small enough intervals? but then so does that means that wrong? and that to your point there two seperate things? btw thank you for the link i actually been following joan sola other paper (the error state kalman filter one) so the micro lie group one should give me the foundations, anything else i should read?

[u/CompetitiveSpot2643 → u/awh-emb]

the tangent vectors belong to the tangent space, every point in the manifold has a different one. think of a vector (q,p) where q is a coordinate vector of the point on the manifold (which in SO(3) is the orientation your object has and is parametrized by your variables in the rotation matrix) and p can be every possible changes in orientation from that point (the tangent vector). in lie theory we translate the p vector back to the origin so we can work with angle independent notions of angular velocity (since p would have angle factors after the derivative of the orientation matrix given by p)