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Maybe I'm asking a very vague question but I'd like to know if there are some visualisation tools available already that explain lie algebra exponential map or logarithm? I'd like to be able to explain the concept of SO3 in my thesis and I have only a basic mathematical knowledge of it, nothing more. If you could pass me a link where things are explained more by visual means than mathematics, it'd help me a lot. I've only been able to find mathematical introductions etc. on this with my search on google.

I apologise if this kind of question has been asked before I didn't find it though.

Kind Regards, Ankur.

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well, those represents rotations in 3-space, and there are many ways to represent those, such as angles ... – kjetil b halvorsen Dec 21 '12 at 2:38
up vote 5 down vote accepted

You might find it helpful that $SO(3)$ is homeomorphic to $\mathbb{R}P^3$. You can easily visualize the latter as a three-ball of radius $\pi$ modulo antipodal identification on the boundary. The identity element corresponds to the origin. The exponential map corresponds to the exponential map of $\mathbb{R}P^3$ as a spherical manifold (since $\mathbb{R}P^3 = \mathbb{S}^3/\pm 1$, and $\mathbb{S}^3\cong \operatorname{Spin}(3)$, the unit quaternions).

You can read about this at Wikipedia. I've also found a good exercise-based walkthrough of this homeomorphism in an analysis book, of all places: Dym & McKean, Fourier Series and Integrals.

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Your link directs the reader back to this webpage. I think you were referring to the Wikipedia page on the hypersphere of rotations:… – Isaac Solomon Dec 21 '12 at 2:50
@IsaacSolomon Ruh-roh! Thanks, fixed! – Neal Dec 21 '12 at 2:56

View it (and prove it!) as the unit tangent bundle $UT(S^2)$ of the sphere. My friend suggested this interpretation: Driving a tank on the boundary of a ball, where the turret of the tank freely rotates 360 degrees around.

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