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What is the 3D rotation equivalent of integrating (or a simpler version of the problem, simply evaluating or enumerating) all the values?

For example in one dimension we have the possibility of an improper integral that covers an infinite range of values, the result can be finite. $\int_{0}^{\infty} \frac{dx}{(x+1)\sqrt{x}} = \pi$

What if instead the domain is a rotation value? Expressed as Euler angles or rotation matrices or quaternions.

Is this possible? How could I approach it? Suppose I want to compute $\int_{rotations}w(q)\ dq$ where I have reason to suspect that the answer is not infinite...

This is weird, also because it is very nontrivial to even try to define a subset of the SO(3) space, whereas that is quite natural to do with a normal integral.

Maybe something can be done with spherical harmonics...

Steven Lu
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  • Can you please remove extraneous writing and pare your question down to its essence? That way we're more likely to help you. – David G. Stork Jan 24 '16 at 00:59
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    Asking about your attempt at solving a problem, instead of asking about the actual problem itself, only complicates things and ends up in a waste of time both for you and for whoever is going to stick around enough to actually answer (cfr. the so-called xy problem). – A.P. Jan 24 '16 at 01:27
  • Yeah this is a low quality question for sure. Hmm. Let me try to fix it – Steven Lu Jan 24 '16 at 01:33
  • If I've understood, the question is "How to numerically approximate an integral over $SO(3)$?" One approach would be to parametrize $SO(3)$ in the same general manner one parametrizes a sphere (using a rectangle) in spherical coordinates. As for how to parametrize so that the resulting $3$-dimensional integral is accurate for a given grid size, that depends on the specifics of your integrand. – Andrew D. Hwang Jan 24 '16 at 01:57
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    Before you even try to evaluate that integral, you should be clear on what it means to integrate something over $SO(3)$. Luckily, most (probably all) introductory books to differential geometry have a section on integration over smooth manifolds. Once you do that, just choose the atlas on $SO(3)$ that best suits your integrand and compute the resulting triple integrals with your favourite numerical method. – A.P. Jan 24 '16 at 11:31

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