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$SO(d)$ is the d square rotation matrix group. I am wondering if we can express all rotation on sphere $S^{d-1}$ as a continuous function operator $f(t)$, i.e. for any $Q\in SO(d)$, there exists a $t\in R$, $Q=f(t)$. Here $f$ is a continuous function with respect to $t$.

Intuitively, I want to try $f(t)=e^{tA}$ with $A$ a fixed matrix.

Also, do you have any reference about this part? I need some refered material to go deep in this area.

Thanks very much.

Yeli Niu
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    A useful point of view is to take SO(d) as a matrix lie group, and then any such A can be viewed as a linear combination of generators, living in so(d). Brian C Hall has a very accessible introduction to matrix lie groups which explains all this and more very nicely. – wnoise Jun 12 '20 at 06:18

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What is the case is that if $A$ is a skew-symmetric matrix, then $\exp(A)$ (as defined by the exponential power series) is an element of $\text{SO}(n)$ and that all elements of $\text{SO}(n)$ can be obtained this way.

So, given $Q\in\text{SO}(n)$ there is a function of the form $f(t)=\exp(tA)$ with $f(0)=I$ and $f(1)=Q$.

Angina Seng
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  • Thanks very much. I want this function $f$ is a mapping from a continuous sub interval of $R$, and mapping onto $SO(d)$. Or from a continuous area of $R^{m}$ onto $SO(d)$ with some integer $m$. – Yeli Niu Jun 12 '20 at 06:04
  • I think it's important to note that this decomposition is not unique. Even picking out the "minimum" A may change discontinuously when changing the final rotation Q, or fail to converge. Nor is computing such an A always easy; you can try computing a matrix logarithm by e.g. power series, but the standard ways only really work near the origin. – wnoise Jun 12 '20 at 06:11