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I am looking for the generalized formula how to find the solid angle for a closed curve in $R^3$ then to generalize it for $R^n$. Thanks for answers and papers or books references and links that are related to the subject.

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I am trying to express the general formula of the projection area ($S$) on the unit sphere as shown in the picture above.

For example how to find the solid angle for the special case?

A curve defined :

$x=3+\cos(t)$

$y=3+\sin(t)$

$z=3$

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Divide your components by the length of $r$: $|r|=\sqrt{(3+\cos(t))^2+(3+\sin(t))^2+3^2}=\sqrt{6\sqrt{2}\sin(t+\pi/4)+28}$

to project them on the unit sphere, e.g. you'll get $ \frac{3+\cos(t)}{\sqrt{6\sqrt{2}\sin(t+\pi/4)+28}} $ for your projected $x$.

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    I know $4\pi$ is total solid angle for 2-sphere. I wonder how we can find the $S$ area on the unit sphere ( $S$ Area I showed in my question ). I need to express it via integral formula. Then I thought the follow the same methods for $R^n$ space.2012-03-27