How to find solid angle of a closed curve?

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. 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 :

$x=3+\cos(t)$

$y=3+\sin(t)$

$z=3$

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 $x$.
• @RossMillikan I'm not sure, if the OP is asking for the voulme formula of $n$-sphere, or what did I miss at the linked page? – draks ... Mar 27 '12 at 15:42
• @draks: I thought OP was asking for the surface area of an n-sphere. You showed how to calculate the area as seen, but OP needs the total area to get to the fraction. It is $4 \pi$ for a 2-sphere, but OP asked about higher dimensions. – Ross Millikan Mar 27 '12 at 17:43
• 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. – Mathlover Mar 27 '12 at 18:39