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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.

enter image description here

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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  • $\begingroup$ Thanks for quick answer.I see what you mean. Then how to define surface integral to find the area on the unit sphere. $\endgroup$ – Mathlover Mar 27 '12 at 14:45
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    $\begingroup$ en.wikipedia.org/wiki/N-sphere gives the formulas you need for any dimension. The formula is explicit in en.wikipedia.org/wiki/Sphere under Generalization to other dimensions. You need to be careful about the number of dimensions-the earth is a 2-sphere, not a 3-sphere. $\endgroup$ – Ross Millikan Mar 27 '12 at 15:34
  • $\begingroup$ @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? $\endgroup$ – draks ... Mar 27 '12 at 15:42
  • $\begingroup$ @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. $\endgroup$ – Ross Millikan Mar 27 '12 at 17:43
  • $\begingroup$ 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. $\endgroup$ – Mathlover Mar 27 '12 at 18:39

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