Im not good in geometric interpretations... any help is very welcome.
Consider the unitary disc $$D=\{(x,y,0)\in\mathbb{R}^3, x^2+y^2\leq1\},$$
parameterized by $$\varphi(r,\theta)=(r\cos\theta,r\sin\theta), (r,\theta)\in[0,1]\times[0,2\pi].$$ Let $\Omega(x,y,z)$ be the solid angle of $\varphi$, viewed from $(x,y,z)$. Consider a closed curve $\gamma:[a,b]\rightarrow\mathbb{R}^3\backslash S$ of class $C^1$, with $$S=\{(x,y,0)\in\mathbb{R}^3, x^2+y^2=1\}.$$ Let $p$ be the number of times that $\gamma$ cuts $D$, coming from $z>0$ to $z<0$, and $q$ the number of times that $\gamma$ cuts $D$, coming from $z<0$ to $z>0$. Use geometric arguments to conclude that $$\int_\gamma d\Omega=4\pi(p-q).$$
PS: if someone wants to know about Solid Angle, take a look at http://en.wikipedia.org/wiki/Solid_angle
or
Hint: Prove that $\int_\gamma d\Omega = \int_{\gamma'} d\Omega$ if $\gamma$ and $\gamma'$ are homotopic in $\mathbb{R}^3\backslash S$ (for some regular enough homotopy, probably you will need $C^2$). Use this to reduce (through deforming and "cutting" the path $\gamma$) the problem to the case where $\gamma$ is for example a circle (or some similarly easy to work with form) cutting $D$ exactly once (through deforming and separating the path $\gamma$).
Here's a related problem (the one I mentioned in the comments) with solution:
Problem (problem 2)
Solution
Ich hoffe, dass du Deutsch lesen kannst ;)