Let us consider a needle of length 1, with one (fixed) end at the coordinate $(x,y,z)=(0,0,0)$. If we make the other end (wich is free) of the needle move, it will describe the unit sphere. We assume the distribution of the free end of the needle is uniform on the sphere, which means it is given by: $\cos(\phi) d\phi d\theta$
There is a light above the vertical axis of the sphere. What is the distribution of the length of the shadow of the needle?
So far, I have considered the hemisphere corresponding to $\phi \in [0,\frac{\pi}{2}[$. Let $t \in ]0,1[$. $P(L