Given two 3D points, say $A = (0, 0, 0)$ and B = $(T, 0, 0)$ and a sphere $S$ of radius $r$ centered at a point $C=(a, b, c)$. The sphere does not cross the $z=0$ plain, say it is located fully below that plain, i.e. $c<-r$.
I am looking for the point $P \in S$ on the sphere such that a ray starting at the point B would arrive at the point A after bouncing off the sphere S at that point $P$. I wonder if analytical answer is possible here.
I think the problem can be written as a constrained optimization, like this: $$\begin{eqnarray} \begin{cases} \sqrt{(T-x)^2 + y^2 + z^2} + \sqrt{x^2 + y^2 + z^2}\rightarrow \min_{x, y, z}\\ (x-a)^2+(y-b)^2+(z-c)^2=r^2 \end{cases} \end{eqnarray}$$ but do not understand what to do with it. Besides, I expect the reflection law to figure in here, but cannot see how to write it in.
Can anyone point me where to look for a solution or a method to find such? Thanks in advance .
