# Find a reflection point on a sphere given source and destination of a ray

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 .

• math.stackexchange.com/questions/1037090/… Commented Dec 2, 2021 at 16:32
• This is Alhazen's problem. Commented Dec 2, 2021 at 16:34
• Thank you Eric, much appreciated. What do I do now to acknowledge that the solution has been pointed? Commented Dec 2, 2021 at 16:57
• Eh... This wasn't a solution/answer. This was just a reference to a more complete body of literature than can reasonably be summarized into an Answer. Commented Dec 2, 2021 at 16:59
• well, it was the answer to my question as stated (point me where to look :). So thanks again. Commented Dec 3, 2021 at 9:28

## 1 Answer

As pointed by Eric Towers in the comments, this is a known problem with a known solution. The article on Alhazen's problem in wikipedia provides a sufficient list of references.

• What you wrote is not an answer. Commented Dec 3, 2021 at 9:33