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I am trying to figure out a way to solve Alhazen's problem for a sphere. I have found a couple of resources online, but I am having issues modifying those to meet my requirements. The basic problem is I have a source of light which reflects off a sphere and is picked up by an observer. I know the location of the source and observer with respect to the center of the sphere, and I would like to find the location of the reflection with respect to the center of the circle.

Here is a drawing of the situation. I am using this to calculate the specular point for the Earth, hence the use of satellites and Earth based symbols.

Figure showing geometry of problem:

image

A couple of notes:

  • Theta 1 is the half angle of RST.
  • Here I know r_R and r_T as well as i_R and i_T.
  • r_E is the radius of the Earth.
  • T refers to the source of light (transmitter)
  • R refers to the observer (reciever)

In some way, I am looking for the latitude and longitude of the point of reflection. I have found a couple of resources that do this, but I am uncertain on how to use them for my case.

The first resource can be found here. This has the issue of presenting a 2D solution. I am uncertain how to bring this to 3D.

The second resource I found is here. The issue I am having here is that it requires you to know the location of the observer and the source with respect to the reflection point. I am not sure how to get this information without first knowing the location of reflection, which is precisely what I am trying to find.

I would also like to add that my background is not in mathematics, but rather engineering. So if the solution to my problem seems obvious that may be why. Thanks in advance for your help!

ACB
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Pfrances
  • 43

1 Answers1

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You can easily transform your 3D problem into a 2D one. Indeed, the point of reflection needs to be in the plane defined by your three points $R$, $T$ and $E$.

The two resources you've found give you two different ways of obtaining the solution. In the second one, I am not sure what is exactly your understanding difficulty, maybe the fact that the unknown $\mathbf{N}$ is being used to define the necessary conditions on it, namely that it has unit length (since that resource has divided all lengths by your $r_E$) and that it respects the symmetry implied by reflection. In order to impose that $\mathbf{N}$ is in the plane of $R$, $T$ and $E$, that document writes $\mathbf{N}$ (your $\overrightarrow{ES}$) as the sum of two vectors, $x\overrightarrow{EL}+y\overrightarrow{ER}$ and solves for $x$ and $y$ under the above two conditions.

Joce
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  • You are entirely correct, my apologies. I took the figures given as defining the vectors, but it seems that is not the case. In case someone finds this in the future, here is another source. You have to run the algorithm twice though (once for latitude and again for longitude). specular point analytic solution – Pfrances Oct 13 '21 at 20:14