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Anonymous at Sun, 20 Oct 2024 09:39:15 UTC No. 16440958

"Now suppose that we have a point at O, at a distance s from the front surface of the glass, and another point O′ at a distance s′ inside the glass, and we desire to arrange the curved surface in such a manner that every ray from O which hits the surface, at any point P, will be bent so as to proceed toward the point O′. For that to be true, we have to shape the surface in such a way that the time it takes for the light to go from O to P, that is, the distance OP divided by the speed of light (the speed here is unity), plus n⋅O′P, which is the time it takes to go from P to O′, is equal to a constant independent of the point P. This condition supplies us with an equation for determining the surface. The answer is that the surface is a very complicated fourth-degree curve, and the student may entertain himself by trying to calculate it by analytic geometry."

Now is this as simple as

1. Set up the coordinate system:

2. Place the point O at (0, 0, -s) on the z-axis.
Place O' at (0, 0, s') on the z-axis.

3. Let P be a general point (x, y, z) on the surface we're trying to determine.


4. Express the distances:

OP = √(x2 + y2 + (z+s)2)
O'P = √(x2 + y2 + (z-s')2)


5. Write the equation based on the constant time condition:
OP + n⋅O'P = k, where k is a constant and n is the refractive index of the glass.
6. Substitute the distances:
√(x2 + y2 + (z+s)2) + n⋅√(x2 + y2 + (z-s')2) = k

7. Continue with this mess

I wonder if there are any practical examples of solving geometries like that? I assume this would be solved computationally so I'm interested in any matlab/python/sage code if anyone knows of stuff like this