Physics Help Forum Shape of refracting surface for focus

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Sep 14th 2018, 08:46 AM   #1
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Shape of refracting surface for focus

So this problem is from the Feynman Lectures V. 1 Ch. 27 Geometrical Optics
27–2 The focal length of a spherical surface

 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.
So I decided to entertain myself, to no avail. Does anyone else know how to solve this?

Cheers

 Sep 17th 2018, 11:12 AM #2 Senior Member     Join Date: Jun 2016 Location: England Posts: 687 I did once see a picture of one of these lenses, they must be terrifically awkward to grind (although perhaps less so now with accurate 3D robotic tools). As to actually sitting down with the maths, My idea of "fun" is far lazier than that... __________________ ~\o/~
Sep 18th 2018, 04:54 AM   #3
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Posts: 2
 Originally Posted by Woody I did once see a picture of one of these lenses, they must be terrifically awkward to grind (although perhaps less so now with accurate 3D robotic tools). As to actually sitting down with the maths, My idea of "fun" is far lazier than that...
Yeah, mine usually is too, but sometimes I get hung up on something.

Anyway, I do think I managed to solve it, I tried posting what I think is the solution, but it seems to have been lost (It said something like pending admin approval).

 Tags feynman, focus, refracting, shape, surface

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