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What shape is formed by the set of all raindrops which create a rainbow to specific observer?

Maybe it's easier to narrow this down and consider it this in parts: what's the set of positions from which a raindrop can refract and reflect a beam of red light in the primary rainbow? What shape does that set form? How about the inner brighter part of the rainbow?

Someone asked me this when I describe why the inner part was brighter, but I'm not sure how to work this out!

Renee Descartes' sketch of how a rainbow is formed

One commenter suggested this sketch as an explanation, which suggests the shape is a flat plane.

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Look for rainbow in wikipedia, there is a sketch by Descartes on that question. –  Georg Oct 29 '11 at 10:29
    
So the shape is a circular plane rather than a three dimensional shape? In other words, for each point of a rainbow, there is only one position a raindrop can be in provide a ray from that point? –  Paul Dixon Oct 29 '11 at 10:37
    
In 1st order analysis its such a plane, if You think of some deviation of angle allowed this plane has some "thickness" in reality of course. I recommend some experimnts with a garden spray nozzle when sun (hopefully) will shine brightly next summer :=) –  Georg Oct 29 '11 at 10:42
    
Just as a thought experiment then: if I had a nozzle which could produce a thin plane of droplets, then there would be only be a fairly narrow range of distances from the observer where a rainbow could be seen? –  Paul Dixon Oct 29 '11 at 10:46
    
Yes, I'd agree. Early 50ties I remember garden spays which produced a thin layer of droplets. But I haven't seen them since :=( This was a simple upward nozzle, and a kind of cone on top. –  Georg Oct 29 '11 at 10:51

2 Answers 2

The raindrops reflecting a certain color are seen as a circle by the observer. See http://www.youtube.com/watch?v=uRmdZVvzMzQ for a simple explanation. There is a nice lecture by Lewin (MIT) on rainbows which goes into the details of rainbow geometry: http://www.youtube.com/watch?v=p9iB2PALVeY

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This lecture is excellent! And: truth is a lot more complicated than the stuff from Highschool. –  Georg Oct 29 '11 at 19:36

The raindrops reflecting a certain color (say, red) are a CONE with your eye at its apex and where the axis of the cone is a line passing through the sun. The raindrops reflecting a different color (say, blue) are a different cone-- with the same apex and axis but a slightly different angle.

In other words, If you look through a straw towards a red point on a rainbow, you are usually looking at many different raindrops -- some closer and some farther, ever changing -- but all of those raindrops are tinted red. That's because the sun is effectively infinitely far away, so the angle from your eye to the raindrop to the sun is the same whether the raindrop is close or far, as long as you are looking through that straw pointed in a fixed direction.

The drawing is misleading. Only one plane is illustrated so that you can see what's going on. But the raindrops in any other plane would also show the rainbow pattern. (Of course, you would not see the pattern from far-away raindrops if they are obstructed by closer-by raindrops.)

If you want to explain the relative brightness of the different parts of the rainbow, you're best off thinking about the spectrum of sunlight, the sensitivity of the human eye to different colors, the tendency for blue and (especially) violet light to get Rayleigh-scattered by air molecules before they reach the raindrop, etc. It's not really related to the first part of your question.

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