Here’s a good resource with some helpful videos:
http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=44.0
Rainbows are puzzling. I heard that rainbows are supposed to be sections of a circle. But I tried to fit the following picture of a rainbow sent to me by family in Duluth, and found that a parabola fit the curve much better than a circle.
I know that parabolas and circles are both conic sections. Would a better fit be an ellipse? I tried it out, and it seems that yes, an ellipse may fit even better than the parabola.
A colleague (thanks Matt!) that told me that if you look at a rainbow from above, it’s a complete circle:
Looking at a circle from an angle, do you see an ellipse? So are rainbows often following precise mathematical elliptical curves when not seen from directly above? What determines the center of a rainbow? Is the rainbow continuing out of sight in a complete circle from another perspective?
Here are the conic sections, from a Google Image search:
“What determines the center of a rainbow? Is the rainbow continuing out of sight in a complete circle from another perspective? I’m now fascinated by these! What does determine the center of a rainbow- and the plane that it’s on? Hurrah for ellipses!”
Another colleague (thanks Natalia!) taught me about an equation for the radius of a rainbow.
“Great problem, Erik! I knew the fact about rainbows being full circles with a center hiding below the horizon, but it never occurred to me to make a math problem out of it! Now I wish I was teaching precalculus just so I could use this problem!
Mathematically, the radius of the rainbow is determined by r = d*tan(42˚), where d is how far the water droplets responsible for forming the rainbow are from your eye. The 40-42˚ angles in the formula come from the refraction index for water. Raindrops reflect sunlight at all angles from 180° (straight back to the sun) down to about 138° (red) to 140° (violet).”
She also provided this picture, not her creation (which I believe has an error within it):
What determines the plane the rainbow is on? Even if it has an error in the picture, it’s still a helpful visual. It’s so cool to see and think about an equation for the radius of the rainbow. Thinking about it now… if the picture is accurate, wouldn’t you have to take half the angle pictured? Wouldn’t it have to be r = d*tan(21˚)? But I think maybe the mistake is in the image itself. Maybe the angle shown in the image should be 84˚?
Here’s a website I just found with some good info, which seems to have the answer to the question about the center of the rainbow:
https://inlightofnature.com/how-rainbows-form/ (Links to an external site.)
Apparently, the center of the rainbow is the shadow of your head. That still doesn’t make total sense to me, but it’s making more sense than it did before!
Here’s a Geogebra thing I made because it didn’t make any sense to me how light separated out from a cloud of droplets. If you Zoom out, you can see that sure enough, the different wavelengths of light do separate out into bands of color. https://www.geogebra.org/m/pnraht55 (Links to an external site.)
Regarding the plane the rainbow is on, it seems from the article that the rainbow is the same no matter where the droplets are. That you can only see a rainbow if the sun is below 42 degrees in the sky and if you are looking at water at the right angle. But if you do see a rainbow, and it doesn’t matter how far away the droplets are from you, then shouldn’t all rainbows appear circular?
Update: it’s all starting to make a little more sense to me now. Rainbows appear circular or as ellipses. That’s because you are seeing the base of a cone. You can see the cone from an angle, and then it’s an ellipse. Now, would it be possible to view another conic section if you were looking just right, or are only elliptical (and circular) curves possible?
InSketchUp:
This shows a circle viewed from the side.