RAINBOW
The bow is a caustic — dD/dθ = 0 on a sphere: red 42.3° outside violet 40.6°, and a dark band above.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A single raindrop, and a shaft of sunlight. Most pictures draw the rainbow as a prism bent into an arc — but the drop is a sphere, and the arc is built from an accident of calculus: the one angle where the exit ray stops moving.
The drop
Light refracts entering the drop, reflects off the far inside wall, and refracts again on the way out. Every impact height gives a different total turn — and traced together, the outgoing rays are not spread evenly at all.
Deviation
Follow one ray. Its deviation is a smooth function of where it struck the drop, and that function has a floor. Rays that strike near the floor all leave at nearly the same angle — they crowd together, and crowding is brightness.
The caustic
The stationary point of the deviation curve is the primary rainbow. Each color crowds at its own angle: red lies on the outside and violet on the inside.
Audit
Everything here is closed form. The stationary angle solves algebraically, Snell holds at both faces to the last bit, the secondary bow inverts the colors, and between the two bows lies a band no ray can reach — Alexander’s dark band, named in 200 AD.