Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== For a ray undergoing one internal reflection inside a sphere (the primary rainbow geometry), the ray-deviation function (in radians) can be written as ===

D(i)  =  π+2i−4r,D(i)
\;=\; \pi + 2 i - 4 r,D(i)=π+2i−4r,
where iii is the incidence angle (outside) and rrr is the refraction angle inside the droplet. Snell’s law gives:

sin⁡i=nsin⁡r⟹r=arcsin⁡ ⁣(sin⁡in).\sin i = n \sin r \quad\Longrightarrow\quad r = \arcsin\!\big(\tfrac{\sin i}{n}\big).sini=nsinr⟹r=arcsin(nsini).
The physically observed bright rainbow corresponds to the minimum of D(i)D(i)D(i) as iii varies; call that Dmin⁡D_{\min}Dmin. The commonly quoted “rainbow zenith angle” (the angular radius from the anti-solar point to the bow) is

β  =  180∘−Dmin⁡(degrees).\beta \;=\; 180^\circ - D_{\min} \quad(\text{degrees}).β=180∘−Dmin(degrees).
Notes: all angles above are treated in degrees where convenient; the formula above is the standard geometric ray-tracing expression for the one-internal-reflection ray.