I’ve been thinking about whether you could have a rainbow with any other shape, trying to come up with a case which is simple enough to analyze. I believe you could get a recognizable circular rainbow from long, cylindrical raindrops falling with random orientation. It’s hard to describe without pictures and equations, so the explanation below is kind of confusing.

With a cylindrical raindrop, ignoring the effect of the ends (which is what I mean by “long” above), the refraction-reflection-refraction path (RRR path) will be specular with respect to the axis of the raindrop, and behave like a spherical raindrop in the plane perpendicular to the raindrop axis. This means the raindrops contributing significantly to an observer will all lie in a plane (the plane is the one a flat mirror reflecting to the observer would be in). This makes the case simple enough to analyze, since the orientation of the raindrop can be described with a single angle, phi.

Consider a global coordinate system, with the Sun at the -Z axis, and with theta[sub]R[/sub] the angle from the +Z axis to the raindrop. Take Phi = 0 to be the orientation when the raindrop is perpendicular to the plane including the Sun, the drop, and the observer. In this case, the RRR path in the cylindrical drop will be similar to one in a spherical drop. If instead the drop is parallel to the Sun, drop, observer plane, the RRR path would be analagous to the path of a spherical drop at the +Z axis, so no visible light is sent to the observer from the RRR path. In general, the RRR path corresponds to that of a spherical drop, but as Phi varies from 0 to 90 degrees, the angle theta[sub]S[/sub] of the corresponding spherical drop varies from theta[sub]R[/sub] to 0.

Assume the drop is at an angle theta[sub]R[/sub] where Red light following the RRR path is going to the observer when Phi = 0. If we allow the orientation angle Phi to change, the color of the light going to the observer will vary from red through all the colors to violet, then no visible light. The change in the color of light will be second order near Phi = 0, so there will be more red following the RRR path, then less of the colors that are farther from red.

If the drop is at a position where a raindrop with Phi = 0 is reflecting yellow light, as Phi changes from zero, light of green, blue, and violet will go to the observer (but not as much as yellow), and there will be no orange or red light. At an angle where violet goes to the observer when Phi = 0, no other colors will follow the RRR path to get there.

So in this case, there will be color variation where you’d see a rainbow from spherical drops, probably the same colors but washed out.