What causes rainbows, and why are they always curved?

Refraction, reflection and dispersion inside raindrops — and the curve is a consequence of geometry rather than of anything about the sky.

What happens in a single drop:

Light enters the drop and refracts — bending because it slows entering water.

Different wavelengths bend by different amounts. This is dispersion: violet bends most, red least, so white light separates into its component colours.

The light reflects off the inside of the back surface of the drop.

It refracts again on exiting, separating the colours further.

The critical angle. Light emerges concentrated at around 42 degrees from the direction it arrived — this is where the intensity peaks, and it is what makes a rainbow visible at all rather than a smear.

Why it is a circle. A rainbow appears wherever drops sit at that 42-degree angle from the line running from the Sun, through your head, to the antisolar point — the point directly opposite the Sun. The set of all directions at a fixed angle from a line forms a cone, and the base of that cone is a circle.

So a rainbow is genuinely a full circle. You normally see only an arc because the ground cuts off the lower half. From an aircraft or a high mountain, complete circular rainbows are visible, which confirms the geometry directly.

Consequences of this:

Every person sees a different rainbow, because it depends on the position of your own eyes. Two people standing side by side see different sets of drops.

You cannot reach the end of one. It moves as you do.

The Sun must be behind you and below about 42 degrees altitude — which is why rainbows appear near dawn and dusk and rarely at midday in summer.

The secondary bow, fainter and outside the first with reversed colours, comes from two internal reflections at around 51 degrees. The dark band between them is Alexander's band.