A rainbow seems to be an arch with two ends touching the ground. That shape is a trick of where we are standing. The familiar primary rainbow is part of a complete circle centred on the direction exactly opposite the Sun. At ground level, the horizon usually hides the lower portion. From an aircraft or high viewpoint, with sunlit droplets below, the whole ring can sometimes be visible.¹,²
To see why a circle appears, it helps to stop imagining the rainbow as a coloured object fixed over a distant field. It is a pattern of directions from which particular colours of light reach your eyes. Move the observer and the set of droplets supplying that pattern changes.
Start with one drop
A round raindrop is a tiny optical system. Sunlight entering it bends, or refracts, at the water surface. Some light reflects from the back of the drop, then bends again as it leaves. Different wavelengths bend by slightly different amounts, separating the white sunlight into colours. For the primary rainbow, light is especially concentrated in directions around 42 degrees from the line directly away from the Sun; red appears toward the outer edge and blue-violet toward the inner edge.¹,³
Not every ray entering a drop reaches your eye. The useful ray must emerge toward you at the right angle. Picture looking at a vast sheet of sunlit droplets. At any instant, only some of those droplets send the strongly concentrated red light to your eyes; a slightly different set sends the blue-violet light. Together, they define coloured rings in your view. The precise angle depends on colour, which is why the bands do not lie on top of one another.
The important geometry is not that droplets arrange themselves into a circle in the sky. They can be scattered through a shower. The circle is the set of viewing directions that make roughly the right angle with the incoming sunlight. Rotate that direction all the way around the line opposite the Sun and it sweeps out a cone; where you look through that cone, you see a circular bow.¹,³
The circle's hidden centre
Imagine a straight line from the Sun, through your head, and on into the sky ahead of you. Its far direction is called the antisolar point. For a primary rainbow, the outer red band appears at an angular radius of about 42 degrees around that point.²,³
This makes a simple prediction. Suppose the Sun is only 10 degrees above the horizon. The antisolar point is 10 degrees below the horizon, so the top of the red ring is roughly 32 degrees above it. Raise the Sun to 30 degrees and the top is only around 12 degrees high. The example ignores small optical and viewing complications, but it shows why rainbows look lower when the Sun is higher. If the Sun rises beyond about 42 degrees, the primary bow lies below the horizon for a person standing on level ground.¹,⁴
The same geometry explains why the “ends” are not true ends. The visible colours simply reach the point where the rest of the ring is blocked by ground or where there are no suitably illuminated droplets in view. No one can walk to an endpoint and stand beneath a permanent strip of colour. As you walk, the line through your own head changes, and so does the set of droplets sending light to you.
This observer-dependence is subtle. Two people standing beside one another share the same shower and sunlight, but each receives light from a slightly different selection of drops. They can agree that they see a rainbow while not literally receiving the same rays from the same drops.²
When can the rest appear?
Height removes the most familiar obstruction. Looking down from an aircraft, a mountaintop or another high position can put sunlit droplets beneath the observer. If drops occupy the needed directions all around the antisolar point, the lower half of the bow has somewhere to appear. NASA has published an example of a full-circle rainbow photographed over Norway, noting that a complete ring is more commonly visible from the air.²
A high viewpoint alone is not sufficient. The Sun must be behind you, light must reach the droplets, and drops must occupy the relevant portion of the cone. Clouds, terrain or the aircraft itself can hide sections. A full circle is possible under suitable conditions, not a guaranteed reward for every flight.
It is also easy to mistake other circular optical effects for a full rainbow. A glory, sometimes seen around an aircraft's shadow on cloud, consists of much smaller coloured rings produced by a different light-scattering process. The National Weather Service describes glories as distinct from the large primary rainbow.⁵ A ring of colour over clouds is therefore not automatically the missing lower half of an ordinary bow.
A circle made by your viewpoint
The everyday arch is a useful example of how an observation can make an underlying pattern look incomplete. The drop-level physics picks out a near-fixed angle. Rotating that angle around the antisolar direction produces a circle. The ground and the available rain decide how much of it you can actually see.
This is why photographs of circular rainbows feel startling even though they obey the same rules as the half-circle over a field. Nothing new has happened to the water. The photographer has gained a view of the directions the ground normally conceals.