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A rainbow appears curved as a semicircle (or more accurately as an arc of a circle) due to a combination of refraction and total internal reflection through spherical raindrops. The raindrops in the air are spherical in shape and act like millions of tiny prisms through which sunlight passes. Due to the differences in density of air and water, the light rays bend slightly when they move from air to water and back from water to air (after total internal reflection). Two things happen when sunlight enters the water droplets. Because different colors defract differently (at different angles), refracted light would be split into beams of different colors, giving multiple colors to the rainbow. Also, the incident light would be refracted in two perpendicular directions due to the spherical shape of a water droplet. So, depending on the location of the observer and the orientation of these water droplets, we observe a rainbow. Technically, the correct orientation and location would give rise to a cone with the rainbow at the circular end and observer at the tip of the cone. We only observe part of the circle because the horizon gets in the way and hence the semicircular shape. This is also the reason that we never see the end of a rainbow, because as soon as we move, the water droplets also change and we see a whole new rainbow.
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