Why is the shape of rainbow curved instead of straight?
First, let's look at what causes a rainbow. The basic concept is the same as a prism, where white light is separated into its separate components by refraction.
In refraction, light is "bent" because of it slows down or speeds up in different media. In our example, light will slow down in water, which (through some advanced math) you can show will change its angle. Well, the angle and the amount that the light slows down is dependent on its wavelength (color), so some wavelengths will be bent more than others.
Because different colors are bent different amounts, when you put white light...
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To answer the question that why the shape of a rainbow curved and not straight, one should first understand that the rainbow is not something that is present physically in the sky. It is the image formed in the eye by refraction, total internal reflection & subsequent refraction of parallel rays of sun light coming from the back of the observer through the rain drops in our front.
If one closely observes the shape of the rainbow, it is like an arc of a circle with the center on the line determined by the observer's eye and the sun. Let us call this line the the rainbow-axis.
Now consider the plain of the rainbow as a plain perpendicular to the rainbow axis with four quadrants.
The rain-drops and sun-rays at an angle of zero will form the image at zero angle in the rainbow plain. Similarly the ones at 45 degrees will form the portion at 45 degrees and so on. Thus the rain drops in each quadrant will form the corresponding image to complete the rainbow.
Rainbow of sizes larger than a semi-circle have been observed in the valleys while standing on a hill. I have personally seen such rainbows with the secondary and tertiary rainbows as well.