### Water drop

The following figure represents a cut of a drop of water through a plane with an incident ray and the ray paths in which that incident ray unfolds. As it is a simpler case, we will initially consider the horizontal incident ray, coming from the left and with a color corresponding to a given frequency and a given index of refraction.

The ray, when it hits the drop at a point \(P_{1}\), emits two rays, one \(C_{1}\), reflecting towards the outside and the other obtained by refraction, inside the drop, uniting \( P_{1}\) to another point \(P_{2}\). Once there, a part of the ray refracts, this time to the outside in a ray \(C_{2}\) of origin \(P_{2}\) and another is reflected at \(P_{2}\) on the tangent to the edge of the drop, until it reaches the point \(P_{3}\) and there again it subdivides into a ray refracted outwards \(C_{3}\) and another reflected \(P_{3}P{4 }\), which refracts on a \(C_{4}\) to the outside. Of course, in each of these subdivisions of a ray into a reflected and a refracted one, there is a loss of intensity for each of the results. It is therefore natural to start by observing the possible contribution of the first four rays emerging from this drop - \(C_{1}\), \(C_{2}\), \(C_{3}\), \(C_{ 4}\) - for any relevant luminous phenomenon observable in the sky. Now, these 4 rays all depend on the point of impact of the ray incident on the drop (the point \(P_{1}\)), which can be parameterized by the ordinate of \(P_{1}\), varying between \(-r\) and \(r\), if we call the radius of the drop \(r\).

• In the following interactive application you can vary the impact point and the color of the incident ray and analyze that variation in the path of the rays \(C_{1}\), \(C_{2}\), \(C_{3}\) and \(C_{4}\).