Vision of several drops

The following question naturally arises: if we fix one of these observers situated on that conical surface and who sees such a bright red droplet, where will all the (other) droplets (illuminated by the sun) that are seen in red under the same conditions by this particular observer be?

The condition is that the vertex angle at the center of each of these drops, between the direction of the sun and the direction of the observer, is \(42.5^{\circ}\), but that angle is precisely equal to vertex angle on the observer, between the (variable) ray joining the observer to each of these drops and the (fixed) ray starting from the sun towards the observer... Therefore, the observer, with his back to the sun, sees as red all the sunlit drops that are in the cone centered on the observer, having as their axis a straight line with the direction of the sun's rays and as a semi-opening the angle of \(42,5^{\circ} \). In the following figure is represented such a drop observer cone seen as red and several cones with vertices in drops seen as red, all of them tangent to the large one.

What was done for red can be repeated for rays \(C_{3}\) of other colors, thus obtaining several cones with smaller apertures and other colors, inside the red one that we considered first. And similar conclusions can be expressed from what has been seen, now for the direction of the rays of class \(C_{4}\), leading to cones with greater opening, with the same axis as the previous ones, this time the red being the one with smallest semi-opening \((50,1^{\circ})\) among all \(C_{4}\). The set of these cones, all having as their axis the straight line from the sun to the observer, is represented in the figure below.

The following figures show the distribution of some of the colors the viewer should see under the best conditions: a set of concentric colored circles (centered on the viewer's shadow, produced by the sun); and an enlargement of a part.

The comparison of those circles with an existing aerial photograph in [3] is suggested.

From the Earth, it is possible to see almost entire rainbows from the tops of mountains, but in general we have to make do with parts of the upper arch, with the concavity facing downwards. The reason lies in the fact that rays from the anterior downward-facing conical surfaces reach the earth's surface before encountering drops illuminated by the sun.