In the topic "Mathematical Training of Lifeguards" some basic properties of the paths of light rays that cross media in which light travels at different speeds, for example air and water (or glass), were discussed. We analyzed particularly simple situations, with rays situated in a plane, and where there were only two media, the separation line being a straight line. It was also mentioned that the path of the light ray minimizes (locally) the time spent and this leads to a similarity with a situation which is qualitatively of the same type: that of a lifeguard helping a bather in trouble... And we concluded that the ratio between the "offsets" of the arrival and departure rays, in relation to the normal at the point of change of the medium, is equal to the ratio between the speeds in the two media. In the case of light, this ratio is called the refractive index of one medium relative to another.

Different Beaches

Let's start by considering some situations similar to those already discussed, but in which either there is more than one change of medium during the course, or there is a separation of media that is done through a curved line (not necessarily a straight line), or both of these conditions occur.

These are examples of topics that could hypothetically be included in a graduate course for lifeguards...


Let's try to understand how sometimes, on days when you see both sun and rain (or at least some area with cloudiness), a dazzling-looking rainbow appears!

• There is a general idea about the reason for the formation of a rainbow, which is more or less present in people who have some basic notions about the phenomena of refraction and who, at the very least, master the notions that were mentioned in the first part of this text and in the text "Mathematical Training of Lifeguards". That idea goes something like this: water droplets in the atmosphere4 cause a scattering of white sunlight into rays of different colors, in a way similar to that observed in the case of the prism, the third example dealt with in the first part of this text. And that is what is at the base of the observed phenomenon. But let's suppose that someone wants to know more details, for example: why the colors are in that regular (and beautiful) arrangement and not in another, why the radii of the arcs of a circumference are those and not others and also for what reason, often, outside the stronger rainbow, another one appears with less intense lights and with the inverse order of colors, in relation to the order of colors of the strongest.

• Since, undoubtedly, the rainbow has, in some way, to do with the effect of the drops on the sun's rays that fall on them, it is this effect that we must begin by analyzing in detail.

4 Whether it's rainwater drops or drops (usually much smaller) forming clouds or cloudiness.

Translated for Atractor by a CMUC team, from its original version in Portuguese. Atractor is grateful for this cooperation.

This work integrates interactive components in CDF format prepared with the Mathematica program. To use these files, you must download them to your computer and access them with the CDF Player, which can be downloaded for free from

This text is a slightly modified version of the following article (in Portuguese) published by Atractor in Gazeta de Matemática

(*) The interactive aplications included throughout this text were developed under a grant by FCT - Fundação para a Ciência e a Tecnologia.

Difficulty level: University