## Mathematical Training of Lifeguards

### Total reflection

We end with a reference to an exemple that underlines a behaviour that is qualitatively different. The next figure shows (in different blue tones) three different paths that go through the same point $$C$$, placed in the separation line between two environments. Each one of them corresponds to one of two possible scenarios, here described in terms of $$A$$ and $$B$$: from $$A$$, one sees point $$B$$, apparently following the direction $$AC$$, or, for an observer in the water, to see $$A$$ from $$B$$, apparently following the direction $$BC$$.

When $$A$$ gets close to the border line of the two environments, $$B$$ gets closer to the limit half-line, with a maximum deviation, represented in orange. What will happen to a light ray emitted by $$B_{a}$$, in a direction with slope greater than the one of the half-line $$CB_{L}$$? It cannot go through the separation surface in order to satisfy the stated condition previously mentioned! In fact, that light ray reflects according to the radius $$CA_{a}$$, as shown (in green) in the same figure. The light ray reflected makes the same angle with the separation environment and, consequently, is also greater than the maximum deviation. Hence, as in the previous case, the same path can represent an observer $$A_{a}$$ seeing an object in $$B_{a},$$ apparently in the direction of $$A_{a}C$$, or an observer in $$B_{a}$$, seeing an object in $$A_{a}$$, apparently in the direction of $$B_{a}C$$. But there is a significant difference from the previous scenario: here, the image obtained has the orientation changed when compared to the original. This phenomenon, called total reflexion, is explicitly shown in the following figure.

The container represented in the first image is full of water and the second image shows us the view from an observer deep in the water, looking to the sign on the wall of the container.