### Time circumference

We will call the set of all points from where the lifeguard can reach the swimmer in exactly the same time \(t\) the "time circumference" of radius \(t\), centered on the swimmer. Is the shape of the figure obtained dependent of the "radius" \(t?\) If we look at this question with a slightly different approach from the one we have been considering (that is, with the lifeguard on the beach), we can also consider the possibility of him being already in the water.

In the previous figure, the "time circumference" is formed by an arc of circumference (in the water), two line segments that join the extremities of the arc to the coast line, and a curve on the beach. In the third image, the fastest path is formed by three line segments; one perpendicular to the external line segment, another in the coast line, and the third one joining the previous one to the position of the swimmer. Hence, the previous conjecture was false. It is not true that, when both are in the water, the best path is always a straight line, as in the fifth image. In fact, the solution presented here is a natural one because if both are close to the coast line but very far away from each other, it is better for the lifeguard to swim a little bit, run along the coast and then go back to the water, reaching the swimmer. Finally, we should note that in the position corresponding to the fourth image, there are two possible minimal paths, starting at the same point.

The following interactive application gives us the "time circumference" for different choices of the distance between the swimmer and the coast line, the ratio between the velocities and the value of \(t\), and, for each one of them and for each one of its points, it allows us to draw optimal "radial" paths from the lifeguard to the swimmer (usually one path, rarely two).

- Choose the distance to the coast, the ratio between velocities and the value of \(t\).
- If you choose to see the radius, you can choose a point on the circumference, manipulating the cursor correspondent to \(\theta\). For a slower variation of \(\theta\), press the Alt key while moving the mouse cursor.
- If you click on the image, you will obtain a zoomed image centered in the point where the cursor is.