The problem in space

We will now face a similar problem, but which, unlike the previous one, cannot be represented on a plane, since it requires a representation in a three dimensional space. Let us suppose we have a lake, where fish swim at different depths, and over which birds fly at different altitudes. And suppose4 that those birds nosedive into the water to catch fish. What would be the best possible choice for the bird? If we assume that the flight velocity (in the air) is constant5 and greater than the velocity of diving (in the water), also constant, we will have a very similar mathematical problem to the previous one: for each pair of positions, we obtain exactly the previous problem. For a certain position of the bird, the following figure represents the several paths to reach different fish, and also shows a surface that we can denote by “spherical over time”, formed by points in the water that the bird can reach in a given time (the same for all). In a certain way, the picture would allow the bird to determine which fish could be reached quickly.

Similarly, the surface in the next picture would allow fish to know which birds are more threatening, the ones that could catch them quickly.

4 This scenario is not a fantasy, on the contrary, it can be often observed in certain regions.
5 For a reason of simplicity of the model, we are not taking into account the changes in the velocity that are dependent on the direction of the flight, and that should be considered, given the effect of gravity. We are also supposing that the fish does not change its position while the bird is nosediving, and that there is no wind in the air nor currents in the water.