### Circular lake

If the beach is located next to a large circular lake, we are faced for the first time with a change of medium, in a non-rectilinear way. When the media separation curve (in this case, a circle) has a tangent at each point (therefore also a normal), we can apply the condition previously found^{1}, but expressed relative to these normals, now with variable direction.

The following interactive application handles this case.

It is important to remember that we are looking for points that minimize time compared to other routes close enough to the one considered. This does not guarantee that they will minimize their time over all possible routes. For example, suppose the swimmer is at a point on the circumference-margin and wants to help someone at the diametrically opposite end. If you follow a straight line, the route is faster than any that are "close enough". But it is enough that the ratio between his running and his swimming speed is greater than \(1.5708\), for there to be faster courses (why?).

^{1}The condition is that of equality between the ratio of velocities in the two media and the ratio between the sines of the angles between the two directions and the normal at the point.