Catastrophe theory applied to the refraction of traffic (Q920827)

Fifty years ago \textit{T. F. Palander} [``Beiträge zur Standortstheorie'', Almqvist \& Wiksell, Uppsala 1935] and \textit{H. von Stackelberg} [Das Brechungsgesetz des Verkehrs, Jahrbücher f. Nationalökonomie u. Statistik 148 (1938), 680 ff.] discovered that transportation routes that cross two different media, like land and water, with different transportation costs obey the same refraction law at the boundary as do light rays passing two media with different refraction indices in optics. The idea of this article is to see what happens when a set of transportation routes from sea pass across for instance a circular coastline. Perhaps caustics, like those observed in a cup coffee can occur? This might be an interesting observation as in the transportation case the counterpart to the highly illuminated caustic would be an extreme concentration of traffic and could be used to explain the formation of communication and settlement patterns. The problem is easy to treat analytically under the idealized assumptions that the coastline is perfectly circular and that transportation routes are straight lines. Catastrophe theory ensures that the results obtained in the exemplificatory case are of general validity.