Delineation of basins and hills by Morse theory and critical nets

Abstract

The delineation of two‐dimensional ascending and descending manifolds represents the theoretical basis for a large number of applications in which functions are used to describe phenomena related to climate, economy, or engineering, to mention only a few. Whereas the applications are related to the pits, passes, peaks, courses, ridges, basins, and hills, of mathematical interest are the corresponding critical points, separatrices as well as two‐dimensional ascending and descending manifolds. The present article demonstrates how the boundaries of the latter, which represent the pre‐images of basins and hills, can be characterized in a graph‐theoretic way. An algorithm for their extraction, which is based on a newly proved theorem, is presented together with its implementation in C#. Finally, the modus operandi of the algorithm is illustrated by two examples, thereby demonstrating how it works even in the case of surfaces with topologically complicated structures.