Two-Dimensional Critical Point Configuration Graphs

The configuration of the critical points of a smooth function of two variables is studied under the assumption that the function is Morse, that is, that all of its critical points are nondegenerate. A critical point configuration graph (CPCG) is derived from the critical points, ridge lines, and course lines of the function. Then a result from the theory of critical points of Morse functions is applied to obtain several constraints on the number and type of critical points that appear on cycles of a CPCG. These constraints yield a catalog of equivalent CPCG cycles containing four entries. The slope districts induced by a critical point configuration graph appear useful for describing the behavior of smooth functions of two variables, such as surfaces, images, and the radius function of three-dimensional symmetric axes.