eJMT Abstract

Interactive geometry programs make it possible to explore critical points of functions of one and many variables, in courses at various levels and in research projects. In this article, we show how easy-to-use Java applets illustrate critical point phenomena for families of curves and surfaces from several different viewpoints. Our methods introduce circular and spherical images of parametric curves and surfaces, leading to results on the tangential degree of a closed curve in the plane, and the Hopf Theorem on the normal degree of a surface in three-space. We also indicate the relationship with the basic critical point theorem of Marston Morse concerning pits, peaks, and passes of a function of two variables.