Morse-Smale complexes on convex polyhedra (Q6607819)

Classical Morse theory investigates the critical points of a smooth function defined on a smooth manifold, according to \textit{J. W. Milnor} [Morse theory. Based on lecture notes by M. Spivak and R. Wells., Annals of Mathematics Studies. No. 51. Princeton, N. J.: Princeton University Press, vi, 153 p. (1963; Zbl 0108.10401)].\par The present paper aims to extend Morse-Smale theory by a geometrical approach: the authors analyze the properties of the radial distance function (measured from an inner point) defined on a polyhedral surface. As a consequence, the notions of Morse-Smale complex and Morse-Smale graph on a polyhedral surface are introduced. The paper, which is motivated by particle shape modelling in geomorphology (see, for example, \textit{G. Domokos} et al. [Math. Geosci. 42, No. 1, 29--47 (2010; Zbl 1185.68593)], includes an explicit algorithm for computing the Smale-Morse complex generated by a generic convex polyhedron.\par A detailed comparison with other generalizations of Morse-Smale theory is also presented, especially regarding the so called discrete problem, piecewise linear problem, point set problem.