Morse-Smale complexes on convex polyhedra

Abstract

Motivated by applications in geomorphology, the aim of this paper is to extend Morse-Smale theory from smooth functions to the radial distance function (measured from an internal point), defining a convex polyhedron in 3dimensional Euclidean space. The resulting polyhedral Morse-Smale complex may be regarded, on one hand, as a generalization of the Morse-Smale complex of the smooth radial distance function defining a smooth, convex body, on the other hand, it could be also regarded as a generalization of the Morse-Smale complex of the piecewise linear parallel distance function (measured from a plane), defining a polyhedral surface. Beyond similarities, our paper also highlights the marked differences between these three problems and it also includes the design, implementation and testing of an explicit algorithm computing the Morse-Smale complex on a convex polyhedron.

Pebbles, graphs and equilibria: higher order shape descriptors for sedimentary particles

“…In [22] we named these vectors candidate gradients at p if they are tangential to the polyhedron P . Candidate gradients were defined the same way at vertices too.…”

“…In this case x is non-degenerate if H ∩ P is the (unique) face, edge or vertex of P that contains x in its relative interior. We have shown in [22] that a point x ∈ P is an equilibrium point if and only if there is no candidate gradient at x. The function r P is polyhedral Morse if all its equilibrium points are non-degenerate.…”

“…The other key concept introduced in [22] beside the extended gradient was the ascending curve. It traces out a path following the extended gradient, similarly to integral curves, but relaxes the requirement on the derivative.…”

“…First order mechanical descriptors proved to be particularly informative [6,11,25] in describing the provenance of particles (see subsection 2.3.2, and see also Figure 3.) Their generalizations, also known as second order mechanical descriptors [8,9,22,24] registered for each particle as a graph (either a so-called Morse-Smale graph or a so-called Reeb-graph, see subsection 2.3.2, see also Figure 4), also appear to be promising tools in sedimentary geomorphology. However, they are fully three-dimensional descriptors and their measurement has been prohibitive until now and they could not be applied in either laboratory or field studies.…”

“…1. Mathematical challenge: There exist at least two different kinds of secondary descriptors: Morse-Smale graphs [8,9,22] and Reeb-graphs [24]. The connection between these two secondary descriptors has not been determined until now: that is, it was not clear how the two classification schemes are related, whether one of them may be regarded as a refinement of the other or not.…”