Morse-Smale complexes on convex polyhedra

TL;DR

This paper extends Morse-Smale theory to convex polyhedra in 3D, providing a new framework for analyzing their geometric structure with an explicit algorithm for computation.

Contribution

It introduces a novel extension of Morse-Smale complexes to convex polyhedra and presents an algorithm for their computation.

Findings

Successfully generalized Morse-Smale complexes to convex polyhedra

Developed an explicit algorithm for computing these complexes

Validated the approach through implementation and testing

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 3-dimensional 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 relates our theory to other methods. Our work includes the design, implementation and testing of an explicit algorithm computing the Morse-Smale complex on a convex polyhedron.