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 includes the design, implementation and testing of an explicit algorithm computing the Morse-Smale complex on a convex polyhedron.

翻译：本文以地貌学的应用为动力,目的是将摩斯-马利理论从光滑功能扩展至射线距离函数(从内部点测量),界定三维欧几里德空间的锥形聚希德龙,由此形成的多面摩斯-Smal综合体可被视为光滑线距离函数摩斯-Smal综合体的概括,确定一个光滑、锥形体,另一方面,它也可被视为片度线性线性平行函数(从平面测量)摩斯-Smal综合体的概括,确定一个多面表面。除了相似之处外,我们的文件还突出这三个问题之间的显著差异,它还包括设计、实施和试验一个明确算算法,计算在康韦克斯多面上的摩斯-男性综合体。