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.

翻译：受地貌学应用的驱动，本文旨在将Morse-Smale理论从光滑函数推广至径向距离函数（从内部点测量），从而定义三维欧氏空间中的凸多面体。所得多面体Morse-Smale复形一方面可视为定义光滑凸体的光滑径向距离函数的Morse-Smale复形的推广，另一方面也可视为定义多面体曲面的分段线性平行距离函数（从平面测量）的Morse-Smale复形的推广。除相似性外，本文亦着重强调了这三个问题之间的显著差异，并将我们的理论与其他方法相关联。我们的工作包括设计、实现并测试一个用于计算凸多面体上Morse-Smale复形的显式算法。