Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds
The authors define the Morse-Smale complex decomposing a piecewise linear 2-manifold and present algorithms for constructing and simplifying this complex. In many physical problems in which some measurements over a space are considered, one obtains the corresponding height function over the space. Then one can do a topological study of the geographic landscape modelled as a height function \(h: D\to\mathbb{R}\) over a two-dimensional domain \(D\). Inspired by some problems arising in this study, the authors present a solution in the piecewise linear domain. They present: (i) an algorithm for constructing a complex whose combinatorial form matches that of the Morse-Smale complex, (ii) an algorithm for deriving the Morse-Smale complex from the complex in (i) via local transformations, (iii) an algorithm for constructing a hierarchy of Morse-Smale complexes, again via local transformations, (iv) the application of the algorithms to concrete problems.
scientific article; zbMATH DE number 1969113