# Morse-Smale complexes on convex polyhedra

@article{Ludmany2021MorseSmaleCO, title={Morse-Smale complexes on convex polyhedra}, author={Bal'azs Ludm'any and Zsolt L'angi and G{\'a}bor Domokos}, journal={ArXiv}, year={2021}, volume={abs/2106.11626} }

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…

## References

SHOWING 1-10 OF 24 REFERENCES

### A solution to some problems of Conway and Guy on monostable polyhedra

- MathematicsBulletin of the London Mathematical Society
- 2022

A convex polyhedron is called monostable if it can rest in stable position only on one of its faces. The aim of this paper is to investigate three questions of Conway, regarding monostable polyhedra,…

### Topological Classification of Morse Functions and Generalisations of Hilbert’s 16-th Problem

- Mathematics
- 2007

The topological structures of the generic smooth functions on a smooth manifold belong to the small quantity of the most fundamental objects of study both in pure and applied mathematics. The problem…

### A USER'S GUIDE TO DISCRETE MORSE THEORY

- Mathematics
- 2002

A number of questions from a variety of areas of mathematics lead one to the problem of analyzing the topology of a simplicial complex. However, there are few general techniques available to aid us…

### A Practical Approach to Morse-Smale Complex Computation: Scalability and Generality

- Computer ScienceIEEE Transactions on Visualization and Computer Graphics
- 2008

A new algorithm and easily extensible framework for computing MS complexes for large scale data of any dimension where scalar values are given at the vertices of a closure-finite and weak topology (CW) complex, therefore enabling computation on a wide variety of meshes such as regular grids, simplicial meshes, and adaptive multiresolution (AMR) meshes is described.

### Hierarchical Morse—Smale Complexes for Piecewise Linear 2-Manifolds

- MathematicsDiscret. Comput. Geom.
- 2003

Algorithm for constructing a hierarchy of increasingly coarse Morse—Smale complexes that decompose a piecewise linear 2-manifold by canceling pairs of critical points in order of increasing persistence is presented.

### Morse theory by perturbation methods with applications to harmonic maps

- Mathematics
- 1981

There are many interesting variational problems for which the PalaisSmale condition cannot be verified. In cases where the Palais-Smale condition can be verified for an approximating integral, and…

### Static Equilibria of Rigid Bodies: Dice, Pebbles, and the Poincare-Hopf Theorem

- MathematicsJ. Nonlinear Sci.
- 2006

By appealing to the Poincare-Hopf Theorem on topological invariants, we introduce a global classification scheme for homogeneous, convex bodies based on the number and type of their equilibria. We…

### A Genealogy of Convex Solids Via Local and Global Bifurcations of Gradient Vector Fields

- MathematicsJ. Nonlinear Sci.
- 2016

1- and 2-parameter families of convex bodies connecting members of adjacent primary and secondary classes are constructed and it is shown that transitions between them can be realized by codimension 1 saddle-node and saddle–saddle bifurcations in the gradient vector fields.

### Balancing polyhedra

- MathematicsArs Math. Contemp.
- 2020

It is proved that the mechanical complexity of a class $(S,U)^E$ is zero if, and only if there exists a convex polyhedron with $S$ faces and $U$ vertices.

### A topological classification of convex bodies

- Mathematics
- 2016

The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted…