In this paper we provide a new characterization of cell de-composition (called slope complex) of a given 2-dimensional continuous surface. Each patch (cell) in the decomposition must satisfy that there exists a monotonic path for any two points in the cell. We prove that any triangulation of such surface is a slope complex and explain how to obtain new slope complexes with a smaller number of slope regions decomposing the surface. We give the minimal number of slope regions by counting certain bounding edges of a triangulation of the surface obtained from its critical points.


Kropatsch, W. G., Casablanca, R. M., Batavia, D., & Gonzalez-Diaz, R. (2019). Computing and Reducing Slope Complexes. In Computational Topology in Image Context (pp. 12–25). https://doi.org/10.1007/978-3-030-10828-1_2