Robust Morse Decompositions of Piecewise Constant Vector Fields Supplementary Material

! Fig. 1. The Morse set containing a large periodic orbit in the diesel engine dataset (7 refinement steps). Fig. 2. Results for a height field on the original 1536-triangle mesh (left) and its subdivided version (three subdivision iterations). Note that only a source and two saddles are visible from this viewpoint. Fig. 3. Morse sets and connecting regions for the figure eight model subdivided three times. In this section, we show that trajectories of a PC vector field, defined on a fine enough mesh and constructed from a good enough vertex-based approximation of a smooth vector field g are close to the trajectories of g. Let g be a smooth vector field on a smooth compact manifold M embedded in R 3 and a triangulated manifold surface M be a G 1-approximation of M , i.e. the vertices of M are close to M and the triangle normals approximate M 's normals at nearby points. The approximate vertex based vector field on M assigns the vector ¯ g(v) = g(π(v)) + µ(v) to a vertex v, where π is a function that maps vertices of M into nearby points of M and µ(v) represents a perturbation, that can include the noise or measurement uncertainty (Figure 4).