Visualization of Morse Connection Graphs for Topologically Rich 2D Vector Fields Supplementary material

The multivalued flow induced by a PC vector field defined on a manifold surface with no boundary is shown to be upper semicontinuous and admissible in [7]. Upper semicontinuity means that the limit of any convergent sequence of trajectory segments defined on a time interval [0,h] is a trajectory segment (defined on the same time interval). This property can be viewed as an extension of flow continuity to the multivalued case. Admissibility (not to be confused with admissibility of a merger operation discussed in Section 3.5 of the paper) means that the set of trajectory segments originating from any point defined on a time interval [0,h] is acyclic (i.e. is nonempty and has trivial reduced homology). In [7], it is shown that this set is contractible (and therefore also acyclic) for multivalued flows defined by PC vector fields on manifold surfaces. Note that this property is trivially satisfied for any single valued flow, if trajectories can be followed indefinitely in time. Because of uniqueness of trajectories, the set of trajectory segments described above contains precisely one trajectory, so it is acyclic. Topological invariants such as the fixed point index and Conley index can be generalized to multivalued flows that are upper semicontinuous and admissible [2, 3, 4]. The PC framework heavily builds on these topological invariants to describe the flow structure for PC vector fields in terms of features that generalize the classical flow features. From practical standpoint, admissibility means that the flow has similar topological properties to the standard single-valued flows. The proof in [7] is restricted to PC vector fields defined on triangulated manifold surfaces without boundary, so it needs to be extended to polygonal mesh domains with a boundary, handled as described in Section 3. The extension to polygonal meshes with convex faces is trivial: the argument in [7] carries over to this case. Alternatively, one could argue that the polygonal faces could be triangulated to directly reduce the polygonal mesh case to the triangle mesh case. The case of non-empty boundary can also be reduced to the manifold surface setting. The idea is to use a PC extension of the flow to the entire plane that makes the domain (call itD) attracting – similar to the extension used to compute the a-index in this paper. Such an extension can be built by adding a quad stripes that follow the boundary of D (Figure 23). Each of the quads is incident to precisely one boundary edge. The vector field value assigned to a quad points perpendicularly toward the boundary edge it is incident to. On edges between the adjacent quads, we set the direction of the flow to point toward D (as shown in the Figure – note that such edges are either exploding or im-