problematic along boundaries of meshes where the underlying geometry has high curvature but few triangles, resulting in few crosses that are actually aligned with the discrete boundary of the triangle mesh. Tangential crossings are problematic because the regions produced via separatrix tracing are no longer guaranteed to be four-sided. Assuming a sufficiently fine triangle mesh along the boundary such that no separatrices exit tangentially, we observe in practice that tangential crossings on the interior typically occur in one of two cases. The first case is when one or more separatrices that approach a limit cycle are traced out for several rotations around the limit cycle. This problem is virtually eliminated by our approach of cutting off separatrices after they cross the same separatrix more than once.

The second case where tangential crossings occur is when there is a very small misalignment of singularities, such that two different separatrices follow virtually the same path. If the two separatrices are heading in opposite directions when the crossing occurs, then this problem can easily be fixed by cutting both separatrices off at the tangential crossing and combining them into a single separatrix, now connecting the two singularities in a heteroclinic orbit. If both 