al. [@knoppel_globally_2013] and Jakob et al. [@jakob_instant_2015], but also has an advantage over these methods on surfaces with boundary, which are common in CAD and meshing for FEM, because it tends to place singularities symmetrically even when the surface is flat (see ).

![Left to right, a comparison of singularity placement from cross fields described in Knöppel et al. [@knoppel_globally_2013], Viertel and Osting [@viertel_approach_2019], and Jakob et al. [@jakob_instant_2015]. Singularities with positive and negative index are shown in blue and red respectively.[]{data-label="fig:disks"}](disks-trimmed.png){width="\linewidth"}

In this section, we extend the diffusion generated method for cross field design to curved surfaces. We use this method of cross field design as the first step in our pipeline (\[alg:ps-overview\]). In \[sec:surfaces\], we first review some tools from the literature that have been used previously to develop cross field design methods on surfaces. In \[sec:discrete-MBO\], we describe the implementation details of our method.

Cross Fields on Surfaces {#sec:surfaces}
------------------------

The main difficulty in extending flat 2D methods to curved surfaces is the lack of a global coordinate system. In this section, we use concepts from differential geometry to formulate the cross field design problem on 2-manifolds. We consider a smooth, orientable 2-manifold $M$ embedded in ${\mathbb 