  A generalization of vector fields, referred to as N-direction fields or cross
fields when N = 4, has been recently introduced and studied for geometry
processing, with applications in quadrilateral (quad) meshing, texture mapping,
and parameterization. We make the observation that cross field design for
two-dimensional quad meshing is related to the well-known Ginzburg-Landau
problem from mathematical physics. This yields a variety of theoretical tools
for efficiently computing boundary-aligned quad meshes, with provable
guarantees on the resulting mesh, such as the number of mesh defects and bounds
on the defect locations. The procedure for generating the quad mesh is to (i)
find a complex-valued "representation" field that minimizes the Ginzburg-Landau
energy subject to a boundary constraint, (ii) convert the representation field
into a boundary-aligned, smooth cross field, (iii) use separatrices of the
cross field to partition the domain into four sided regions, and (iv) mesh each
of these four-sided regions using standard techniques. Leveraging the
Ginzburg-Landau theory, we prove that this procedure can be used to produce a
cross field whose separatrices partition the domain into four sided regions. To
minimize the Ginzburg-Landau energy for the representation field, we use an
extension of the Merriman-Bence-Osher (MBO) threshold dynamics method,
originally conceived as an algorithm to simulate mean curvature flow. Finally,
we demonstrate the method on a variety of test domains.
