continuum, streamlines can only cross each other orthogonally [@viertel_approach_2019], numerical inaccuracies often lead to tangential crossings of streamlines. This is especially true near singularities, where large changes in direction occur over arbitrarily small lengths.

We make three primary contributions:

1.  We extend the diffusion generated method for cross field design in [@viertel_approach_2019] to curved surfaces. This method which has previously only been described in 2D tends to have good singularity placement in locations where singularities occur because of boundary curvature rather than Gaussian curvature of the surface, a common scenario for CAD surfaces.

2.  We prove that near singularities, streamlines of a cross field are hyperbolic under a conformal map. This results in a simple method to compute streamlines in the neighborhood of a singularity that prevents tangential crossings.

3.  We describe a novel algorithm for simplifying a quad partition with T-junctions such that the number of partition components strictly decreases and the number of T-junctions decreases monotonically.

A triangle mesh $T$ representing a surface $M$ in ${\mathbb R}^3$.

A quad layout with T-junctions $Q$ partitioning $M$ into four-sided regions.

1\. Compute a smooth cross field on $\mathrm{T}$. 2. Construct a quad layout with T-junctions by tracing out separatrices of 