@gould_automated_2012], computing the Morse-Smale complex [@dong_spectral_2006], surface foliations [@lei_generalized_2017], dual loops [@campen_dual_2012], simplifying an existing quad mesh [@tarini_simple_2011; @bommes_global_2011], and more recently, cross field based approaches [@kowalski_pde_2013; @bommes_integer-grid_2013; @razafindrazaka_perfect_2015; @fogg_automatic_2015; @campen_quantized_2015; @zhang_automatic_2016; @pietroni_tracing_2016]. While in this discussion we focus on cross field based streamline tracing approaches, Campen \[4\] provides an excellent literature review on quad patching algorithms.

In cross field based streamline tracing approaches, irregular node placement is determined by computing singular points of the cross field, where simple singularities of positive or negative index correspond to irregular nodes of valence 3 or 5 respectively. The connectivity of the layout is then determined by either tracing out raw separatrices of the cross field, or by using the isolines of an underlying parameterization. A common problem with separatrix tracing approaches is that on a discrete geometry, singularities are never perfectly aligned. In practice, this frequently causes limit cycles and very thin regions to occur within the quad layout. This is problematic for meshing because very small mesh elements are required. Further, the *base complex* of the mesh is often far more complicated than necessary, mitigating the benefits of a multi-block decomposition. Another problem is that despite the fact that in the 