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Quadrilateral layouts on surfaces are valuable in texture mapping, andessential in generation of quadrilateral meshes and in fitting splines.Previous work has characterized such layouts as a special metric on a surfaceor as a meromorphic quartic differential with finite trajectories. In thiswork, a surface quadrilateral layout is alternatively characterized as aspecial immersion of a cut representation of the surface into the Euclideanplane. We call this a quad layout immersion. This characterization, while posedin smooth topology, naturally generalizes to piecewise-linear representations.As such, it mathematically describes and generalizes integer grid maps, whichare common in computer graphics settings. Finally, the utility of therepresentation is demonstrated by computationally extracting quadrilaterallayouts on surfaces of interest.

The Quad Layout Immersion: A Mathematically Equivalent Representation of a Surface Quadrilateral Layout