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