Quadrilateral layouts on surfaces are essential for defining quadrilateral meshes, for fitting splines for engineering design and analysis, and for texture mapping in computer graphics. Previous work has characterized such layouts as a special mapping with extensive integer constraints, 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 represent the current state-of-the-art. Due to its generality, the proposed representation supports a range of different computational methods to extract quadrilateral layouts. This is demonstrated by several computational examples, including the extraction of quadrilateral layouts on a 1996 Dodge Neon.
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