The quad layout immersion: a mathematically equivalent representation of a surface quadrilateral layout

Keywords

quadrilateral layout, mesh, splines, quad layout immersion

Abstract

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.

Original Publication Citation

K. M. Shepherd, R. R. Hiemstra, and T. J. R. Hughes. “The quad layout immersion: a mathematically equivalent representation of a surface quadrilateral layout,” Preprint available at arXiv:2012.09368, 2020.

Document Type

Peer-Reviewed Article

Publication Date

2020-12-17

Publisher

arXiv

Language

English

College

Ira A. Fulton College of Engineering

Department

Civil and Environmental Engineering

University Standing at Time of Publication

Graduate Student

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