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

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.

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Document Details

Document Type
Technical Report
Publication Date
Dec 01, 2020
Accession Number
AD1146616

Entities

People

  • Kendrick M. Shepherd
  • RenĂ© R. Hiemstra
  • Thomas J.R. Hughes

Organizations

  • University of Texas at Austin

Tags

Communities of Interest

  • Air Platforms

DTIC Thesaurus Topics

  • Algebraic Topology
  • Cellular Structures
  • Computational Fluid Dynamics
  • Computational Science
  • Computations
  • Computer-Aided Design
  • Coordinate Systems
  • Curvature
  • Differential Geometry
  • Geometric Forms
  • Geometry
  • Lines (Geometry)
  • Materials
  • Mathematical Analysis
  • Theorems
  • Topology
  • Two Dimensional

Readers

  • Finite Element Method (FEM) for solving Partial Differential Equations (PDEs)
  • Integrated Circuit Design and Technology.
  • Linear Algebra