research-article

N-symmetry direction field design

Authors:

Bruno LévyAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 27, Issue 2

Article No.: 10, Pages 1 - 13

https://doi.org/10.1145/1356682.1356683

Published: 08 May 2008 Publication History

Abstract

Many algorithms in computer graphics and geometry processing use two orthogonal smooth direction fields (unit tangent vector fields) defined over a surface. For instance, these direction fields are used in texture synthesis, in geometry processing or in nonphotorealistic rendering to distribute and orient elements on the surface. Such direction fields can be designed in fundamentally different ways, according to the symmetry requested: inverting a direction or swapping two directions might be allowed or not.

Despite the advances realized in the last few years in the domain of geometry processing, a unified formalism is still lacking for the mathematical object that characterizes these generalized direction fields. As a consequence, existing direction field design algorithms are limited to using nonoptimum local relaxation procedures.

In this article, we formalize N-symmetry direction fields, a generalization of classical direction fields. We give a new definition of their singularities to explain how they relate to the topology of the surface. Specifically, we provide an accessible demonstration of the Poincaré-Hopf theorem in the case of N-symmetry direction fields on 2-manifolds. Based on this theorem, we explain how to control the topology of N-symmetry direction fields on meshes. We demonstrate the validity and robustness of this formalism by deriving a highly efficient algorithm to design a smooth field interpolating user-defined singularities and directions.

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Cited By

Capouellez RZorin D(2024)Seamless Parametrization in Penner CoordinatesACM Transactions on Graphics10.1145/365820243:4(1-13)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658202
Corman E(2024)Curvature-Driven Conformal DeformationsACM Transactions on Graphics10.1145/365814543:4(1-16)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658145
Simons LAmenta N(2024)Anisotropy and Cross FieldsComputer Graphics Forum10.1111/cgf.1513243:5Online publication date: 5-Aug-2024
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Index Terms

N-symmetry direction field design

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Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 27, Issue 2

April 2008

54 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/1356682

Issue’s Table of Contents

Copyright © 2008 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 08 May 2008

Accepted: 01 February 2008

Revised: 01 October 2007

Received: 01 May 2007

Published in TOG Volume 27, Issue 2

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Cited By

Capouellez RZorin D(2024)Seamless Parametrization in Penner CoordinatesACM Transactions on Graphics10.1145/365820243:4(1-13)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658202
Corman E(2024)Curvature-Driven Conformal DeformationsACM Transactions on Graphics10.1145/365814543:4(1-16)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658145
Simons LAmenta N(2024)Anisotropy and Cross FieldsComputer Graphics Forum10.1111/cgf.1513243:5Online publication date: 5-Aug-2024
https://doi.org/10.1111/cgf.15132
Zhang CYu ZShi JLi YXu WGuo ZZhang HZhu ZQiang S(2024)A complex model decomposition algorithm based on 3D frame fields and featuresEngineering Computations10.1108/EC-01-2023-003741:1(237-258)Online publication date: 9-Feb-2024
https://doi.org/10.1108/EC-01-2023-0037
Wang JWestermann RGao XWu J(2024)Design and optimization of functionally-graded triangular lattices for multiple loading conditionsComputer Methods in Applied Mechanics and Engineering10.1016/j.cma.2024.117335432(117335)Online publication date: Dec-2024
https://doi.org/10.1016/j.cma.2024.117335
Zhang YJi YZhu C(2024)Multi-patch parameterization method for isogeometric analysis using singular structure of cross-fieldComputers & Mathematics with Applications10.1016/j.camwa.2024.03.001162(61-78)Online publication date: May-2024
https://doi.org/10.1016/j.camwa.2024.03.001
Zheng WWu HXu GLing RGu R(2024)Feature-preserving quadrilateral mesh Boolean operation with cross-field guided layout blendingComputer Aided Geometric Design10.1016/j.cagd.2024.102324111(102324)Online publication date: Jun-2024
https://doi.org/10.1016/j.cagd.2024.102324
Dotse KMouysset VPernet S(2024)Quadrilateral Mesh of Non-simply Connected Domain and Non-planar Surfaces From a Given Cross-FieldSIAM International Meshing Roundtable 202310.1007/978-3-031-40594-5_14(293-312)Online publication date: 21-Mar-2024
https://doi.org/10.1007/978-3-031-40594-5_14
Coiffier GCorman E(2023)The Method of Moving Frames for Surface Global ParametrizationACM Transactions on Graphics10.1145/360428242:5(1-18)Online publication date: 20-Sep-2023
https://dl.acm.org/doi/10.1145/3604282
Levi Z(2023)Seamless Parametrization with Cone and Partial Loop ControlACM Transactions on Graphics10.1145/360008742:5(1-22)Online publication date: 30-Aug-2023
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