research-article

Green Coordinates

Authors:

Daniel Cohen-OrAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 27, Issue 3

Pages 1 - 10

https://doi.org/10.1145/1360612.1360677

Published: 01 August 2008 Publication History

Abstract

We introduce Green Coordinates for closed polyhedral cages. The coordinates are motivated by Green's third integral identity and respect both the vertices position and faces orientation of the cage. We show that Green Coordinates lead to space deformations with a shape-preserving property. In particular, in 2D they induce conformal mappings, and extend naturally to quasi-conformal mappings in 3D. In both cases we derive closed-form expressions for the coordinates, yielding a simple and fast algorithm for cage-based space deformation. We compare the performance of Green Coordinates with those of Mean Value Coordinates and Harmonic Coordinates and show that the advantage of the shape-preserving property is not achieved at the expense of speed or simplicity. We also show that the new coordinates extend the mapping in a natural analytic manner to the exterior of the cage, allowing the employment of partial cages.

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Cited By

Thiery JMichel ÉChen J(2024)Biharmonic Coordinates and their Derivatives for Triangular 3D CagesACM Transactions on Graphics10.1145/365820843:4(1-17)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658208
de Goes FDesbrun M(2024)Stochastic Computation of Barycentric CoordinatesACM Transactions on Graphics10.1145/365813143:4(1-13)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658131
Hu JHui KLiu ZZhang HFu C(2024)CNS-Edit: 3D Shape Editing via Coupled Neural Shape OptimizationACM SIGGRAPH 2024 Conference Papers10.1145/3641519.3657412(1-12)Online publication date: 13-Jul-2024
https://dl.acm.org/doi/10.1145/3641519.3657412
Show More Cited By

Index Terms

Green Coordinates
1. Computing methodologies
  1. Artificial intelligence
    1. Computer vision
      1. Computer vision problems
        Interest point and salient region detections
      2. Image and video acquisition
        Epipolar geometry
  2. Computer graphics

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Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 27, Issue 3

August 2008

844 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/1360612

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Copyright © 2008 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 August 2008

Published in TOG Volume 27, Issue 3

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Cited By

Thiery JMichel ÉChen J(2024)Biharmonic Coordinates and their Derivatives for Triangular 3D CagesACM Transactions on Graphics10.1145/365820843:4(1-17)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658208
de Goes FDesbrun M(2024)Stochastic Computation of Barycentric CoordinatesACM Transactions on Graphics10.1145/365813143:4(1-13)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658131
Hu JHui KLiu ZZhang HFu C(2024)CNS-Edit: 3D Shape Editing via Coupled Neural Shape OptimizationACM SIGGRAPH 2024 Conference Papers10.1145/3641519.3657412(1-12)Online publication date: 13-Jul-2024
https://dl.acm.org/doi/10.1145/3641519.3657412
Ströter DThiery JHormann KChen JChang QBesler SMueller‐Roemer JBoubekeur TStork AFellner D(2024)A Survey on Cage‐based Deformation of 3D ModelsComputer Graphics Forum10.1111/cgf.1506043:2Online publication date: 30-Apr-2024
https://doi.org/10.1111/cgf.15060
Guo JZhang WYe CFu X(2024)Robust Coarse Cage Construction With Small Approximation ErrorsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2023.325520730:7(4234-4245)Online publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1109/TVCG.2023.3255207
Huang ZLiu SFu X(2024)High-order shape interpolationComputer Aided Geometric Design10.1016/j.cagd.2024.102301111(102301)Online publication date: Jun-2024
https://doi.org/10.1016/j.cagd.2024.102301
Leng BHuang JShen GWang B(2024)Shape embedding and retrieval in multi-flow deformationComputational Visual Media10.1007/s41095-022-0315-310:3(439-451)Online publication date: 8-Feb-2024
https://doi.org/10.1007/s41095-022-0315-3
Straub ASadlo FErtl T(2024)Feature-based deformation for flow visualizationJournal of Visualization10.1007/s12650-024-00963-5Online publication date: 13-May-2024
https://doi.org/10.1007/s12650-024-00963-5
Dodik AStein OSitzmann VSolomon J(2023)Variational Barycentric CoordinatesACM Transactions on Graphics10.1145/361840342:6(1-16)Online publication date: 5-Dec-2023
https://dl.acm.org/doi/10.1145/3618403
Araújo CVining NBurla SDe Oliveira MRosales ESheffer A(2023)Slippage-Preserving Reshaping of Human-Made 3D ContentACM Transactions on Graphics10.1145/361839142:6(1-18)Online publication date: 5-Dec-2023
https://dl.acm.org/doi/10.1145/3618391
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