article

Mean value coordinates

Author:

Michael S. FloaterAuthors Info & Claims

Computer Aided Geometric Design, Volume 20, Issue 1

Pages 19 - 27

Published: 01 March 2003 Publication History

Abstract

We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its neighbouring vertices. The coordinates are motivated by the Mean Value Theorem for harmonic functions and can be used to simplify and improve methods for parameterization and morphing.

References

[1]

Eck, M., DeRose, T., Duchamp, T., Hoppe, H., Lounsbery, M., Stuetzle, W., 1995. Multiresolution analysis of arbitrary meshes. In: Computer Graphics Proceedings, SIGGRAPH 95, pp. 173-182.

[2]

Floater, M.S., 1997. Parametrization and smooth approximation of surface triangulations. Computer Aided Geometric Design 14, 231-250.

Digital Library

[3]

Floater, M.S., 1998. Parametric filings and scattered data approximation. Intern. J. Shape Modeling 4, 165-182.

[4]

Floater, M.S., t.a. One-to-one piecewise linear mappings over triangulations. Math. Comp., to appear.

[5]

Floater, M.S., Gotsman, C., 1999. How to morph tilings injectively. J. Comput. Appl. Math. 101, 117-129.

Digital Library

[6]

Gotsman, C., Surazhsky, V., 2001. Guaranteed intersection-free polygon morphing. Computers and Graphics 25-1, 67-75.

[7]

Iserles, A., 1996. A First Course in Numerical Analysis of Differential Equations. Cambridge University Press.

[8]

Loop, C., DeRose, T., 1989. A multisided generalization of Bézier surfaces. ACM Trans. Graph. 8, 204-234.

Digital Library

[9]

Meyer, M., Lee, H., Barr, A., Desbrun, M., 2001. Generalizing barycentric coordinates to irregular n-gons. Preprint. Caltech.

[10]

Pinkall, U., Polthier, K., 1993. Computing discrete minimal surfaces and their conjugates. Exp. Math. 2, 15-36.

[11]

Sibson, R., 1981. A brief description of natural neighbour interpolation. In: Barnett, V. (Ed.), Interpreting Multivariate Data. John Wiley, Chichester, pp. 21-36.

[12]

Surazhsky, V., Gotsman, C., t.a. Intrinsic morphing of compatible triangulations. Preprint.

[13]

Tutte, W.T., 1963. How to draw a graph. Proc. London Math. Soc. 13, 743-768.

[14]

Wachspress, E., 1975. A Rational Finite Element Basis. Academic Press.

[15]

Warren, J., 1996. Barycentric coordinates for convex polytopes. Adv. Comp. Math. 6, 97-108.

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
Segall ARen JVaxman ASorkine-Hornung O(2024)Fabric Tessellation: Realizing Freeform Surfaces by SmockingACM Transactions on Graphics10.1145/365815143:4(1-20)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658151
Aigerman NGroueix T(2024)Generative Escher MeshesACM SIGGRAPH 2024 Conference Papers10.1145/3641519.3657452(1-11)Online publication date: 13-Jul-2024
https://dl.acm.org/doi/10.1145/3641519.3657452
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Index Terms

Mean value coordinates

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Published In

cover image Computer Aided Geometric Design

Computer Aided Geometric Design Volume 20, Issue 1

March 2003

57 pages

ISSN:0167-8396

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Publisher

Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 01 March 2003

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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
Segall ARen JVaxman ASorkine-Hornung O(2024)Fabric Tessellation: Realizing Freeform Surfaces by SmockingACM Transactions on Graphics10.1145/365815143:4(1-20)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658151
Aigerman NGroueix T(2024)Generative Escher MeshesACM SIGGRAPH 2024 Conference Papers10.1145/3641519.3657452(1-11)Online publication date: 13-Jul-2024
https://dl.acm.org/doi/10.1145/3641519.3657452
Tian XGünther T(2024)A Survey of Smooth Vector Graphics: Recent Advances in Repr esentation, Creation, Rasterization, and Image VectorizationIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2022.322057530:3(1652-1671)Online publication date: 1-Mar-2024
https://dl.acm.org/doi/10.1109/TVCG.2022.3220575
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
Bunge AAlexa MBotsch MKim J(2023)Discrete Laplacians for General Polygonal and Polyhedral MeshesSIGGRAPH Asia 2023 Courses10.1145/3610538.3614620(1-49)Online publication date: 6-Dec-2023
https://dl.acm.org/doi/10.1145/3610538.3614620
Wang YGuo MSolomon J(2023)Variational quasi-harmonic maps for computing diffeomorphismsACM Transactions on Graphics10.1145/359210542:4(1-26)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592105
van de Ruit MEisemann E(2023)Metameric: Spectral Uplifting via Controllable Color ConstraintsACM SIGGRAPH 2023 Conference Proceedings10.1145/3588432.3591565(1-10)Online publication date: 23-Jul-2023
https://dl.acm.org/doi/10.1145/3588432.3591565
Michel ÉThiery J(2023)Polynomial 2D Green Coordinates for Polygonal CagesACM SIGGRAPH 2023 Conference Proceedings10.1145/3588432.3591499(1-9)Online publication date: 23-Jul-2023
https://dl.acm.org/doi/10.1145/3588432.3591499
Yuan YSun YLai YMa YJia RKobbelt LGao L(2023)Interactive NeRF Geometry Editing With Shape PriorsIEEE Transactions on Pattern Analysis and Machine Intelligence10.1109/TPAMI.2023.331506845:12(14821-14837)Online publication date: 1-Dec-2023
https://dl.acm.org/doi/10.1109/TPAMI.2023.3315068
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