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Edge subdivision schemes and the construction of smooth vector fields

Authors:

Mathieu Desbrun,

Peter SchröderAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 25, Issue 3

Pages 1041 - 1048

https://doi.org/10.1145/1141911.1141991

Published: 01 July 2006 Publication History

Abstract

Vertex- and face-based subdivision schemes are now routinely used in geometric modeling and computational science, and their primal/dual relationships are well studied. In this paper, we interpret these schemes as defining bases for discrete differential 0- resp. 2-forms, and complete the picture by introducing edge-based subdivision schemes to construct the missing bases for discrete differential 1-forms. Such subdivision schemes map scalar coefficients on edges from the coarse to the refined mesh and are intrinsic to the surface. Our construction is based on treating vertex-, edge-, and face-based subdivision schemes as a joint triple and enforcing that subdivision commutes with the topological exterior derivative. We demonstrate our construction for the case of arbitrary topology triangle meshes. Using Loop's scheme for 0-forms and generalized half-box splines for 2-forms results in a unique generalized spline scheme for 1-forms, easily incorporated into standard subdivision surface codes. We also provide corresponding boundary stencils. Once a metric is supplied, the scalar 1-form coefficients define a smooth tangent vector field on the underlying subdivision surface. Design of tangent vector fields is made particularly easy with this machinery as we demonstrate.

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References

[1]

Arnold, D. N., Falk, R. S., and Winther, R. 2006. Finite Element Exterior Calculus, Homological Techniques, and Applications. Acta Numerica 15.

[2]

Biermann, H., Levin, A, and Zorin, D. 2000. Piecewise Smooth Subdivision Surfaces with Normal Control. Comp. Graphics (ACM/SIGGRAPH Proc.), 113--120.

Digital Library

[3]

Bossavit, A 1998. Computational Electromagnetism. Academic Press, Boston.

[4]

Catmull, E, and Clark, J. 1978. Recursively Generated B-Spline Surfaces on Arbitrary Topological Meshes. Comp. Aid Des. 10, 6, 350--355.

[5]

Desbrun, M., Kanso. E., and Tong, Y. 2005. Discrete Differential Forms for Computational Modeling. In Discrete Differential Geometry, E. Grinspun, P. Schröder, and M. Desbrun, Eds., Course Notes. ACM SIGGRAPH.

[6]

Doo, D., and Sabin, M. 1978. Analysis of the Behaviour of Recursive Division Surfaces near Extraordinary Points. Comp. Aid. Des. 10, 6, 356--360.

[7]

Dyn, N., Gregory, J. A. and Levin, D. 1987. A 4-point Interpolatory Subdivision Scheme for Curve Design, Comput. Aided Geom. Des. 4, 4, 257--268.

Digital Library

[8]

Elcott, S., Tong, Y, Kanso, E., Schröder P., and Desbrun, M. 2005. Stable, Circulation-Preserving, Simplicial Fluids. Submitted for Publication.

[9]

Grinspun. E., Krysl, P. and Schröder P 2002, CHARMS: A Simple Framework for Adaptive Simulation. ACM Trans. on Graph. 21, 3, 281--290.

Digital Library

[10]

Halstead, M., Kass, M., and DeRose, T. 1993. Efficient, Fair Interpolation using Catmull-Clark Surfaces. Comp. Graphics (ACM/SIGGRAPH Proc.), 35--44.

Digital Library

[11]

Hiptmair, R. 2001. Higher-Order Whitney Forms. Progress in Electromagnetics Research 32, 271--299.

[12]

Khodakovsky, A, Schröder P, and Sweldens, W 2000. Progressive Geometry Compression. Comp. Graphics (ACM/SIGGRAPH Proc.), 271--278.

Digital Library

[13]

Kobbelt, L. 2000. √3 Subdivision. Comp. Graphics (ACM/SIGGRAPH Proc.), 103--112.

Digital Library

[14]

Litke, N., Levin, A, and Schröder P 2001. Fitting Subdivision Surfaces. In Visualization '01, 319--324.

Digital Library

[15]

Loop, C. 1987. Smooth Subdivision Surfaces Based on Triangles. Master's thesis, University of Utah, Department of Mathematics.

[16]

Oswald, P, and Schröder, P. 2003. Composite Primal/Dual √3 Subdivision Schemes. Comput. Aided Geom. Des. 20, 5, 135--164.

Digital Library

[17]

Prautzsch, H., Boehm W. and Paluszny, M. 2002. Bézier and B-Spline Techniques. Springer.

Digital Library

[18]

Reif. U. 1995. A Unified Approach to Subdivision Algorithms Near Extraordinary Points. Comput. Aided Geom. Des. 12, 153--174.

Digital Library

[19]

Reif, U. 1999. Analyse und Konstruktion von Subdivisionsalgorithmen für Freiform-flächen beliebiger Topologie. Shaker Verlag. Habilitationsschrift.

[20]

Reynolds, C. W. 1999. Steering Behaviors For Autonomous Characters. In Game Developers Conference.

[21]

Schlick, C. 1994. An Inexpensive BRDF Model for Physically-Based Rendering. Comput. Graph. Forum 13, 3, 233--246.

[22]

Shi, L., and Yu, Y. 2005. Taming Liquids for Rapidly Changing Targets. In ACM/EG Symp. on Comp. Anim., 229--236.

Digital Library

[23]

Shiue, L.-J., Alliez. P, Ursu, R., and Kettner, L., 2005. A Tutorial on CGAL Polyhedron for Subdivision Algorithms.

[24]

Stam, J. 1998. Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter Values. Comp. Graphics (ACM/SIGGRAPH Proc.), 395--404.

Digital Library

[25]

Tong, Y, Lombeyda, S., Hirani, A. N., and Desbrun, M. 2003. Discrete Multiscale Vector Field Decomposition. ACM Trans. on Graph. 22, 3, 445--452.

Digital Library

[26]

Turk, G. 2001. Texture Synthesis on Surfaces. Comp. Graphics (ACM/SIGGRAPH Proc.), 347--354.

Digital Library

[27]

Wang, K. 2006. Online Companion to "Edge Subdivision Schemes and the Con struction of Smooth Vector Fields". Tech. rep., Caltech.

Digital Library

[28]

Warren, J., and Weimer H. 2001. Subdivision Methods for Geometric Design: A Constructive Approach, first ed. Morgan Kaufman Publishers.

Digital Library

[29]

Whitney. H. 1957. Geometric Integration Theory. Princeton University Press.

[30]

Zorin, D., and Kristjansson, D. 2002. Evaluation of Piecewise Smooth Subdivision Surfaces. Visual Computer 18, 5--6, 299--315.

[31]

Zorin, D., and Schröder P., Eds. 2000. Subdivision for Modeling and Animation. Course Notes. ACM SIGGRAPH.

[32]

Zorin, D. 2000. A Method for Analysis of C1-Continuity of Subdivision Surfaces. SIAM J. Numer. Anal. 37, 5, 1677--1708.

Digital Library

[33]

Zorin, D. 2000. Smoothness of Subdivision on Irregular Meshes. Constructive Approximation 16, 3, 359--397.

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Edge subdivision schemes and the construction of smooth vector fields

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Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 25, Issue 3

July 2006

742 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/1141911

Issue’s Table of Contents

Copyright © 2006 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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Publication History

Published: 01 July 2006

Published in TOG Volume 25, Issue 3

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Saleh HTeixeira F(2024)A Study on Particle Trajectory Error in Finite-Element Particle-in-Cell AlgorithmsIEEE Transactions on Plasma Science10.1109/TPS.2024.338836952:4(1546-1554)Online publication date: Apr-2024
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