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Discrete differential forms and applications to surface tiling

Published: 05 June 2006 Publication History

Abstract

The geometry of manifolds has been extensively studied for centuries - though almost exclusively from a differential point of view. Unfortunately, well-established theoretical geometric foundations do not directly translate to discrete meshes: discretizations of inherently-continuous notions such as curvatures and geodesics may lose their geometric and/or variational properties.In this talk, we will introduce the notion of discrete differential forms and show how they provide differential, yet readily discretizable computational foundations [1]. We will describe how key geometric properties built into their description can more readily yield robust numerical computations which are true to the underlying continuous equations: they exactly preserve invariants of continuous models in the discrete computational realm.These discrete forms will be put to good use, first for surface flows and conformal parameterizations, then for the design of pure quadrilateral tiling of arbitrary 2-manifolds [2]. We will also briefly mention other applications (fluid animation, vector field design) benefiting greatly from this principled, discrete approach to geometry and computations.

References

[1]
Mathieu Desbrun, Eva Kanso, Yiying Tong, Discrete Differential Forms for Computational Modeling, in ACM SIGGRAPH '05/'06 Course Notes of Discrete Differential Geometry.
[2]
Yiying Tong, Pierre Alliez, David Cohen-Steiner, Mathieu Desbrun, Designing Quadrangulations with Discrete Harmonic Forms, Preprint (2006).

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cover image ACM Conferences
SCG '06: Proceedings of the twenty-second annual symposium on Computational geometry
June 2006
500 pages
ISBN:1595933409
DOI:10.1145/1137856
  • Program Chairs:
  • Nina Amenta,
  • Otfried Cheong
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

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Published: 05 June 2006

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  • (2016)Vector field processing on triangle meshesACM SIGGRAPH 2016 Courses10.1145/2897826.2927303(1-49)Online publication date: 24-Jul-2016
  • (2016)On the implementation of discrete exterior calculus2016 IEEE 11th Colombian Computing Conference (CCC)10.1109/ColumbianCC.2016.7750797(1-5)Online publication date: Sep-2016
  • (2010)Supine and Prone Colon Registration Using Quasi-Conformal MappingIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2010.20016:6(1348-1357)Online publication date: 1-Nov-2010
  • (2009)Generalized Koebe's method for conformal mapping multiply connected domains2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling10.1145/1629255.1629267(89-100)Online publication date: 5-Oct-2009
  • (2009)Surface Quasi-Conformal Mapping by Solving Beltrami EquationsProceedings of the 13th IMA International Conference on Mathematics of Surfaces XIII10.1007/978-3-642-03596-8_23(391-408)Online publication date: 27-Aug-2009
  • (2009)Recent Advances in Computational Conformal GeometryProceedings of the 13th IMA International Conference on Mathematics of Surfaces XIII10.1007/978-3-642-03596-8_11(189-221)Online publication date: 27-Aug-2009

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