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Discrete differential operators on polygonal meshes

de Goes, Fernando and Butts, Andrew and Desbrun, Mathieu (2020) Discrete differential operators on polygonal meshes. ACM Transactions on Graphics, 39 (4). Art. No. 110. ISSN 0730-0301. doi:10.1145/3386569.3392389.

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Geometry processing of surface meshes relies heavily on the discretization of differential operators such as gradient, Laplacian, and covariant derivative. While a variety of discrete operators over triangulated meshes have been developed and used for decades, a similar construction over polygonal meshes remains far less explored despite the prevalence of non-simplicial surfaces in geometric design and engineering applications. This paper introduces a principled construction of discrete differential operators on surface meshes formed by (possibly non-flat and non-convex) polygonal faces. Our approach is based on a novel mimetic discretization of the gradient operator that is linear-precise on arbitrary polygons. Equipped with this discrete gradient, we draw upon ideas from the Virtual Element Method in order to derive a series of discrete operators commonly used in graphics that are now valid over polygonal surfaces. We demonstrate the accuracy and robustness of our resulting operators through various numerical examples, before incorporating them into existing geometry processing algorithms.

Item Type:Article
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Desbrun, Mathieu0000-0003-3424-6079
Additional Information:© 2020 Copyright held by the owner/author(s). We thank Athena Xenakis, Laura Hainke, and Colin Thompson for help with Figures 1 and 12, and Mark Meyer for proof-reading. Meshes are courtesy of Bay Raitt (big guy), Keenan Crane (blub, spot), David Bommes (buddha), Mario Botsch (loki), Stanford 3D Scanning Repository (bunny), and Viewpoint Animation Engineering / Sun Microsystems (cow). All other meshes and images are copyrighted by Disney/Pixar. Finally, MD gratefully acknowledges the hospitality of ShanghaiTech University during his sabbatical.
Subject Keywords:differential operators, geometry processing, polygonal meshes
Issue or Number:4
Record Number:CaltechAUTHORS:20201218-142602531
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Official Citation:Fernando De Goes, Andrew Butts, and Mathieu Desbrun. 2020. Discrete differential operators on polygonal meshes. ACM Trans. Graph. 39, 4, Article 110 (July 2020), 14 pages. DOI:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:107207
Deposited By: Tony Diaz
Deposited On:18 Dec 2020 23:08
Last Modified:16 Nov 2021 19:00

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