Biharmonic Coordinates and their Derivatives for Triangular 3D Cages
Article No.: 138, Pages 1 - 17
Abstract
As a natural extension to the harmonic coordinates, the biharmonic coordinates have been found superior for planar shape and image manipulation with an enriched deformation space. However, the 3D biharmonic coordinates and their derivatives have remained unexplored. In this work, we derive closed-form expressions for biharmonic coordinates and their derivatives for 3D triangular cages. The core of our derivation lies in computing the closed-form expressions for the integral of the Euclidean distance over a triangle and its derivatives. The derived 3D biharmonic coordinates not only fill a missing component in methods of generalized barycentric coordinates but also pave the way for various interesting applications in practice, including producing a family of biharmonic deformations, solving variational shape deformations, and even unlocking the closed-form expressions for recently-introduced Somigliana coordinates for both fast and accurate evaluations.
Supplementary Material
supplemental
- Download
- 165.50 MB
References
[1]
Mirela Ben-Chen, Ofir Weber, and Craig Gotsman. 2009. Variational harmonic maps for space deformation. ACM ToG 28, 3 (2009), 1--11.
[2]
Mario Botsch and Leif Kobbelt. 2004. An intuitive framework for real-time freeform modeling. ACM Transactions on Graphics (TOG) 23, 3 (2004), 630--634.
[3]
Christopher Brandt, Elmar Eisemann, and Klaus Hildebrandt. 2018. Hyper-reduced projective dynamics. ACM Transactions on Graphics (TOG) 37, 4 (2018), 1--13.
[4]
Qingjun Chang, Chongyang Deng, and Kai Hormann. 2023. Maximum Likelihood Coordinates. In Computer Graphics Forum, Vol. 42. Wiley Online Library, e14908.
[5]
Jiong Chen, Fernando de Goes, and Mathieu Desbrun. 2023. Somigliana Coordinates: an elasticity-derived approach for cage deformation. In Special Interest Group on Computer Graphics and Interactive Techniques Conference Conference Proceedings (SIGGRAPH '23 Conference Proceedings). 8 pages.
[6]
Lu Chen, Jin Huang, Hanqiu Sun, and Hujun Bao. 2010. Cage-based deformation transfer. Computers & Graphics 34, 2 (2010), 107--118.
[7]
Fabrizio Corda, Jean Marc Thiery, Marco Livesu, Enrico Puppo, Tamy Boubekeur, and Riccardo Scateni. 2020. Real-Time Deformation with Coupled Cages and Skeletons. Computer Graphics Forum (to appear) (2020).
[8]
Ana Dodik, Oded Stein, Vincent Sitzmann, and Justin Solomon. 2023. Variational Barycentric Coordinates. ACM Transactions on Graphics (TOG) 42, 6 (2023), 1--16.
[9]
Michael S Floater. 2003. Mean value coordinates. Computer aided geometric design 20, 1 (2003), 19--27.
[10]
Kai Hormann and Natarajan Sukumar. 2008. Maximum entropy coordinates for arbitrary polytopes. In Computer Graphics Forum, Vol. 27. Wiley Online Library, 1513--1520.
[11]
Alec Jacobson, Ilya Baran, Ladislav Kavan, Jovan Popović, and Olga Sorkine. 2012a. Fast automatic skinning transformations. ACM Transactions on Graphics (TOG) 31, 4 (2012), 1--10.
[12]
Alec Jacobson, Ilya Baran, Jovan Popović, and Olga Sorkine. 2011. Bounded Biharmonic Weights for Real-Time Deformation. ACM Transactions on Graphics (proceedings of ACM SIGGRAPH) 30, 4 (2011), 78:1--78:8.
[13]
Alec Jacobson, Elif Tosun, Olga Sorkine, and Denis Zorin. 2010. Mixed Finite Elements for Variational Surface Modeling. Computer Graphics Forum (proceedings of EUROGRAPHICS/ACM SIGGRAPH Symposium on Geometry Processing) 29, 5 (2010), 1565--1574.
[14]
Alec Jacobson, Tino Weinkauf, and Olga Sorkine. 2012b. Smooth Shape-Aware Functions with Controlled Extrema. Computer Graphics Forum (proceedings of EUROGRAPHICS/ACM SIGGRAPH Symposium on Geometry Processing) 31, 5 (2012), 1577--1586.
[15]
Pushkar Joshi, Mark Meyer, Tony DeRose, Brian Green, and Tom Sanocki. 2007. Harmonic coordinates for character articulation. ACM TOG 26, 3 (2007), 71--es.
[16]
Tao Ju, Scott Schaefer, and Joe Warren. 2005. Mean value coordinates for closed triangular meshes. In ACM Siggraph 2005 Papers. 561--566.
[17]
Torsten Langer, Alexander Belyaev, and Hans-Peter Seidel. 2006. Spherical barycentric coordinates. In Symposium on Geometry Processing. 81--88.
[18]
Yaron Lipman, Johannes Kopf, Daniel Cohen-Or, and David Levin. 2007. GPU-assisted positive mean value coordinates for mesh deformations. In SGP. 117--123.
[19]
Yaron Lipman, David Levin, and Daniel Cohen-Or. 2008. Green coordinates. ACM ToG 27, 3 (2008), 1--10.
[20]
Élie Michel and Jean-Marc Thiery. 2023. Polynomial 2D Green Coordinates for Polygonal Cages. In ACM SIGGRAPH 2023 Conference Proceedings. 1--9.
[21]
August Ferdinand Möbius. 1827. Der barycentrische Calcul; ein neues Hülfsmittel zur analytischen Behandlung der Geometrie... Mit vier Kupfertafeln. Barth.
[22]
Oded Stein, Eitan Grinspun, Alec Jacobson, and Max Wardetzky. 2019. A mixed finite element method with piecewise linear elements for the biharmonic equation on surfaces. arXiv preprint arXiv:1911.08029 (2019).
[23]
Jean-Marc Thiery and Tamy Boubekeur. 2022. Green Coordinates for Triquad Cages in 3D. In SIGGRAPH Asia 2022 Conference Papers. 1--8.
[24]
Jean-Marc Thiery, Pooran Memari, and Tamy Boubekeur. 2018. Mean value coordinates for quad cages in 3D. ACM ToG 37, 6 (2018), 1--14.
[25]
Philipp von Radziewsky, Elmar Eisemann, Hans-Peter Seidel, and Klaus Hildebrandt. 2016. Optimized subspaces for deformation-based modeling and shape interpolation. Computers & Graphics 58 (2016), 128--138.
[26]
Christoph Von-Tycowicz, Christian Schulz, Hans-Peter Seidel, and Klaus Hildebrandt. 2015. Real-time nonlinear shape interpolation. ACM Transactions on Graphics (TOG) 34, 3 (2015), 1--10.
[27]
Yu Wang, Alec Jacobson, Jernej Barbič, and Ladislav Kavan. 2015. Linear subspace design for real-time shape deformation. ACM Transactions on Graphics (TOG) 34, 4 (2015), 1--11.
[28]
Ofir Weber, Mirela Ben-Chen, Craig Gotsman, et al. 2009. Complex barycentric coordinates with applications to planar shape deformation. In Computer Graphics Forum, Vol. 28. Citeseer, 587.
[29]
Ofir Weber, Roi Poranne, and Craig Gotsman. 2012. Biharmonic coordinates. In Computer Graphics Forum, Vol. 31. Wiley Online Library, 2409--2422.
[30]
Juyong Zhang, Bailin Deng, Zishun Liu, Giuseppe Patanè, Sofien Bouaziz, Kai Hormann, and Ligang Liu. 2014. Local barycentric coordinates. ACM Transactions on Graphics (TOG) 33, 6 (2014), 1--12.
Index Terms
- Biharmonic Coordinates and their Derivatives for Triangular 3D Cages
Recommendations
Green Coordinates for Triquad Cages in 3D
SA '22: SIGGRAPH Asia 2022 Conference PapersWe introduce Green coordinates for triquad cages in 3D. Based on Green’s third identity, Green coordinates allow defining the harmonic deformation of a 3D point inside a cage as a linear combination of its vertices and face normals. Using appropriate ...
Polynomial 2D Green Coordinates for Polygonal Cages
SIGGRAPH '23: ACM SIGGRAPH 2023 Conference ProceedingsCage coordinates are a powerful means to define 2D deformation fields from sparse control points. We introduce Conformal polynomial Coordinates for closed polyhedral cages, enabling segments to be transformed into polynomial curves of any order. ...
Mean value coordinates for quad cages in 3D
Space coordinates offer an elegant, scalable and versatile framework to propagate (multi-)scalar functions from the boundary vertices of a 3-manifold, often called a cage, within its volume. These generalizations of the barycentric coordinate system ...
Comments
Information & Contributors
Information
Published In
Copyright © 2024 Copyright is held by the owner/author(s). Publication rights licensed to 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 the author(s) 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].
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 19 July 2024
Published in TOG Volume 43, Issue 4
Check for updates
Author Tags
Qualifiers
- Research-article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 133Total Downloads
- Downloads (Last 12 months)133
- Downloads (Last 6 weeks)24
Reflects downloads up to 06 Oct 2024
Other Metrics
Citations
View Options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in