Omri Azencot
Mirela Ben-Chen
Abstract
Vector fields on surfaces are fundamental in various applications in computer graphics and geometry processing. In many cases, in addition to representing vector fields, the need arises to compute their derivatives, for example, for solving partial differential equations on surfaces or for designing vector fields with prescribed smoothness properties. In this work, we consider the problem of computing the Levi-Civita covariant derivative, that is, the tangential component of the standard directional derivative, on triangle meshes. This problem is challenging since, formally, tangent vector fields on polygonal meshes are often viewed as being discontinuous, hence it is not obvious what a good derivative formulation would be. We leverage the relationship between the Levi-Civita covariant derivative of a vector field and the directional derivative of its component functions to provide a simple, easy-to-implement discretization for which we demonstrate experimental convergence. In addition, we introduce two linear which provide access to additional constructs in Riemannian geometry that are not easy to discretize otherwise, including the parallel transport operator which can be seen simply as a certain matrix exponential. Finally, we show the applicability of our operator to various tasks, such as fluid simulation on curved surfaces and vector field design, by posing algebraic constraints on the covariant derivative operator.
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- A. H. Al-Mohy and N. J. Higham. 2011. Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33, 2, 488--511. Google Scholar
Digital Library
- D. N. Arnold, R. S. Falk, and R. Winther. 2006. Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1, 1--155.Google ScholarCross Ref
- O. Azencot, M. Ben-Chen, F. Chazal, and M. Ovsjansikov. 2013. An operator approach to tangent vector field processing. Comput. Graph. Forum 32, 73--82. Google ScholarDigital Library
- G. K. Batchelor. 2000. An Introduction to Fluid Dynamics. Cambridge University Press.Google ScholarCross Ref
- M. Ben-Chen, A. Butscher, J. Solomon, and L. Guibas. 2010. On discrete Killing vector fields and patterns on surfaces. In Proceedings of the EUROGRAPHICS Symposium on Geometric Processing (SGP'10). Vol. 29. 1701--1711.Google Scholar
- D. Bommes, M. Campen, H.-C. Ebke, P. Alliez, and L. Kobbelt. 2013. Integer-grid maps for reliable quad meshing. ACM Trans. Graph. 32, 4. Google ScholarDigital Library
- D. Bommes, H. Zimmer, and L. Kobbelt. 2009. Mixed-integer quadrangulation. ACM Trans. Graph. 28, 77. Google ScholarDigital Library
- M. Botsch, L. Kobbelt, M. Pauly, P. Alliez, and B. Levy. 2010. Polygon Mesh Processing. AK Peters.Google Scholar
- M. Campen, D. Bommes, and L. Kobbelt. 2012. Dual loops meshing: Quality quad layouts on manifolds. ACM Trans. Graph. 31, 4. Google ScholarDigital Library
- K. Crane, M. Desbrun, and P. Schroder. 2010. Trivial connections on discrete surfaces. Comput. Graph. Forum 29, 1525--1533.Google ScholarCross Ref
- M. P. Do Carmo. 1976. Differential Geometry of Curves and Surfaces. Prentice-Hall.Google Scholar
- M. P. Do Carmo. 1992. Riemannian Geometry. Birkhauser.Google Scholar
- J. R. Dormand and P. J. Prince. 1980. A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6, 1, 19--26.Google ScholarCross Ref
- G. Dziuk and C. M. Elliott. 2013. Finite element methods for surface PDEs. Acta Numerica 22, 289--396.Google ScholarCross Ref
- S. Elcott, Y. Tong, E. Kanso, P. Schroder, and M. Desbrun. 2007. Stable, circulation-preserving, simplicial fluids. ACM Trans. Graph. 26, 1. Google ScholarDigital Library
- M. Fisher, P. Schroder, M. Desbrun, and H. Hoppe. 2007. Design of tangent vector fields. ACM Trans. Graph. 26, 3. Google ScholarDigital Library
- A. N. Hirani. 2003. Discrete exterior calculus. Ph.D. thesis, California Institute of Technology. http://thesis.library.caltech.edu/1885/3/thesis_hirani.pdf. Google ScholarDigital Library
- F. Kalberer, M. Nieser, and K. Polthier. 2007. Quadcover-surface parameterization using branched coverings. Comput. Graph. Forum 26, 375--384.Google ScholarCross Ref
- F. Knoppel, K. Crane, U. Pinkall, and P. Schroder. 2013. Globally optimal direction fields. ACM Trans. Graph. 32, 4, 59:1--59:10. Google ScholarDigital Library
- I. Kolar. 1993. Natural Operations in Differential Geometry. Springer.Google Scholar
- Y.-K. Lai, M. Jin, X. Xie, Y. He, J. Palacios, E. Zhang, S.-M. Hu, and X. Gu. 2010. Metric-driven RoSy field design and remeshing. IEEE Trans. Visual. Comput. Graph. 16, 1, 95--108. Google ScholarDigital Library
- L. M. Lui, Y. Wang, and T. F. Chan. 2005. Solving PDEs on manifolds with global conformal parameterization. In Variational, Geometric, and Level Set Methods in Computer Vision, Springer, 307--319. Google ScholarDigital Library
- A. J. Majda and A. L. Bertozzi. 2001. Vorticity and Incompressible Flow, Vol. 27, Cambridge University Press.Google Scholar
- M. Meyer, M. Desbrun, P. Schroder, and A. H. Baar. 2002. Discrete differential-geometry operators for triangulated 2-manifolds. In Proceedings of the VisMath Conference (VisMath'02). 35--57.Google Scholar
- C. Moler and C. Van Loan. 2003. Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45, 1, 3--49.Google ScholarDigital Library
- S. Morita. 2001. Geometry of Differential Forms. American Mathematical Society, Providence, RI.Google Scholar
- A. Myles, N. Pietroni, and D. Zorin. 2014. Robust field-aligned global parameterization. ACM Trans. Graph. 33, 4. Google ScholarDigital Library
- A. Myles and D. Zorin. 2013. Controlled-distortion constrained global parameterization. ACM Trans. Graph. 32, 4, 105:1--105:14. Google ScholarDigital Library
- P. K. Newton. 2001. The N-Vortex Problem: Analytical Techniques. Vol. 145, Springer.Google Scholar
- I. Nitschke, A. Voigt, and J. Wensch. 2012. A finite element approach to incompressible two-phase flow on manifolds. J. Fluid Mechan. 708, 418.Google ScholarCross Ref
- D. Panozzo, E. Puppo, M. Tarini, and O. Sorkine-Hornung. 2014. Frame fields: Anisotropic and non-orthogonal cross fields. ACM Trans. Graph. 33, 4. Google ScholarDigital Library
- P. Petersen. 2006. Riemannian Geometry. Springer, New York.Google Scholar
- K. Polthier. 2005. Computational aspects of discrete minimal surfaces. Global Theory Minim. Surf. 2, 65--111.Google Scholar
- K. Polthier and E. Preuss. 2003. Identifying vector field singularities using a discrete Hodge decomposition. Visual. Math. 3, 113--134.Google ScholarCross Ref
- K. Polthier and M. Schmies. 1998. Straightest geodesics polyhedral surfaces. In Mathematical Visualization, Springer, 135--150.Google ScholarCross Ref
- H. Pottmann, Q. Huang, B. Deng, A. Schiftner, M. Kilian, L. Guibas, and J. Wallner. 2010. Geodesic patterns. ACM Trans. Graph. 29, 3. Google ScholarDigital Library
- N. Ray and D. Sokolov. 2013. Tracing cross-free polylines oriented by a n-symmetry direction field on triangulated surfaces. http://arxiv.org/pdf/1306.0706.pdf.Google Scholar
- P. G. Saffman 1992. Vortex Dynamics. Cambridge University Press.Google Scholar
- L. Shi and Y. Yu. 2004. Inviscid and incompressible fluid simulation on triangle meshes. Comput. Animat. Virtual Worlds 15, 3--4, 173--181. Google ScholarDigital Library
- J. Stam. 1999. Stable fluids. In Proceedings of the 26th Annual ACM SIGGRAPH Conference on Computer Graphics and Interactive Techniques (SIGGRAPH'99). 121--128. Google ScholarDigital Library
- A. Szymczak and E. Zhang. 2012. Robust Morse decomposition of piecewise constant vector fields. IEEE Trans. Visual. Comput. Graph. 18, 6, 938--951. Google ScholarDigital Library
- M. Taylor 1996. Partial Differential Equations, Vol. III. Springer.Google Scholar
- M. Wardetzky. 2006. Discrete differential operators on polyhedral surfaces -- Convergence and approximation. Ph.D. thesis, Freie Universitat Berlin.Google Scholar
- M. Wardetzky, S. Mathur, F. Kalberer, and E. Grinspun. 2007. Discrete Laplace operators: No free lunch. In Proceedings of the Symposium on Geometry Processing (SGP'07). 33--37. Google ScholarDigital Library
- E. Zhang, K. Mischaikow, and G. Turk. 2006. Vector field design on surfaces. ACM Trans. Graph. 25, 4, 1294--1326. Google ScholarDigital Library
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- Discrete Derivatives of Vector Fields on Surfaces -- An Operator Approach
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