“Discrete Connection and Covariant Derivative for Vector-Field Analysis and Design” by Wang, Goes, Tong and Desbrun

  • ©Beibei Wang, Fernando de Goes, Yiying Tong, and Mathieu Desbrun




    Discrete Connection and Covariant Derivative for Vector-Field Analysis and Design

Session/Category Title: MESHES & FIELDS




    In this article, we introduce a discrete definition of connection on simplicial manifolds, involving closed-form continuous expressions within simplices and finite rotations across simplices. The finite-dimensional parameters of this connection are optimally computed by minimizing a quadratic measure of the deviation to the (discontinuous) Levi-Civita connection induced by the embedding of the input triangle mesh, or to any metric connection with arbitrary cone singularities at vertices. From this discrete connection, a covariant derivative is constructed through exact differentiation, leading to explicit expressions for local integrals of first-order derivatives (such as divergence, curl, and the Cauchy-Riemann operator) and for L2-based energies (such as the Dirichlet energy). We finally demonstrate the utility, flexibility, and accuracy of our discrete formulations for the design and analysis of vector, n-vector, and n-direction fields.


    1. R. Abraham, J. E. Marsden, and R. Ratiu. 1988. Manifolds, Tensor Analysis, and Applications, 2nd ed. Springer-Verlag. 
    2. David Bommes, Henrik Zimmer, and Leif Kobbelt. 2009. Mixed-integer quadrangulation. ACM Trans. Graph. 28, 3 (2009), 77:1–77:10. 
    3. Keenan Crane, Mathieu Desbrun, and Peter Schröder. 2010. Trivial connections on discrete surfaces. Comp. Graph. Forum 29, 5 (2010), 1525–1533.
    4. Fernando de Goes, Beibei Liu, Max Budninskiy, Yiying Tong, and Mathieu Desbrun. 2014. Discrete 2-tensor fields on triangulations. Comp. Graph. Forum 33, 5 (2014), 13–24. 
    5. Mathieu Desbrun, Eva Kanso, and Yiying Tong. 2008. Discrete differential forms for computational modeling. In Discrete Differential Geometry, A. I. Bobenko, P. Schröder, J. M. Sullivan, and G. M. Ziegler (Eds.). Oberwolfach Seminars, Vol. 38. Birkhäuser Basel, 287–324.
    6. Matthew Fisher, Peter Schröder, Mathieu Desbrun, and Hugues Hoppe. 2007. Design of tangent vector fields. ACM Trans. Graph. 26, 3 (2007), 56:1–56:9. 
    7. Cindy M. Grimm and John F. Hughes. 1995. Modeling surfaces of arbitrary topology using manifolds. In Proc. ACM SIGGRAPH Conf. 359–368. 
    8. Aaron Hertzmann and Denis Zorin. 2000. Illustrating smooth surfaces. In Proc. ACM SIGGRAPH Conf. 517–526. 
    9. Scott Kircher and Michael Garland. 2008. Free-form motion processing. ACM Trans. Graph. 27, 2 (2008), 12:1–12:13. 
    10. M. S. Knebelman. 1951. Spaces of relative parallelism. Annals of Mathematics (1951), 387–399.
    11. Felix Knöppel, Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2013. Globally optimal direction fields. ACM Trans. Graph. 32, 4 (2013), 59:1–59:10. 
    12. Yaron Lipman, Olga Sorkine, David Levin, and Daniel Cohen-Or. 2005. Linear rotation-invariant coordinates for meshes. ACM Trans. Graph. 24, 3 (2005), 479–487. 
    13. Beibei Liu, Yanlin Weng, Jiannan Wang, and Yiying Tong. 2013. Orientation field guided texture synthesis. J. Comp. Sci. Tech. 28, 5 (2013), 827–835.
    14. Kishore Marathe. 2010. Topics in Physical Mathematics. Springer Science.
    15. Ashish Myles, Nico Pietroni, and Denis Zorin. 2014. Robust field-aligned global parametrization. ACM Trans. Graph. 33, 4 (2014), 135:1–135:8. 
    16. Ashish Myles and Denis Zorin. 2013. Controlled-distortion constrained global parametrization. ACM Trans. Graph. 32, 4 (2013), 105:1–105:14. 
    17. Jonathan Palacios and Eugene Zhang. 2007. Rotational symmetry field design on surfaces. ACM Trans. Graph. 26, 3 (2007), 55:1–55:10. 
    18. Daniele Panozzo, Yaron Lipman, Enrico Puppo, and Denis Zorin. 2012. Fields on symmetric surfaces. ACM Trans. Graph. 31, 4 (2012), 111:1–111:12. 
    19. Konrad Polthier and Eike Preuß. 2000. Variational approach to vector field decomposition. In Data Visualization. 147–155.
    20. Konrad Polthier and Eike Preuß. 2003. Identifying vector field singularities using a discrete Hodge decomposition. In Vis. and Math. III, Hans-Christian Hege and Konrad Polthier (Eds.). Springer Verlag, 113–134.
    21. Nicolas Ray and Dmitry Sokolov. 2014. Robust polylines tracing for n-symmetry direction field on triangle surfaces. ACM Trans. Graph. 33, 3 (2014), 30:1–30:8. 
    22. Nicolas Ray, Bruno Vallet, Laurent Alonso, and Bruno Levy. 2009. Geometry-aware direction field processing. ACM Trans. Graph. 29, 1 (2009), 1:1–1:11. 
    23. Nicolas Ray, Bruno Vallet, Wan Chiu Li, and Bruno Lévy. 2008. N-symmetry direction field design. ACM Trans. Graph. 27, 2 (2008), 10:1–10:13. 
    24. Michael Spivak. 1979. A Comprehensive Introduction to Differential Geometry. Vol. II, 2nd ed. Publish or Perish.
    25. Holger Theisel. 2002. Designing 2d vector fields of arbitrary topology. Comp. Graph. Forum 21, 3 (2002), 595–604.
    26. Yiying Tong, Santiago Lombeyda, Anil N. Hirani, and Mathieu Desbrun. 2003. Discrete multiscale vector field decomposition. ACM Trans. Graph. 22, 3 (2003), 445–452. 
    27. Ke Wang, Weiwei, Yiying Tong, Mathieu Desbrun, and Peter Schröder. 2006. Edge subdivision schemes and the construction of smooth vector fields. ACM Trans. Graph. 25, 3 (2006), 1041–1048. 
    28. Y. Wang, B. Liu, and Y. Tong. 2012. Linear surface reconstruction from discrete fundamental forms on triangle meshes. Comp. Graph. Forum 31, 8 (2012), 2277–2287. 
    29. H. Whitney. 1957. Geometric Integration Theory. Princeton University Press.
    30. Eugene Zhang, Konstantin Mischaikow, and Greg Turk. 2006. Vector field design on surfaces. ACM Trans. Graph. 25, 4 (2006), 1294–1326.

ACM Digital Library Publication:

Overview Page: