“Form-finding with polyhedral meshes made simple” by Tang, Sun, Gomes, Wallner and Pottmann

  • ©Chengcheng Tang, Xiang Sun, Alexandra Gomes, Johannes Wallner, and Helmut Pottmann




    Form-finding with polyhedral meshes made simple


Session Title: Geometry Processing



    We solve the form-finding problem for polyhedral meshes in a way which combines form, function and fabrication; taking care of user-specified constraints like boundary interpolation, planarity of faces, statics, panel size and shape, enclosed volume, and last, but not least, cost. Our main application is the interactive modeling of meshes for architectural and industrial design. Our approach can be described as guided exploration of the constraint space whose algebraic structure is simplified by introducing auxiliary variables and ensuring that constraints are at most quadratic. Computationally, we perform a projection onto the constraint space which is biased towards low values of an energy which expresses desirable “soft” properties like fairness. We have created a tool which elegantly handles difficult tasks, such as taking boundary-alignment of polyhedral meshes into account, planarization, fairing under planarity side conditions, handling hybrid meshes, and extending the treatment of static equilibrium to shapes which possess overhanging parts.


    1. Block, P., and Ochsendorf, J. 2007. Thrust network analysis: A new methodology for three-dimensional equilibrium. J. Int. Assoc. Shell and Spatial Structures 48, 3, 167–173.Google Scholar
    2. Block, P. 2009. Thrust Network Analysis: Exploring Three-dimensional Equilibrium. PhD thesis, M.I.T.Google Scholar
    3. Bouaziz, S., Schwartzburg, Y., Weise, T., and Pauly, M. 2012. Shape-up: Shaping discrete geometry with projections. Comp. Graph. Forum 31, 1657–1667. Proc. SGP. Google ScholarDigital Library
    4. de Goes, F., Alliez, P., Owhadi, H., and Desbrun, M. 2013. On the equilibrium of simplicial masonry structures. ACM Trans. Graph. 32, 4, #93, 1–10. Proc. SIGGRAPH. Google ScholarDigital Library
    5. Deng, B., Bouaziz, S., Deuss, M., Zhang, J., Schwartzburg, Y., and Pauly, M. 2013. Exploring local modifications for constrained meshes. Comp. Graph. Forum 32, 2, 11–20. Proc. Eurographics.Google ScholarCross Ref
    6. Deng, B., Bouaziz, S., Deuss, M., Kaspar, A., Schwartzburg, Y., and Pauly, M. 2014. Interactive design exploration for constrained meshes. Computer-Aided Design. to appear. Google ScholarDigital Library
    7. Fraternali, F. 2010. A thrust network approach to the equilibrium problem of unreinforced masonry vaults via polyhedral stress functions. Mechanics Res. Comm. 37, 2, 198–204.Google ScholarCross Ref
    8. Glymph, J., Shelden, D., Ceccato, C., Mussel, J., and Schober, H. 2004. A parametric strategy for free-form glass structures using quadrilateral planar facets. Automation in Construction 13, 2, 187–202.Google ScholarCross Ref
    9. Gründig, L., Moncrieff, E., Singer, P., and Ströbel, D. 2000. A history of the principal developments and applications of the force density method in Germany 1970–1999. In 4th Int. Coll. Computation of Shell & Spatial Structures.Google Scholar
    10. Heyman, J. 1998. Structural Analysis: A Historical Approach. Cambridge University Press.Google ScholarCross Ref
    11. Kaspar, A., and Deng, B. 2013. Realtime deformation of constrained meshes using GPU. In Symposium on GPU Computing and Applications. to appear.Google Scholar
    12. Kotnik, T., and Weinstock, M. 2012. Material, form and force. Architectural Design 82, 104–111.Google ScholarCross Ref
    13. Linkwitz, K., and Schek, H.-J. 1971. Einige Bemerkungen zur Berechnung von vorgespannten Seilnetzkonstruktionen. Ingenieur-Archiv 40, 145–158.Google ScholarCross Ref
    14. Liu, Y., Pottmann, H., Wallner, J., Yang, Y.-L., and Wang, W. 2006. Geometric modeling with conical meshes and developable surfaces. ACM Trans. Graph. 25, 3, 681–689. Proc. SIGGRAPH. Google ScholarDigital Library
    15. Liu, Y., Wang, W., Lévy, B., Sun, F., Yan, D.-M., Lu, L., and Yang, C. 2009. On centroidal Voronoi tessellation — energy smoothness and fast computation. ACM Trans. Graph. 28, 4, #101, 1–17. Google ScholarDigital Library
    16. Liu, Y., Xu, W., Wang, J., Zhu, L., Guo, B., Chen, F., and Wang, G. 2011. General planar quadrilateral mesh design using conjugate direction field. ACM Trans. Graph. 30, 6, #140, 1–10. Proc. SIGGRAPH Asia. Google ScholarDigital Library
    17. Liu, Y., Pan, H., Snyder, J., Wang, W., and Guo, B. 2013. Computing self-supporting surfaces by regular triangulation. ACM Trans. Graph. 32, 4, #92, 1–10. Proc. SIGGRAPH. Google ScholarDigital Library
    18. Panozzo, D., Block, P., and Sorkine-Hornung, O. 2013. Designing unreinforced masonry models. ACM Trans. Graph. 32, 4, #91, 1–12. Proc. SIGGRAPH. Google ScholarDigital Library
    19. Peng, C.-H., Barton, M., Jiang, C., and Wonka, P. 2014. Exploring quadrangulations. ACM Trans. Graph. 33, 1, #12, 1–12. Google ScholarDigital Library
    20. Poranne, R., Chen, R., and Gotsman, C., 2013. On linear spaces of polyhedral meshes. ArXiv:1303.4110.Google Scholar
    21. Poranne, R., Ovreiu, E., and Gotsman, C. 2013. Interactive planarization and optimization of 3D meshes. Comp. Graph. Forum 32, 1, 152–163.Google ScholarCross Ref
    22. Pottmann, H., Huang, Q.-X., Yang, Y.-L., and Hu, S.-M. 2006. Geometry and convergence analysis of algorithms for registration of 3D shapes. Int. J. Computer Vision 67, 3, 277–296. Google ScholarDigital Library
    23. Pottmann, H., Liu, Y., Wallner, J., Bobenko, A., and Wang, W. 2007. Geometry of multi-layer freeform structures for architecture. ACM Trans. Graph. 26, 3, #65, 1–11. Proc. SIGGRAPH. Google ScholarDigital Library
    24. Schiftner, A., and Balzer, J. 2010. Statics-sensitive layout of planar quadrilateral meshes. In Advances in Architectural Geometry 2010. Springer, 221–236.Google Scholar
    25. Schiftner, A., Leduc, N., Bompas, P., Baldassini, N., and Eigensatz, M. 2012. Architectural geometry from research to practice — the Eiffel Tower Pavilions. In Advances in Architectural Geometry 2012. Springer, 213–228.Google Scholar
    26. Takayama, K., Panozzo, D., Sorkine-Hornung, A., and Sorkine-Hornung, O. 2013. Sketch-based generation and editing of quad meshes. ACM Trans. Graph. 32, 4, #97, 1–8. Google ScholarDigital Library
    27. Umetani, N., Igarashi, T., and Mitra, N. J. 2012. Guided exploration of physically valid shapes for furniture design. ACM Trans. Graph. 31, 4, #86, 1–11. Proc. SIGGRAPH. Google ScholarDigital Library
    28. Vaxman, A. 2012. Modeling polyhedral meshes with affine maps. Comp. Graph. Forum 31, 1647–1656. Proc. SGP. Google ScholarDigital Library
    29. Vouga, E., Höbinger, M., Wallner, J., and Pottmann, H. 2012. Design of self-supporting surfaces. ACM Trans. Graph. 31, 4, #87, 1–11. Proc. SIGGRAPH. Google ScholarDigital Library
    30. Yang, Y., Yang, Y., Pottmann, H., and Mitra, N. 2011. Shape space exploration of constrained meshes. ACM Trans. Graph. 30, 6, #124, 1–11. Proc. SIGGRAPH Asia. Google ScholarDigital Library
    31. Zadravec, M., Schiftner, A., and Wallner, J. 2010. Designing quad-dominant meshes with planar faces. Comp. Graph. Forum 29, 5, 1671–1679. Proc. SGP.Google ScholarCross Ref

ACM Digital Library Publication: