Variational Tangent Plane Intersection for Planar Polygonal Meshing

  • Henrik Zimmer
  • Marcel Campen
  • Ralf Herkrath
  • Leif Kobbelt


Several theoretical and practical geometry applications are based on polygon meshes with planar faces. The planar panelization of freeform surfaces is a prominent example from the field of architectural geometry. One approach to obtain a certain kind of such meshes is by intersection of suitably distributed tangent planes. Unfortunately, this simple tangent plane intersection (TPI) idea has a number of limitations. It is restricted to the generation of hexagon-dominant meshes: as vertices are in general defined by three intersecting planes, the resulting meshes are basically duals of triangle meshes. Furthermore, the explicit computation of intersection points requires dedicated handling of special cases and degenerate constellations to achieve robustness on freeform surfaces. Another limitation is the small number of degrees of freedom for incorporating design parameters.

Using a variational re-formulation, we equip the concept of TPI with additional degrees of freedom and present a robust, unified approach for creating polygonal structures with planar faces that is readily able to integrate various objectives and constraints needed in different applications scenarios. We exemplarily demonstrate the abilities of our approach on three common problems in geometry processing.


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  1. ALEXA, M., AND WARDETZKY, M. 2011. Discrete Laplacians on general polygonal meshes. ACM Trans. Graph. 30, 4, 102:1–102:10.CrossRefGoogle Scholar
  2. BOBENKO, A., AND SURIS, Y. 2008. Discrete Differential Geometry: Integrable Structure. Graduate Studies in Mathematics. American Mathematical Society.Google Scholar
  3. BOBENKO, A. I., HOFFMANN, T., AND SPRINGBORN, B. A. 2003. Minimal surfaces from circle patterns: Geometry from combinatorics. Annals of Mathematics 164, 231–264.CrossRefGoogle Scholar
  4. BOMMES, D., ZIMMER, H., AND KOBBELT, L. 2010. Practical Mixed-Integer Optimization for Geometry Processing. In Curves and Surfaces, Springer, J.-D. Boissonnat, P. Chenin, A. Cohen, C. Gout, T. Lyche, M.-L. Mazure, and L. L. Schumaker, Eds., vol. 6920 of Lecture Notes in Computer Science, 193–206.Google Scholar
  5. COHEN-STEINER, D., ALLIEZ, P., AND DESBRUN, M. 2004. Variational shape approximation. ACM Trans. Graph. 23, 3, 905–914.CrossRefGoogle Scholar
  6. CUTLER, B., AND WHITING, E. 2007. Constrained planar remeshing for architecture. In Proceedings of Graphics Interface 2007, ACM, GI’ 07, 11–18.Google Scholar
  7. EIGENSATZ, M., KILIAN, M., SCHIFTNER, A., MITRA, N. J., POTTMANN, H., AND PAULY, M. 2010. Paneling architectural freeform surfaces. ACM Trans. Graph. 29, 4, 45:1–45:10.CrossRefGoogle Scholar
  8. FU, C.-W., LAI, C.-F., HE, Y., AND COHEN-OR, D. 2010. K-set tilable surfaces. ACM Trans. Graph. 29, 44:1–44:6.CrossRefGoogle Scholar
  9. 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.CrossRefGoogle Scholar
  10. MÖBIUS, J., AND KOBBELT, L. 2010. OpenFlipper: An Open Source Geometry Processing and Rendering Framework. In Curves and Surfaces, Springer, J.-D. Boissonnat, P. Chenin, A. Cohen, C. Gout, T. Lyche, M.-L. Mazure, and L. L. Schumaker, Eds., vol. 6920 of Lecture Notes in Computer Science, 488–500.Google Scholar
  11. MÜLLER, C. 2011. Conformal hexagonal meshes. Geometriae Dedicata 154, 27–46.CrossRefGoogle Scholar
  12. POTTMANN, H., BRELL-COKCAN, S., AND WALLNER, J. 2007. Discrete surfaces for architectural design. In Curves and Surface Design: Avignon 2006, P. Chenin, T. Lyche, and L. L. Schumaker, Eds. Nashboro Press, 213–234.Google Scholar
  13. 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, 65:1–65:11.Google Scholar
  14. SINGH, M., AND SCHAEFER, S. 2010. Triangle surfaces with discrete equivalence classes. ACM Trans. Graph. 29, 46:1–46:7.CrossRefGoogle Scholar
  15. TRAUTZ, M., AND HERKRATH, R. 2009. The application of folded plate principles on spatial structures with regular, irregular and free-form geometries. In Proc. IASS, 1019–1031.Google Scholar
  16. TROCHE, C. 2008. Planar hexagonal meshes by tangent plane intersection. In Advances in Architectural Geometry.Google Scholar
  17. WÄCHTER, A., AND BIEGLER, L. T. 2006. On the implementation of an interiorpoint filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming 106, 25–57.CrossRefGoogle Scholar
  18. WANG, W., LIU, Y., YAN, D., CHAN, B., LING, R., AND SUN, F. 2008. Hexagonal meshes with planar faces. Tech. Rep. TR-2008-13, Department of Computer Science, The University of Hong Kong.Google Scholar
  19. YANG, Y.-L., YANG, Y.-J., POTTMANN, H., AND MITRA, N. J. 2011. Shape space exploration of constrained meshes. ACM Trans. Graph. 30, 6, 124:1–124:12.Google Scholar
  20. ZADRAVEC, M., SCHIFTNER, A., AND WALLNER, J. 2010. Designing quaddominant meshes with planar faces. Comput. Graph. Forum 29, 5, 1671–1679.CrossRefGoogle Scholar
  21. ZIMMER, H., CAMPEN, M., BOMMES, D., AND KOBBELT, L. 2012. Rationalization of Triangle-Based Point-Folding Structures. Computer Graphics Forum 31, 2, 611–620.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag/Wien 2013

Authors and Affiliations

  • Henrik Zimmer
    • 1
  • Marcel Campen
    • 1
  • Ralf Herkrath
    • 2
  • Leif Kobbelt
    • 1
  1. 1.Computer Graphics GroupRWTH Aachen UniversityAachenGermany
  2. 2.RWTH Aachen UniversityAachenGermany

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