Optimized Ruled Surfaces with an Application to Thin-Walled Concrete Shells

  • Kevin Noack
  • Daniel LordickEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 809)


For lightweight structures in the field of architecture and civil engineering, concrete shells with negative Gaussian curvature are frequently used. One class of such surfaces are the skew ruled surfaces. To model such surfaces for the purpose of form-finding, we use the line geometry model of the Study sphere in the space of dual vectors. It allows the mapping of lines of the three-dimensional Euclidean space into points of the four-dimensional model space. The correspondence of minimal ruled surfaces, which are the helicoids, with geodesics on the dual unit sphere can be handled with the dual Rodrigues formula. This paper presents a proof of the formula and extends it to a general form, which avoids exceptions like parallel rulings. This approach also speeds up the interpolation algorithms for form-finding. The line geometry model, as implemented in Rhinoceros3D’s plug-in Grasshopper, was used to design a small thin-walled footbridge of concrete in cooperation with the TU Berlin. The formwork was prepared with a hot-wire foam cutter at the TU Dresden.


Geodesic interpolation Dual numbers Ruled surfaces Rodrigues formula Exponential mapping Footbridge Carbon-reinforced concrete 



This work is part of the research project “Thin-walled Concrete Structures with Line Geometry” funded by the German Research Foundation (DFG) as part of the SPP 1542. Two theses at the TU Berlin by Jakob Grave (master thesis) and Jonas Klages (bachelor thesis) contributed to the realization of the footbridge prototype. The formwork was prepared at the Makerspace of the Saxon State and University Library in Dresden (SLUB).


  1. 1.
    Sprott, K., Ravani, B.: Kinematic generation of ruled surfaces. Adv. Comput. Math. 17(1–2), 115–133 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Hagemann, M., Klawitter, D., Lordick, D.: Force driven ruled surfaces. J. Geom. Graph. 17(2), 193–204 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Pott, M., Lordick, D.: Dual spherical energy minimizer with applications to smoothing splines. In: 17th International Conference on Geometry and Graphics, Beijing (2016)Google Scholar
  4. 4.
    Odehnal, B.: Hermite interpolation of ruled surfaces and channel surfaces (2017)Google Scholar
  5. 5.
    Osman Letelier, J.P., Goldack, A., Schlaich, M., Lordick, D., Grave, J.: Shape optimization of concrete shells with ruled surface geometry using line geometry. In: International Association for Shells and Spatial Structures: IASS Annual Symposium, Hamburg (2017)Google Scholar
  6. 6.
    Hagemann, M., Klawitter, D.: Discretisation of light-weight concrete elements using a line-geometric model. In: Proceedings of the 9th fib International PhD Symposium in Civil Engineering, pp. 269–274. KIT Scientific Publishing, Karlsruhe (2012)Google Scholar
  7. 7.
    Klawitter, D., Hagemann, M., Odehnal, D.: Curve flows on ruled surfaces. J. Geom. Graph. 17(2), 129–140 (2013)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Varano, V., Gabriele, S., Teresi, L., Dryden, I.L., Puddu, P.E., Torromeo, C., Piras, P.: The TPS direct transport: a new method for transporting deformations in the size-and-shape space. Int. J. Comput. Vis. 124(3), 384–408 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Pottmann, H., Wallner, J.: Computational Line Geometry. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  10. 10.
    Lordick, D.: Intuitive design and meshing of non-developable ruled surfaces. In: Proceedings of the Design Modelling Symposium Berlin, pp. 248–261, University of the Arts Berlin (2009). URL
  11. 11.
    Lordick, D., Klawitter, D., Hagemann, M.: Liniengeometrie für den Leichtbau. In: Scheerer, S., Curbach, M. (Hrsg.): Leicht Bauen mit Beton—Forschung im Schwerpunktprogramm 1542 Förderphase I, pp. 224–235. TU Dresden, Dresden (2014)Google Scholar
  12. 12.
    Schlaich, J.: Conceptual design of light structures. J. Int. Assoc. Shells Spat. Struct.: IASS 45, 157–168 (2004)Google Scholar
  13. 13.
    Odehnal, B.: Subdivision algorithms for ruled surfaces. J. Geom. Graph. 12(1), 1–18 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Firl, M.: Optimal shape design of shell structures. Dissertation, München (2010). URL
  15. 15.
    Bletzinger, K.-U., Wüchner, R., Daoud, F., Camprubí, N.: Computational methods for form finding and optimization of shells and membranes. Comput. Methods Appl. Mech. Eng. 194(30), 3438–3452 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ramm, E., Bletzinger, K.-U.: Computational form finding and optimization. In: Adriaenssens, S., Block, P., Veenendaal, D., Williams, C. (Hrsg.): Shell Structures for Architecture. Form Finding and Optimization, pp. 45–55. Taylor, Hoboken (2014)Google Scholar
  17. 17.
    Pottmann, H., Peternell, M., Ravani, B.: Introduction to line geometry with applications. CAD Comput. Aided Des. 31, 3–16 (1999)CrossRefGoogle Scholar
  18. 18.
    Kemmler, R.: Große Verschiebungen und Stabilität in der Topologie- und Formoptimierung. Dissertation, Stuttgart (2004). URL
  19. 19.
    Hofer, M., Pottmann, H.: Energy-minimizing splines in manifolds. ACM Trans. Graph. 23(3), 284 (2004)CrossRefGoogle Scholar
  20. 20.
    Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization algorithms on matrix manifolds. Princeton University Press, Princeton, NJ (2008)CrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Technische Universität DresdenDresdenGermany

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