Abstract
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Sprott, K., Ravani, B.: Kinematic generation of ruled surfaces. Adv. Comput. Math. 17(1–2), 115–133 (2002)
Hagemann, M., Klawitter, D., Lordick, D.: Force driven ruled surfaces. J. Geom. Graph. 17(2), 193–204 (2013)
Pott, M., Lordick, D.: Dual spherical energy minimizer with applications to smoothing splines. In: 17th International Conference on Geometry and Graphics, Beijing (2016)
Odehnal, B.: Hermite interpolation of ruled surfaces and channel surfaces (2017)
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)
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)
Klawitter, D., Hagemann, M., Odehnal, D.: Curve flows on ruled surfaces. J. Geom. Graph. 17(2), 129–140 (2013)
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)
Pottmann, H., Wallner, J.: Computational Line Geometry. Springer, Heidelberg (2001)
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 http://lordick.dgfgg.de/docs/DMSB2009-Lordick-150.pdf
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)
Schlaich, J.: Conceptual design of light structures. J. Int. Assoc. Shells Spat. Struct.: IASS 45, 157–168 (2004)
Odehnal, B.: Subdivision algorithms for ruled surfaces. J. Geom. Graph. 12(1), 1–18 (2008)
Firl, M.: Optimal shape design of shell structures. Dissertation, München (2010). URL http://mediatum.ub.tum.de/doc/981720/512948.pdf
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)
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)
Pottmann, H., Peternell, M., Ravani, B.: Introduction to line geometry with applications. CAD Comput. Aided Des. 31, 3–16 (1999)
Kemmler, R.: Große Verschiebungen und Stabilität in der Topologie- und Formoptimierung. Dissertation, Stuttgart (2004). URL https://www.ibb.uni-stuttgart.de/publikationen/fulltext_new/2004/kemmler-2004.pdf
Hofer, M., Pottmann, H.: Energy-minimizing splines in manifolds. ACM Trans. Graph. 23(3), 284 (2004)
Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization algorithms on matrix manifolds. Princeton University Press, Princeton, NJ (2008)
Acknowledgements
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).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Noack, K., Lordick, D. (2019). Optimized Ruled Surfaces with an Application to Thin-Walled Concrete Shells. In: Cocchiarella, L. (eds) ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics. ICGG 2018. Advances in Intelligent Systems and Computing, vol 809. Springer, Cham. https://doi.org/10.1007/978-3-319-95588-9_27
Download citation
DOI: https://doi.org/10.1007/978-3-319-95588-9_27
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-95587-2
Online ISBN: 978-3-319-95588-9
eBook Packages: EngineeringEngineering (R0)