Discrete Geometric Structures Motivated by Applications and Architecture

Geometry Supporting the Realization of Freeform Architecture

Many of today's most striking buildings are nontraditional freeform shapes. Their fabrication is a big challenge, but also a rich source of research topics in geometry. Project A08 addresses key questions such as: "How can we most efficiently represent and explore the variety of manufacturable designs?" or "Can we do this even under structural constraints such as force equilibrium?" Answers to these questions are expected to support the development of next generation modelling tools which combine shape design with key aspects of function and fabrication.

Scientific Details+

The emergence of freeform structures in contemporary architecture raises numerous challenging research problems, most of which are related to the actual fabrication and are of a mathematical nature. The present project will take up those fundamental research problems raised by demands in architecture which cannot be addressed by specialized practitioners or current software, and which are also beyond the scope of application oriented research projects. While motivation comes from architecture, we focus on research topics that are also of substantial interest from a purely mathematical perspective. These topics evolve around important classes of discrete surfaces, 3-dimensional discrete structures, discrete problems in the geometry of webs and relations between those. Our main research directions are closely linked and can be summarized as follows:

• Webs are arrangements of curves which cover surfaces in a certain combinatorial arrangement. A main web type to be investigated are the so-called hexagonal webs, which correspond to a triangular or tri-hex decomposition of a surface. We are interested in such discrete webs which are formed by either smooth or discrete curves of a specified type (planar, geodesic, principal curvature line, etc.). Going beyond the currently available numerical optimization methods for constructing such functional webs in an approximate manner, we are interested in existence results, characterizations and direct constructions. These are expected to provide important information about the flexibility in the design spaces for various types of functional webs. Our web geometric studies will also include studies of volumetric structures.

• Meshes for architectural applications may be highly constrained and thus the traditional handle-based editing techniques used mainly to manipulate triangle meshes are not necessarily appropriate. We plan to study the moduli spaces of important classes of constrained meshes such as circular or conical meshes and of discrete functional webs. We aim at a better understanding of the flexibility of these meshes in view of the architectural application. This amounts to the study of properties of a space of meshes in the neighborhood of a given mesh, i.e., the local structure of the moduli space. This project should also lead to coordinates for the moduli spaces of constrained meshes that are adapted to applications of different variational principles. One part will be concerned with curves - discrete and continuous - in the moduli space that will single out families of meshes related by flows or transformations.

Discrete 3D structures motivated by architecture. So far, the relation between Discrete Differential Geometry and architecture has been exploited in form of discretizations of surfaces. We plan to analyze transformations of discrete nets and their consistency to obtain fully 3D structures, which are relevant to applications as well. In particular 3D structures all whose elements are easy to produce are interesting for architectural applications.

Discrete nets with differentiable extensions. Smooth architectural freeform hulls are still a big challenge. Since they have to be composed of panels and the production of the panels needs to be feasible, one has to compose smooth surfaces from simple types of surface patches. A good example is provided by the fact that a discrete circular net (quad mesh with circular quads) can be extended to a continuously differentiable surface by appropriately filling the quads with patches of Dupin cyclides. We will investigate similar constructions where the underlying nets and the inserted surface patches are composed of simple curve types such as straight lines and conics (in particular circles).

Structures in equilibrium. It is well-known that certain discretizations of special surface classes such as surfaces of constant negative Gaussian curvature or minimal surfaces are networks in static equilibrium. This is connected to graphic statics and the concept of reciprocal force diagrams. Recent studies of self-supporting freeform structures are based on discrete structures (thrust networks) which are in equilibrium under the application of vertical loads. We plan to link this work to discrete differential geometry and to provide a more complete study of discrete structures in equilibrium. This includes a study of the variety of thrust networks in a given shell that confirm its action as a self-supporting structure.


  • Alexander I. Bobenko, Carl O. R. Lutz, Helmut Pottmann, and Jan Techter.
    Non-Euclidean Laguerre geometry and incircular nets.
    preprint, September 2020.
  • Thilo Rörig and Gudrun Szewieczek.
    The Ribaucour families of discrete R-congruences.
    preprint, April 2020.
  • Caigui Jiang, Klara Mundilova, Florian Rist, Johannes Wallner, and Helmut Pottmann.
    Curve-pleated Structures.
    ACM Trans. Graph., 38(6):169:1–169:13, November 2019.
    doi:10.1145/3355089.3356540, dgd:608.
  • Hui Wang, Davide Pellis, Florian Rist, Helmut Pottmann, and Christian Müller.
    Discrete Geodesic Parallel Coordinates.
    ACM Trans. Graph., 38(6):173:1–173:13, November 2019.
    doi:10.1145/3355089.3356541, dgd:607.
  • Alexander I. Bobenko, Tim Hoffmann, and Thilo Rörig.
    Orthogonal ring patterns.
    preprint, November 2019.
    arXiv:1911.07095, dgd:588.
  • Alexander I Bobenko, Helmut Pottmann, and Thilo Rörig.
    Multi-Nets. Classification of discrete and smooth surfaces with characteristic properties on arbitrary parameter rectangles.
    Discrete Comput. Geom., May 2019.
    arXiv:1802.05063, doi:10.1007/s00454-019-00101-1.
  • Andrew O. Sageman-Furnas, Albert Chern, Mirela Ben-Chen, and Amir Vaxman.
    Chebyshev Nets from Commuting PolyVector Fields.
    ACM Trans. Graphics, 38(6):172:1–172:16, 2019. Proc. SIGGRAPH Asia.
  • Michael R. Jimenez, Christian Müller, and Helmut Pottmann.
    Discretizations of Surfaces with Constant Ratio of Principal Curvatures.
    Discrete Comput. Geom., 2019. accepted for publication.
    URL: http://www.geometrie.tuwien.ac.at/ig/publications/constratio/constratio.pdf, doi:10.1007/s00454-019-00098-7.
  • D. Pellis, M. Kilian, F. Dellinger, J. Wallner, and H. Pottmann.
    Visual Smoothness of polyhedral surfaces.
    ACM Trans. Graphics, 2019.
    URL: http://hdl.handle.net/10754/653104.
  • Davide Pellis and Helmut Pottmann.
    Aligning principal stress and curvature directions.
    Advances in Architectural Geometry, pages 34–53, 2018.
  • Amir Vaxman, Christian Müller, and Ofir Weber.
    Canonical Möbius Subdivision.
    ACM Trans. Graphics (Proc. SIGGRAPH ASIA), 2018.
    URL: http://www.geometrie.tuwien.ac.at/geom/ig/publications/moebiussubdivision/moebiussubdivision.pdf.
  • Eike Schling, Martin Kilian, Hui Wang, Denis Schikore, and Helmut Pottmann.
    Design and construction of curved support structures with repetitive parameters.
    In Lars Hesselgren, Axel Kilian, Samar Malek, Karl-Gunnar Olsson, Olga Sorkine-Hornung, and Chris Williams, editors, Advances in Architectural Geometry, pages 140–165. Klein Publishing Ltd, 2018.
  • Chi-Han Peng, Helmut Pottmann, and Peter Wonka.
    Designing patterns using triangle-quad hybrid meshes.
    ACM Trans. Graphics, 37(4):14, 2018. Proc. SIGGRAPH.
  • Changyeob Baek, Andrew O. Sageman-Furnas, Mohammad K. Jawed, and Pedro M. Reis.
    Form finding in elastic gridshells.
    Proceedings of the National Academy of Sciences, 115(1):75–80, 2018.
    URL: https://www.pnas.org/content/115/1/75, doi:10.1073/pnas.1713841115.
  • Alexander I Bobenko, Emanuel Huhnen-Venedey, and Thilo Rörig.
    Supercyclidic nets.
    International Mathematics Research Notices, 2017(2):323–371, February 2017.
    arXiv:1412.7422, doi:10.1093/imrn/rnv328.
  • Martin Kilian, Davide Pellis, Johannes Wallner, and Helmut Pottmann.
    Material-minimizing forms and structures.
    ACM Trans. Graphics, 36(6):article 173, 2017. Proc. SIGGRAPH Asia.
  • Amir Vaxman, Christian Müller, and Ofir Weber.
    Regular meshes from polygonal patterns.
    ACM Transactions on Graphics (TOG), 36(4):113, 2017.
  • Christian Müller.
    Planar discrete isothermic nets of conical type.
    Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry, 57(2):459–482, June 2016.
  • Chengcheng Tang, Martin Kilian, Pengbo Bo, Johannes Wallner, and Helmut Pottmann.
    Analysis and design of curved support structures.
    In Sigrid Adriaenssens, Fabio Gramazio, Matthias Kohler, Achim Menges, and Mark Pauly, editors, Advances in Architectural Geometry 2016, pages 8–23. VDF Hochschulverlag, ETH Zürich, 2016.
  • Helmut Pottmann and Johannes Wallner.
    Geometry and freeform architecture.
    In Wolfgang König, editor, Mathematics and Society, pages 131–151. EMS, 2016.
  • Wolfgang Carl.
    A Laplace Operator on Semi-Discrete Surfaces.
    Foundations of Computational Mathematics, pages 1–36, 2015.
  • Helmut Pottmann, Michael Eigensatz, Amir Vaxman, and Johannes Wallner.
    Architectural Geometry.
    Computers and Graphics, 47:145–164, 2015.
    URL: http://www.geometrie.tugraz.at/wallner/survey.pdf, doi:10.1016/j.cag.2014.11.002.
  • Amir Vaxman, Christian Müller, and Ofir Weber.
    Conformal mesh deformations with Möbius transformations.
    ACM Transactions on Graphics (TOG), 34(4):55, 2015.
    URL: http://www.geometrie.tuwien.ac.at/geom/ig/publications/2015/conformal2015/conformal2015.pdf.
  • Christian Müller.
    Semi-discrete constant mean curvature surfaces.
    Mathematische Zeitschrift, 279(1-2):459–478, 2015.
    URL: http://www.geometrie.tuwien.ac.at/geom/ig/publications/2015/sdcmc2015/sdcmc.pdf, doi:10.1007/s00209-014-1377-4.
  • Thilo Rörig, Stefan Sechelmann, Agata Kycia, and Moritz Fleischmann.
    Surface panelization using periodic conformal maps.
    In Philippe Block, Jan Knippers, Niloy Mitra, and Wenping Wang, editors, Advances in Architectural Geometry 2014. Springer, September 2014. Best Paper Award.
  • Emanuel Huhnen-Venedey and Thilo Rörig.
    Discretization of asymptotic line parametrizations using hyperboloid surface patches.
    Geometriae Dedicata, 168(1):265–289, February 2014.
    arXiv:1112.3508, doi:10.1007/s10711-013-9830-9.
  • Florian Käferböck.
    Affine arc length polylines and curvature continuous uniform B-splines.
    Computer-Aided Geom. Design, 2014.
  • Helmut Pottmann, Caigui Jiang, Mathias Höbinger, Jun Wang, Philippe Bompas, and Johannes Wallner.
    Cell packing structures.
    Computer-Aided Design, 2014. to appear. Special issue on Material Ecology.
  • Chengcheng Tang, Xiang Sun, Alexandra Gomes, Johannes Wallner, and Helmut Pottmann.
    Form-finding with Polyhedral Meshes Made Simple.
    ACM Trans. Graphics, 33(4):$#$70,1–9, 2014. Proc. SIGGRAPH.
  • Caigui Jiang, Jun Wang, Johannes Wallner, and Helmut Pottmann.
    Freeform Honeycomb Structures.
    Comput. Graph. Forum, 33(5):185–194, 2014. Proc. Symposium Geometry Processing.
  • Caigui Jiang, Chengcheng Tang, Marko Tomičić, Johannes Wallner, and Helmut Pottmann.
    Interactive modeling of architectural freeform structures - combining geometry with fabrication and statics.
    In P. Block, J. Knippers, and W. Wang, editors, Advances in Architectural Geometry. Springer, 2014.
  • Christian Müller.
    On Discrete Constant Mean Curvature Surfaces.
    Discrete Comput. Geom., 51(3):516–538, 2014.
  • Oleg Karpenkov and Johannes Wallner.
    On offsets and curvatures for discrete and semidiscrete surfaces.
    Beitr. Algebra Geom., 55:207–228, 2014.
  • Ling Shi, Jun Wang, and Helmut Pottmann.
    Smooth surfaces from rational bilinear patches.
    Comput. Aided Geom. Design, 31(1):1–12, 2014.
  • Wolfgang Carl and Johannes Wallner.
    Variational Laplacians for semidiscrete surfaces.
    submitted, 2014.
    URL: http://www.geometrie.tugraz.at/carl/gradients.pdf.
  • J. Wang, C. Jiang, P. Bompas, J. Wallner, and H. Pottmann.
    Discrete Line Congruences for Shading and Lighting.
    Computer Graphics Forum, 32(5):53–62, 2013. Proc. Symposium Geometry Processing.
  • Stefan Sechelmann, Thilo Rörig, and Alexander I. Bobenko.
    Quasiisothermic Mesh Layout.
    In Lars Hesselgren, Shrikant Sharma, Johannes Wallner, Niccolo Baldassini, Philippe Bompas, and Jacques Raynaud, editors, Advances in Architectural Geometry 2012, pages 243–258. Springer Vienna, 2013.
  • Florian Käferböck and Helmut Pottmann.
    Smooth surfaces from bilinear patches: discrete affine minimal surfaces.
    Computer-Aided Geom. Design, 30:476–489, 2013.
  • Elisa Lafuente Hernández, Stefan Sechelmann, Thilo Rörig, and Christoph Gengnagel.
    Topology Optimisation of Regular and Irregular Elastic Gridshells by Means of a Non-linear Variational Method.
    In Lars Hesselgren, Shrikant Sharma, Johannes Wallner, Niccolo Baldassini, Philippe Bompas, and Jacques Raynaud, editors, Advances in Architectural Geometry 2012, pages 147–160. Springer Vienna, 2013.
  • Simon Flöry, Yukie Nagai, Florin Isvoranu, Helmut Pottmann, and Johannes Wallner.
    Ruled Free Forms.
    In Lars Hesselgren, Shrikant Sharma, Johannes Wallner, Niccolo Baldassini, Philippe Bompas, and Jacques Raynaud, editors, Advances in Architectural Geometry 2012, pages 57–66. Springer, 2012.

PhD thesis
  • Emanuel Huhnen-Venedey.
    Cyclidic and hyperbolic nets: A piecewise smooth discretization of orthogonal and asymptotic nets in discrete differential geometry.
    Dissertation, TU Berlin, 2014.

  • Christoph Seidel, Thilo Rörig, and Stefan Sechelmann.
    Planar quad layout on NURBS-surfaces from symmetric conjugate curves.
    September 2014. Presented at Advances in Architectural Geometry 2014.


Prof. Dr. Alexander I. Bobenko   +

Projects: A01, A02, C01, B02, Z, CaP, II
University: TU Berlin, Institut für Mathematik, MA 881
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31424655
E-Mail: bobenko[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~bobenko/

Prof. Dr. Helmut Pottmann   +

Projects: C01
University: TU Wien
E-Mail: pottmann[at]geometrie.tuwien.ac.at
Website: http://www.dmg.tuwien.ac.at/pottmann/
University: King Abdullah University of Science and Technology
E-Mail: helmut.pottmann[at]kaust.edu.sa

Prof. Dr. Johannes Wallner   +

Projects: C01
University: TU Graz
E-Mail: j.wallner[at]tugraz.at
Website: http://www.geometrie.tugraz.at/wallner/

Leonardo Alese   +

Projects: C01
University: TU Graz
E-Mail: alese[at]tugraz.at

Felix Dellinger   +

Projects: C01
University: TU Wien, Institute of Discrete Mathematics and Geometry, DA 07 G22
Address: Wiedner Hauptstrasse 8–10, 1040 Vienna, AUSTRIA
Tel: +43 1 58801104683
E-Mail: felix.dellinger[at]tuwien.ac.at
Website: https://dmg.tuwien.ac.at/fg6/dellinger/home.html
University: TU Graz, Institut für Geometrie
Address: Kopernikusgasse 24/IV, 8010 Graz, AUSTRIA
E-Mail: f.dellinger[at]tugraz.at
Website: https://online.tugraz.at/tug_online/visitenkarte.show_vcard?pPersonenId=D50E0828A11111B3&pPersonenGruppe=3

Dr. Martin Kilian   +

Projects: C01
University: TU Wien
E-Mail: kilian[at]geometrie.tuwien.ac.at

Dr. Christian Müller   +

Projects: C01
University: TU Wien
E-Mail: cmueller[at]geometrie.tuwien.ac.at

Dr. Andrew O'Shea Sageman-Furnas   +

Projects: C01
University: TU Berlin
E-Mail: aosafu[at]math.tu-berlin.de