Discrete Geometric Structures Motivated by Applications and Architecture

Geometry Supporting the Realization of Freeform Architecture

Contemporary architecture of large-scale buildings makes it necessary to subdivide large surfaces into smaller segments, the panels. While the building industry, architects and engineers are mainly interested in design aspects, manufacturing and statics, we will systematically investigate the related purely mathematical problems together with computational aspects. In particular we will investigate diagonal parametrizations, ring pattern structures, and discrete structures in sphere geometries.


This project studies discrete geometric structures that arise in the context of applications. We will mainly focus on the investigation of nets in Lie sphere geometry and their relation to integrable structures.

Scientific Details+

Our investigation includes the following items:

Diagonal parametrizations:
Smooth and discrete curvature line, asymptotic and conformal parametrizations play a crucial role in discrete differential geometry and are also important for applications. We plan to investigate structure preserving discretizations of the parametrizations that are in a way “diagonal” to the classical ones. The main example we have in mind is the parametrization with equal normal curvatures, in particular mean curvature parametrization. Diagonal discretizations can be combined into consistent webs and layers. By extending our work on quadrics we plan to investigate this problem in a more general setup more appropriate for applications.

Ring patterns:
We plan to develop a comprehensive theory of ring patterns based on our work on orthogonal ring patterns in a plane. Many remarkable results about circle patterns can be probably generalized to the case of rings. This includes a relation to discrete integrable systems, hyperbolic geometry interpretation with a geometric variational principle and conformal patterns. Especially promising in this aspect is the investigation of orthogonal ring patterns in the sphere and hyperbolic plane. Similarly to circle patterns we plan to use ring patterns in architectural design.

Discrete structures in sphere geometries:
We plan to investigate nets in Lie sphere geometry and thereby particularly focusing on invariant face elements. We will study (triangular, quad, hexagonal) nets in the projective models of Lie sphere geometry, e.g., describing smooth checkerboard surfaces (black = sphere; white = Dupin cyclide), nets from pairs of spherical caps as envelopes of discrete sphere congruences, conical nets in conformal models of elliptic and hyperbolic space, nets with support structure elements from circular cones, nets with spherical nodes, etc.

Furthermore, we will investigate subdivision methods in the space of circles and spheres building up on our recent Möbius invariant approach. In this context we aim at a thorough smoothness analysis. We will further explore a promising analogue of the four-point-scheme for sequences of circles and spheres and investigate possible generalizations in particular to discrete sphere congruences.

Extending our previous work on differentiable extensions described by integrable systems (Dupin cyclide, hyperbolic, supercyclide surface patches), we plan to investigate a larger class of extensions that are not necessarily differentiable.


  • 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, B02, C01, CaP, Z
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

Dr. Christian Müller   +

Projects: C01
University: TU Wien, Institute of Discrete Mathematics and Geometry, 104
Address: Wiedner Hauptstr. 8-10, 1040 Vienna, AUSTRIA
Tel: +43 1 58801104465
Fax: +43 1 5880110493
E-Mail: cmueller[at]geometrie.tuwien.ac.at
Website: https://www.dmg.tuwien.ac.at/geom/ig/mueller/index.php

Dr. Martin Kilian   +

Projects: C01
University: TU Wien, Institute of Discrete Mathematics and Geometry, 104
Address: Wiedner Hauptstr. 8-10, 1040 Vienna, AUSTRIA
Tel: +43 1 58801104469
E-Mail: kilian[at]geometrie.tuwien.ac.at
Website: https://www.dmg.tuwien.ac.at/geom/ig/kilian/index.html

Dr. Andrew O'Shea Sageman-Furnas   +

Projects: C01
University: TU Berlin, Institut für Mathematik, MA 879
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31429486
E-Mail: aosafu[at]math.tu-berlin.de
Website: https://page.math.tu-berlin.de/~aosafu/