# C01Discrete 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.

#### Publications+

##### Papers
###### A Laplace Operator on Semi-Discrete Surfaces

Author: Carl, Wolfgang
Journal: Foundations of Computational Mathematics, pages 1-36
Date: 2015
DOI: 10.1007/s10208-015-9271-y

###### Architectural Geometry

Authors: Pottmann, Helmut and Eigensatz, Michael and Vaxman, Amir and Wallner, Johannes
Journal: Computers and Graphics, 47:145-164
Date: 2015
DOI: 10.1016/j.cag.2014.11.002

###### Supercyclidic nets

Authors: Bobenko, Alexander I. and Huhnen-Venedey, Emanuel and Rörig, Thilo
Journal: Int. Math. Res. Not.
Note: accepted, preprint at arxiv
Date: 2015

###### Surface panelization using periodic conformal maps

Authors: Rörig, Thilo and Sechelmann, Stefan and Kycia, Agata and Fleischmann, Moritz
In Proceedings: Advances in Architectural Geometry 2014, Springer
Note: Best Paper Award
Date: Sep 2014

###### Discretization of asymptotic line parametrizations using hyperboloid surface patches

Authors: Huhnen-Venedey, Emanuel and Rörig, Thilo
Journal: Geometriae Dedicata, 168(1):265-289
Date: Feb 2014
DOI: 10.1007/s10711-013-9830-9

###### Affine arc length polylines and curvature continuous uniform B-splines

Author: Käferböck, Florian
Journal: Computer-Aided Geom. Design, 31
Date: 2014

###### Cell packing structures

Authors: Pottmann, Helmut and Jiang, Caigui and Höbinger, Mathias and Wang, Jun and Bompas, Philippe and Wallner, Johannes
Journal: Computer-Aided Design
Note: to appear. Special issue on Material Ecology
Date: 2014

###### Form-finding with Polyhedral Meshes Made Simple

Authors: Tang, Chengcheng and Sun, Xiang and Gomes, Alexandra and Wallner, Johannes and Pottmann, Helmut
Journal: ACM Trans. Graphics, 33(4):$\#$70,1-9
Note: Proc. SIGGRAPH
Date: 2014
DOI: 10.1145/2601097.2601213

###### Freeform Honeycomb Structures

Authors: Jiang, Caigui and Wang, Jun and Wallner, Johannes and Pottmann, Helmut
Journal: Comput. Graph. Forum, 33(5):185-194
Note: Proc. Symposium Geometry Processing
Date: 2014
DOI: 10.1111/cgf.12444

###### Interactive modeling of architectural freeform structures - combining geometry with fabrication and statics

Authors: Jiang, Caigui and Tang, Chengcheng and Tomičić, Marko and Wallner, Johannes and Pottmann, Helmut
In Collection: Advances in Architectural Geometry, Springer
Date: 2014

###### On Discrete Constant Mean Curvature Surfaces

Author: Müller, Christian
Journal: Discrete Comput. Geom., 51(3):516--538
Date: 2014
DOI: 10.1007/s00454-014-9577-6

###### On offsets and curvatures for discrete and semidiscrete surfaces

Authors: Karpenkov, Oleg and Wallner, Johannes
Journal: Beitr. Algebra Geom., 55:207-228
Date: 2014
DOI: 10.1007/s13366-013-0146-6

###### Smooth surfaces from rational bilinear patches

Authors: Shi, Ling and Wang, Jun and Pottmann, Helmut
Journal: Comput. Aided Geom. Design, 31(1):1--12
Date: 2014
DOI: 10.1016/j.cagd.2013.11.001

###### Variational Laplacians for semidiscrete surfaces

Authors: Carl, Wolfgang and Wallner, Johannes
Note: submitted
Date: 2014

###### Discrete Line Congruences for Shading and Lighting

Authors: Wang, J. and Jiang, C. and Bompas, P. and Wallner, J. and Pottmann, H.
Journal: Computer Graphics Forum, 32(5):53-62
Note: Proc. Symposium Geometry Processing
Date: 2013
DOI: 10.1111/cgf.12172

###### Quasiisothermic Mesh Layout

Authors: Sechelmann, Stefan and Rörig, Thilo and Bobenko, Alexander I.
In Collection: Advances in Architectural Geometry 2012, Springer Vienna
Date: 2013
DOI: 10.1007/978-3-7091-1251-9_20
ISBN: 978-3-7091-1250-2

###### Smooth surfaces from bilinear patches: discrete affine minimal surfaces

Authors: Käferböck, Florian and Pottmann, Helmut
Journal: Computer-Aided Geom. Design, 30:476-489
Date: 2013

###### Topology Optimisation of Regular and Irregular Elastic Gridshells by Means of a Non-linear Variational Method

Authors: Lafuente Hernández, Elisa and Sechelmann, Stefan and Rörig, Thilo and Gengnagel, Christoph
In Collection: Advances in Architectural Geometry 2012, Springer Vienna
Date: 2013
DOI: 10.1007/978-3-7091-1251-9_11
ISBN: 978-3-7091-1250-2

###### Ruled Free Forms

Authors: Flöry, Simon and Nagai, Yukie and Isvoranu, Florin and Pottmann, Helmut and Wallner, Johannes
In Collection: Advances in Architectural Geometry 2012, Springer
Date: 2012
DOI: 10.1007/978-3-7091-1251-9_4

##### PhD thesis
###### Cyclidic and hyperbolic nets: A piecewise smooth discretization of orthogonal and asymptotic nets in discrete differential geometry

Author: Huhnen-Venedey, Emanuel
Date: 2014

##### Posters
###### Planar quad layout on NURBS-surfaces from symmetric conjugate curves

Authors: Seidel, Christoph and Rörig, Thilo and Sechelmann, Stefan
Note: Presented at Advances in Architectural Geometry 2014
Date: Sep 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) 314 24655
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/

#### 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

#### Wolfgang Carl   +

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

#### Dr. Martin Kilian   +

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

#### Florian Käferböck   +

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

#### Dr. Christian Müller   +

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

#### Dr. Thilo Rörig   +

Projects: C01
University: TU Berlin
E-Mail: roerig[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~thilosch

#### Andrew O'Shea Sageman-Furnas   +

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