Technische Universität Berlin
Fakultät II: Institut für Mathematik, MA 8-3,
Straße des 17. Juni 136
10623 Berlin
Germany
Room: | MA 879 |
Phone: | +49-30-314-29486 |
Fax: | +49-30-314-79282 |
Email: | roerig"AT"math.tu-berlin.de |
Member of the SFB/TR 109: Discretization in Geometry and Dynamics
Junior Faculty of the Berlin Mathematical School
Co-author and developer of the software package polymake and VaryLab .
Organizer of KisSem (Keep it simple) Graduate Seminar of SFB/TR 109.
We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on larger parameter rectangles. For discrete planar quadrilateral nets, circular nets, Q^{*}-nets and conical nets we obtain a characterization of the corresponding discrete multi-nets. In the limit these discrete nets lead to some classical classes of smooth surfaces. Furthermore, we propose to use the characterized discrete nets as discrete extensions for the nets to obtain structure preserving subdivision schemes.
Supercyclides are surfaces with a characteristic conjugate parametrization consisting of two families of conics. Patches of supercyclides can be adapted to a Q-net (a discrete quadrilateral net with planar faces) such that neighboring surface patches share tangent planes along common boundary curves. We call the resulting patchworks 'supercyclidic nets' and show that every Q-net in RP^{3} can be extended to a supercyclidic net. The construction is governed by a multidimensionally consistent 3D system. One essential aspect of the theory is the extension of a given Q-net in RP^{N} to a system of circumscribed discrete torsal line systems. We present a description of the latter in terms of projective reflections that generalizes the systems of orthogonal reflections which govern the extension of circular nets to cyclidic nets by means of Dupin cyclide patches.
We present a new method to obtain periodic conformal parameterizations of surfaces with cylinder topology and describe applications to architectural design and rationalization of surfaces. The method is based on discrete conformal maps from the surface mesh to a cylinder or cone of revolution. It accounts for a number of degrees of freedom on the boundary that can be used to obtain a variety of alternative panelizations. We illustrate different choices of parameters for nurbs surface designs. Further, we describe how our parameterization can be used to get a periodic boundary aligned hex-mesh on a doubly-curved surface and show the potential on an architectural facade case study. Here we optimize an initial mesh in various ways to consist of a limited number of planar regular hexagons that panel a given surface.
Two-dimensional affine A-nets in 3-space are quadrilateral meshes that discretize surfaces parametrized along asymptotic lines. The characterizing property of A-nets is planarity of vertex stars, so for generic A-nets the elementary quadrilaterals are skew. We classify the simply connected affine A-nets that can be extended to continuously differentiable surfaces by gluing hyperboloid surface patches into the skew quadrilaterals. The resulting surfaces are called "hyperbolic nets" and are a novel piecewise smooth discretization of surfaces parametrized along asymptotic lines. It turns out that a simply connected affine A-net has to satisfy one combinatorial and one geometric condition to be extendable - all vertices have to be of even degree and all quadrilateral strips have to be "equi-twisted". Furthermore, if an A-net can be extended to a hyperbolic net, then there exists a 1-parameter family of such C^{1}-surfaces. It is briefly explained how the generation of hyperbolic nets can be implemented on a computer. The article uses the projective model of Plücker geometry to describe A-nets and hyperboloids.
Gridshells composed of elastically-bent profiles offer significant cost and time advantages during the production, transport and construction processes. Nevertheless, the shaping of the initially flat grid also generates important bending stresses on the structures, reducing therewith their bearing capacity against external loads. An optimisation of the grid topology in order to minimise the profiles curvature and, with it, the initial stresses is therefore crucial. In this paper a non-linear variational method for optimising topologies of elastic gridshells with regular and irregular meshes is presented. Different case studies of double-curved gridshells show the advantages and capacity of this method.
The quality of a quad-mesh depends on the shape of the individual quadrilaterals. The ideal shape from an architectural point of view is the planar square or rectangles with fixed aspect ratio. A parameterization that divides a surface into such shapes is called isothermic, i.e., angle-preserving and curvaturealigned. Such a parameterization exists only for the special class of isothermic surfaces. We extend this notion and introduce quasiisothermic parameterizations for arbitrary triangulated surfaces. We describe an algorithm that creates quasiisothermic meshes. Interestingly many surfaces appearing in architecture are close to isothermic surfaces, namely those coming from form finding methods and physical simulation. For those surfaces our method works particularly well and gives a high quality and robust mesh layout. We show how to optimize such meshes further to obtain disk packing representations. The quadrilaterals of these meshes are planar and possess touching incircles.
Konvexe Polytope, also konvexe Hüllen endlich vieler Punkte im euklidischen Raum, stehen an der Schnittstelle zwischen Geometrie und Kombinatorik. Ziel dieses Textes ist es, ein Ergebnis vorzustellen, zu motivieren und zu visualisieren, das darauf abzielt, Polytope zu klassifizieren, die sich auf besonders einfache Weise in Teilpolytope zerlegen lassen (in Herrmann und Joswig, Discrete Comput. Geom. 44:149-166, 2010). Um diese Klassifikation formulieren und illustrieren zu können, beginnt der Text mit einer Einführung in die Theorie der Sekundärfächer.
We investigate necessary conditions for the existence of projections of polytopes that preserve the full k-skeleta. More precisely, given the combinatorics of a polytope and the dimension e of the target space, what are obstructions to the existence of a geometric realization of a polytope with the given combinatorial type such that a linear projection to e-space strictly preserves the k-skeleton. Building on the work of Sanyal (2009), we develop a general framework to calculate obstructions to the existence of such realizations using topological combinatorics. Our obstructions take the form of graph colorings and linear integer programs. We focus on polytopes of product type and calculate the obstructions for products of polygons, products of simplices, and wedge products of polytopes. Our results show the limitations of constructions for the deformed products of polygons of Sanyal & Ziegler (2009) and the wedge product surfaces of Rörig & Ziegler (2009) and complement their results.
Gridshell structures made of highly elastic materials provide significant advantages thanks to their cost-effective and rapid erection process, whereby the initially in-plane grid members are progressively bent elastically until the desired structural geometry is achieved. Despite the strong growing interest that architects and engineers have in such structures, the complexity of generating grid configurations that are developable into free-form surfaces and the limitation of suitable materials restrict the execution of elastically bent gridshells. Over the past ten years, several research studies have focused on methodologies to generate developable grid configurations and to calculate their resulting geometry after the erection process. However, the same curved shell surface can be reproduced by various developable grid configurations which, in combination with their material properties, exhibit different structural behaviours not only during the shaping process but also on the gridshell load-bearing capacity. In this paper, the structural consequences of the choice of the grid configuration for an anticlastic surface have been analysed by means of FEM-Modelling combined with an geometrical optimisation of the initial bending stresses. In addition, the potential of using natural fibre-reinforced composites as a lightweight and environmentally friendly alternative has been investigated.
We introduce the wedge product of two polytopes. The wedge product is described in terms of inequality systems, in terms of vertex coordinates as well as purely combinatorially, from the corresponding data of its constituents. The wedge product construction can be described as an iterated "subdirect product" as introduced by McMullen (1976); it is dual to the "wreath product" construction of Joswig and Lutz (2005). One particular instance of the wedge product construction turns out to be especially interesting: The wedge products of polygons with simplices contain certain combinatorially regular polyhedral surfaces as subcomplexes. These generalize known classes of surfaces "of unusually large genus" that first appeared in works by Coxeter (1937), Ringel (1956), and McMullen, Schulz, and Wills (1983). Via "projections of deformed wedge products" we obtain realizations of some of the surfaces in the boundary complexes of 4-polytopes, and thus in R^{3}. As additional benefits our construction also yields polyhedral subdivisions for the interior and the exterior, as well as a great number of local deformations ("moduli") for the surfaces in R3. In order to prove that there are many moduli, we introduce the concept of "affine support sets" in simple polytopes. Finally, we explain how duality theory for 4-dimensional polytopes can be exploited in order to also realize combinatorially dual surfaces in R^{3} via dual 4-polytopes.
This note wants to explain how to obtain meaningful pictures of (possibly high-dimensional) convex polytopes, triangulated manifolds, and other objects from the realm of geometric combinatorics such as tight spans of finite metric spaces and tropical polytopes. In all our cases we arrive at specific, geometrically motivated, graph drawing problems. The methods displayed are implemented in the software system polymake.
There are d-dimensional zonotopes with n zones for which a 2-dimensional central section has \Omega(n^{d-1}) vertices. For d=3 this was known, with examples provided by the "Ukrainian easter eggs" by Eppstein et al. Our result is asymptotically optimal for all fixed d>=2.
We introduce the wedge product of two polytopes which is dual to the wreath product of Joswig and Lutz. The wedge product of a p-gon and a (q-1)-simplex contains many p-gon faces of which we select a subcomplex corresponding to a surface. This surface is regular of type {p,2q}, that is, all faces are p-gons, all vertices have degree 2q, and the combinatorial automorphism group acts transitively on the flags of the surface. We show that for certain choices of parameters $p$ and $q$ there exists a realization of the wedge product such that the surface survives the projection to R^4. For a different choice of parameters such a realization does not exist.
We prove that the neighborly cubical polytopes studied by Günter M. Ziegler and the first author arise as a special case of neighborly cubical spheres constructed by Babson, Billera, and Chan. By relating the two constructions we obtain an explicit description of a non-polytopal neighborly cubical sphere and, further, a new proof of the fact that the cubical equivelar surfaces of McMullen, Schulz, and Wills can be embedded into R^3.
We derive a non-recursive combinatorial description of the cubical spheres constructed by Babson, Billera, and Chan. This enables us to deduce many interesting properties of these spheres, for example neighborliness and (non)- polytopality. The ingredients needed for this construction are the mirror complex, the cubical fissure and BBC sequences.
This model complements the above article on Zonotopes with large 2D-cuts.