of pixels
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Enhancing Free-Form Architecture with Conical Panels
Bernhard Blaschitz, Benjamin Schneider and René Ziegler
Abstract Enhancing a given architectural free-form shape to feasibility often involves the approximation by a developable surface, either as an intermediate step for further planarization or to build single curved panels. Over the last years, many theoretical approaches have been presented that require optimization frameworks or mathematical equation solvers that are not readily available for the practitioner.
The new approach we present in this work takes a family of curves and constructs strips of conical and cylindrical panels between them. Two panels touch along a common ruling, which almost follows the conjugate direction. Planarization and development of the strip into the plane follow naturally. An easy, step-by-step algorithm that can be scripted in any CAD software is presented, as well as examples from architecture.
Bernhard Blaschitz, Benjamin Schneider and René Ziegler
Waagner-Biro Stahlbau AG, Vienna, Austria
Fig. 1 Design Study based on the algorithm of this work. The family of curves shown as steel beams was approximated with strips of conical panels, which are easier to manufacture in glass than general double curved elements.
1 Introduction
The evolution of a digital shape, described as a NURBS surface, into a discrete model and its further development into a built structure made out of many components with material-immanent characteristics, poses new challenges for architects, structural engineers and building firms. A close interdisciplinary collaboration, sometimes even with mathematicians and computer scientists, an exact understanding of the parties involved in the project and a consistent parametric data model are therefore the key to a successful project.
This has been enabled by the developments of digital modeling software and fabrication techniques. Computers have helped to push the boundaries that define which architectural forms can be realized. Because of new complex shapes which can be generated, and due to the development of new tools, which can rationalize and refine the complex geometries into easier buildable and more cost beneficial solutions. This enables the client to afford freeform architecture, which in the past would have been vastly more expensive.
1.1 Previous Work
Experience in building free-formed envelopes show that single curved elements are much easier to design, to produce and to install than general, double curved panels (Shelden 2002) or (Glaeser and Gruber 2007) even though there has been great improvement in the design of double curved molds (Raun et al 2012) and their repetitive use (Eigensatz et al 2010).
The focus of research over the last ten years has been planar panels, which can either be constructed directly (Glymph et al 2004), optimized from curve networks (Liu et al 2011) or by a physics based approach (Piker 2012). All optimization frameworks share a strong dependency on a good starting value, which determines aesthetics and convergence. Note that the four corners of the conical panels presented in this work already lie in a plane, so planarization is an inherent feature of the method.
The class of developable surfaces subsumes general cones, cylinders and tangent surfaces to a space curve. Numerous papers have dealt with the task of approximating a general shape with a developable surface (e.g. Aumann 2004, Chu and Séquin 2002, Frey 2004 or Liu et al 2006), but most of them require a computational framework to solve non-linear equations or insights into mathematical optimization, (Pottmann et al 2008) or (Zadravec et al 2010) that are not readily available for the practitioner. In computer graphics most papers use triangular meshes to compute developable surfaces (Rose et al 2007), (Julius et al 2005) or (Wang and Tang 2004) but ruling directions play no significant role.
1.2 Contribution of the present paper
In contrast to most of these methods, this paper does not directly approximate a given surface or mesh, but rather assumes a family of curves that outline a surface. Through two consecutive curves, strips of developable patches are spanned that allow the user to control the direction of the edges.
Fig. 2 Architectural study constructed with the methods of this work. Not only curvature lines can be used for panel alignment, but also general curves.
After showing the differences between single curved and double curved glass panels in design and production in section 2, we review all geometrical concepts necessary for understanding this work in section 3.
The main contribution of this paper, an approach to approximate given boundary curves with panels made of general cylinders and general cones, can be found in section 4. No further optimization is employed. All results can be scripted in standard CAD software without the need of external libraries or mathematical software. Since the desired application is architecture, the presented algorithm also allows the user to enter tolerances regarding the distance of panels. Examples and architectural studies are shown in section 5 and we conclude with a summary in section 6.
2 Application to architectural design with bent glass panels
Due to the rise of complex free-form shapes in architecture the glass industry had to develop new strategies and adapt their production lines. Now we differentiate between two main bending methods, the Annealed Glass Bending and the Tempered Glass Bending. These two techniques differ in the type of glass used, bending tools or machines and application of the fabricated elements. Based on many associated parameters of the glass, like edge curvature or panel geometry (Fig. 3) the costs for producing cylindrical and conical glass is quite the same, but costs for double curved glass are much more higher.
For double curved panels one needs to build a mould for every shape. (Eigensatz et al 2010) propose a method for computing different types of curved panels for given panel edges. Their algorithm depends on the choice of a coordinate system in the panel’s centre, which in turn determines the direction of curvature lines and thus reflection properties. Furthermore, the panel’s edge curves have to be given.
Fig. 3 Special glass panel types: a) cylindrical, b) cylindrical, edges not aligned,