Survey PaperArchitectural geometry
Introduction
Free forms constitute one of the major trends within contemporary architecture. In its earlier days a particularly important figure was Frank Gehry, with his design approach based on digital reconstruction of physical models, resulting in shapes which are not too far away from developable surfaces and thus ideally suited for his preferred characteristic metal cladding [94]. Nowadays we see an increasing number of landmark buildings involving geometrically complex freeform skins and structures (Fig. 1).
While the modeling of freeform geometry with current tools is well understood, the actual fabrication on the architectural scale is a challenge. One has to decompose the skins into manufacturable panels, provide appropriate support structures, meet structural constraints and last, but not least make sure that the cost does not become excessive. Many of these practically highly important problems are actually of a geometric nature and thus the architectural application attracted the attention of the geometric modeling and geometry processing community. This research area is now called Architectural Geometry. It is the purpose of the present survey to provide an overview of this field from the Computer Graphics perspective. We are not addressing here the many beautiful designs which have been realized by engineers with a clever way of using state of the art software, but we are focusing on research contributions which go well beyond the use of standard tools. This research direction has also been inspired by the work of the smart geometry group (www.smartgeometry.com), which promoted the use of parametric design and scripting for mastering geometric complexity in architecture.
From a methodology perspective, it turned out that the probably two most important ingredients for the solution of Architectural Geometry problems are Discrete Differential Geometry (DDG) [16], [84] and Numerical Optimization. In order to keep this survey well within Graphics, we will be rather short in discussing the subject from the DDG perspective and only mention those insights which are essential for a successful implementation. It is a fact that understanding a problem from the DDG viewpoint is often equivalent to understanding how to successfully initialize and solve the numerical optimization problems which are more directly related to the questions at hand.
In general, the approximation of an ideal design surface by a surface which is suitable for fabrication is called rationalization in Architecture. This often means panelization, i.e. finding a collection of smaller elements covering the design surface, but it can also mean replacing the design surface by a surface which has a simple generation like a ruled surface. Often, rationalization is harder than the 3D modeling of a surface. A digital modeling tool which automatically generates only buildable structures of a certain type (fabrication-aware design) is probably more efficient than the still prevalent approach based on rationalization. For research, both rationalization and fabrication-aware design are interesting, but the latter poses more unsolved problems, at the same time going far beyond architecture.
The solution of the above-mentioned problems may become easier if the shape under consideration has special properties, in which case we do not call it truly freeform. E.g. a surface generated by translation is easily rationalized into flat quadrilaterals (see Fig. 6). Special shapes have been extensively and very successfully employed, but this paper focuses on properties and algorithms relating to arbitrary (freeform) shapes. Remark The reader is advised that we use the word design in its purely technical sense, meaning that a designer uses available tools (drawings, software) to convert ideas to a geometric representation. We never refer to those aspects of design which touch cognitive science or artificial intelligence.
This paper is a survey, discussing a wide range of topics. It is divided into sections as follows: Section 2: Flat panels; Section 3: Developable panels; Section 4: Smooth double-curved skins; Section 5: Paneling; Section 6: Support structures; Section 7: Repetition; Section 8: Patterns; Section 9: Statics; Section 10: Shading and other functional aspects; Section 11: Design exploration. Within each section we address the following points:
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We point out why a certain topic gets addressed and which practical aspects are motivating it.
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We discuss the most essential and interesting aspects of the methodology for its solution.
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Results are provided along with a discussion which is based on real projects wherever possible.
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We address open problems and directions of future research.
Section snippets
Polyhedral surfaces – structures from flat panels
In order to realize a freeform surface in architecture, one often breaks it into smaller elements, called panels. Certainly, flat panels are the easiest and cheapest to produce and thus surfaces composed of flat panels, the so-called polyhedral surfaces or polyhedral meshes play a key role in Architectural Geometry. In this section, we discuss their various types, with a focus on meshes from planar quads. They have turned out to be the most interesting species of panel from the viewpoint of
Developable surfaces as limits of PQ meshes
Developable surfaces, also known as single-curved surfaces, can be unfolded into the plane without stretching or tearing. Thus it is easy to cover them with panels from metal or other materials with a similar behavior. They are characterized by containing a family of straight lines, each of which possesses a constant tangent plane – see Fig. 10 for an illustration of this fact and a limit process which transforms a sequence of planar quads into a developable strip. Their developability is not
Smooth double-curved skins
Realizing a double-curved freeform surface as a large and completely smooth (tangent plane continuous) architectural skin is a great challenge, and for some materials is only affordable when deviating from perfect smoothness or when accepting restrictions on the possible shapes. In this section we only address perfect smoothness; the successful play with tolerances is discussed in connection with the paneling problem in Section 5.
Obviously, we need smooth double-curved panels which for most
Geometric and algorithmic aspects of panelings
The paneling problem refers to realizing double curved architectural freeform surfaces by rationalization, i.e. replacing the surface by a union of panels. This section treats paneling in the narrow sense, where the panels are curved, and are to approximate the reference shape smoothly, up to tolerances. Paneling can be seen as two tasks, which are not independent:
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Segmentation of the reference shape into smaller pieces, which are called segments.
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Approximation of each segment by a panel which
Geometric support structures
The term support structure can denote different things. In Sections 6.1 and 6.2 it denotes a technical term in geometry which is more properly called torsion-free support structure. We return to the general meaning of the word in Section 6.3.
Repetitive elements
When paneling freeform surfaces, tolerances are often large enough to allow for repeating panel shapes (see Section 5). For structural elements like nodes, beams, and frames, however, the tolerances are often tighter and the geometry of these structures is often more complex than that of the outer skin. Therefore optimizing freeform structures for repetitive elements is highly challenging and sometimes impossible. This complicates logistics and increases production cost, and is a typical
Patterns
Geometric patterns have fascinated mankind since ancient times. Artists had an excellent understanding of this subject and studied patterns and tilings thoroughly. This is especially true for the islamic world. In the context of freeform architecture, patterns can arise in many ways, including the arrangement of panels, the subconstruction, in addition functional layers such as shading systems or simply as textures. We here point to the few research contributions which look at these patterns
Self-supporting masonry
Naturally, stability is of paramount importance in all architectural designs. However it is only recently that it can be taken into account during the design process in an automatic way. The complex nature of the question of stability and the involvement of many factors besides geometry makes statics-aware design feasible only in such situations where specifying the geometry already allows for statics analysis, and the designer is not held up by having to additionally specify structural
Shading and lighting systems
The distribution of light within a building is of great importance, but so far has received little attention from the Architectural Geometry community. Several of the geometric structures discussed in this paper are relevant for shading, e.g. torsion-free support structures (see Fig. 51), which can be optimized for shading effect. A more general treatment of this topic has been based on interpreting them as discrete line congruences and using basic insights from line geometry for the
Interactive design systems and design exploration
Technical papers about the geometric problems related with freeform architecture sometimes create the impression that those problems are solved by an optimization process which has a unique solution. Such a situation of course would be at odds with the creative processes instrumental to both design and architecture. Obeying constraints is of course nothing new to designers, and architects are well accustomed to statics and properties of materials. However, within the context of architectural
Conclusion
Our aim was to compile an overview of recent research around freeform architecture in an effort to identify core tasks and results, illustrate the discussion by real world examples and to outline some of the many directions for future research. Partially those go well beyond architecture, which probably constitutes the most important message of this survey: Next generation geometric modeling systems should be much easier to use and contribute to a shorter product development cycle. One way of
Acknowledgments
This research was supported by the Austrian Science Fund (FWF) through Grant P23735-N13 and by the DFG-Collaborative Research Center, TRR 109 Discretization in Geometry and Dynamics, through FWF Grants I705-N26 and I706-N26. The authors would like to thank Philippe Bompas and Jacques Raynaud for their helpful discussions. We are very grateful to the anonymous reviewers for their extensive and helpful comments.
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