Geometric computing for freeform architecture
Abstract
Geometric computing has recently found a new field of applications, namely the various geometric problems which lie at the heart of rationalization and constructionaware design processes of freeform architecture. We report on our work in this area, dealing with meshes with planar faces and meshes which allow multilayer constructions (which is related to discrete surfaces and their curvatures), triangles meshes with circlepacking properties (which is related to conformal uniformization), and with the paneling problem. We emphasize the combination of numerical optimization and geometric knowledge.
Keywords
Triangle Mesh Circle Packing Quadrilateral Mesh Polyhedral Surface Unit Vector Field1 Background
The time of writing this survey paper coincides with the summing up of a sixyear socalled national research network entitled Industrial geometry which was funded by the Austrian Science Fund (FWF). By serendipity at the same point in time where the first Ph.D. students started their work in this project, a whole new direction of research in applied geometry turned up: meshes and threedimensional geometric structures which are relevant for rationalization and constructionaware design in freeform architecture. It turned out to be fruitful and rewarding, and of course it is also a topic which perfectly fits the heading of ‘Industrial Geometry’.
It seems that everybody who is in the business of actually realizing freeform architectural designs as a steelglass construction, or in concrete, or by means of a wooden paneling, quickly encounters the limits of the tools which are commercially available. Some of the problems whose solutions are on top of the list of desiderata are in fact very hard. As a consequence there is great demand for a systematic approach and, most importantly, a full understanding of the geometric possibilities and obstructions inherent in obstacles which present themselves.

The discrete differential geometry of quadrilateral meshes and the sphere geometries of Möbius, Laguerre, and Lie, are closely related to the realization of freeform shapes as steelglass constructions with socalled torsion free nodes.

Conformal uniformization appears in connection with circlepacking meshes and derived triangle and hexagonal meshes.

Optimization using various ideas ranging from combinatorial optimization to image processing is instrumental in solving the paneling problem, that is, the rationalization of freeform shapes via decomposition into simple and repetitive elements.
Our research is part of the emerging interdisciplinary field of architectural geometry. The interested reader is referred to the proceedings volume [1] which collects recent contributions from different areas (mathematics, engineering, architecture), the textbook [2], and the articles [3, 4, 5]. Our aim is to convince the reader that many issues in freeform architecture can be dealt with by meshes or other geometric structures with certain local properties. Further, that we are capable of formulating target functionals for optimization which  if successful  achieve these properties. It is however important to know that in many cases optimization without additional geometric knowledge (utilized, for example, by way of initialization) does not succeed.
2 Multilayer structures
In this section we deal with the remarkable interrelation between the discrete differential geometry of polyhedral surfaces on the one hand, and problems regarding multilayer structures and torsionfree nodes in steelglass constructions on the other hand.
2.1 Torsionfree nodes
Definition Assume that all edges of a mesh are equipped with a plane which contains that edge. A vertex where the intersection of planes associated with adjacent edges is a straight line is called a node without torsion, and that line is called the node axis.
Problem Is it possible to find meshes (and associated planes) such that vertices do not exhibit torsion, possibly by minimally changing an existing mesh?
2.2 Meshes with planar faces
Very often a steel construction is required to have planar faces for the simple reason that its faces have to be covered by planar glass panels. The planarity is of course easy to fulfill in case of triangle meshes (which do not admit torsionfree nodes), but this is not the case for quad meshes. From the architects’ side quad meshes have therefore become attractive (see, for example, [8, 9]), but actual designs relied on simple constructions of meshes, such as parallel translation of one polyline along another polyline. The following problem turned out to be not so easy:
Problem Approximate a given surface by a quad mesh$v:{\mathbb{Z}}^{2}\to {\mathbb{R}}^{3}$with planar faces. The same question is asked for slightly more general combinatorics (quaddominant meshes).
where the symbol ‘∨’ means the straight line spanned by two points. In practice this target functional has to be augmented by terms which penalize deviation from the reference surface and by a regularization term (for example, one which penalizes deviation of 2nd order differences from their previous values).
Caveat. Design of freeform architecture does not work such that an amorphous ‘shape’ is created, and this shape is subsequently approximated by a PQ mesh for the purpose of making a steelglass structure. The edges of such a decomposition into planar parts are highly visible and therefore must be part of the original design process. Nowadays it is possible to incorporate the PQ property already in the design phase, for instance by a plugin for the widely used software Rhino (see [13]).
2.3 Meshes with offsets
For multilayer constructions the following question is relevant:
Problem Find an offset pair M, ${M}^{\prime}$of PQ meshes which approximate a given surface (meaning these meshes are at constant distance from each other).
The distance referred to here can be measured between planes (which are then parallel), leading to a face offset pair of meshes; or it can be measured between edges (an edge offset pair, implying the same parallelity) or between vertices (if corresponding edges are parallel, we call this a vertex offset pair). For a systematic treatment of this topic we refer to [15]. A weaker requirement is the existence of a parallel mesh${M}^{\prime}$ which is combinatorially equivalent to M but whose edges are parallel to their respective corresponding edge in M (here translated and scaled copies of M do not count).

A polyhedral surface is capable of torsionfree nodes essentially if and only if it has a nontrivial parallel mesh. This is illustrated by Figure 6.

A polyhedral surface has a face/edge/vertex offset if and only if there is a parallel mesh whose faces/edges/vertices are tangent to the unit sphere.
 A PQ mesh has a face offset if and only if in each vertex the two sums of opposite angles between edges are equal. Optimization of a quadrilateral mesh such that its faces become planar, and such that in addition it has a faceoffset, is done by augmenting (2) further by the functional (see Figure 7):$\sum _{\mathrm{angles}\phantom{\rule{0.25em}{0ex}}{\omega}_{1},\dots ,{\omega}_{4}\phantom{\rule{0.25em}{0ex}}\mathrm{a}\mathrm{t}\phantom{\rule{0.25em}{0ex}}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{x}}{({\omega}_{1}{\omega}_{2}+{\omega}_{3}{\omega}_{4})}^{2}.$(3)

Similarly, a PQ mesh with convex faces has a vertex offset if and only if in each face the sums of opposite angles are equal (these sums then equal π).

Any surface can be approximated by a PQ mesh which has vertex offsets, and the same for face offsets: initialize optimization from the network of principal curves. The class of meshes with edge offsets is more restricted. For more details see [12, 15].
2.4 Curvatures of polyhedral surfaces
where H, K are Gaussian and mean curvatures, respectively. It therefore makes sense to call ${H}_{f}$, ${K}_{f}$ in (4) the mean curvature and Gaussian curvature of the face f (w.r.t. to the offset ${M}^{\prime}$).
This definition is remarkable in so far as notable constructions of discrete minimal surfaces such as [16] turn out to have zero mean curvature in this sense. For details and further developments we refer to [11, 17, 18].
3 Conformal uniformization
Uniformization in general refers to finding a list of model domains and model surfaces such that ‘all’ domains/surfaces under consideration can be conformally mapped to one of the models. The unit disk and the unit sphere serve this purpose for the simply connected surfaces with boundary and for the simply connected closed surfaces without boundary, respectively.
Surfaces which are topologically equivalent to an annulus are conformally equivalent to a special annulus of the form $\{z\in \mathbb{C}\mid {r}_{1}<z<{r}_{2}\}$, but the ratio ${r}_{1}:{r}_{2}$ is a conformal invariant, and there is a continuum of annuli which are mutually nonequivalent via conformal mappings. A classical theorem states that planar domains with n holes are conformally equivalent to a circular domain with n circular holes, and that domain is unique up to Möbius transformations. A similar result, whose statement requires the concept of Riemann surface, is true for surfaces of higher genus with finitely many boundary components.
3.1 Circlepacking meshes
It is very interesting how the previous paragraph is related to the following question, which for designers of freeform architecture is interesting to know the answer to:
Problem Find a covering (within tolerance) of a surface by a circle pattern of mainly regularhexagonal combinatorics.
It turns out that this and similar questions can be answered if one can solve the following:
Problem Given is a triangle mesh M. Find a triangle mesh${M}^{\prime}$such that all pairs of neighbouring incircles of${M}^{\prime}$have a common point, but such that${M}^{\prime}$approximates M.
3.2 Discrete conformal mappings
If we had optimized triangular subdivisions of planar domains instead of spatial triangle meshes, the reason for the behaviour of numerical optimization mentioned in the previous paragraph would be clear. This is because in the planar case the CP meshes and associated ball packings are the same as the circle packings as studied by [20], and for these much is known: Two combinatorially equivalent circle packings constitute a discreteconformal mapping of domains, and one can show convergence to the classical conformal mappings when packings are refined (in fact, this approach to conformal mappings was used in the proof of an extension of the Koebe normal form theorem to more general domains by He and Schramm in [21]).
To sum up, if a planar CP mesh M covers a domain D, then the conformal equivalence class of D is stored in M’s combinatorics. We can optimize a general triangle mesh which overs D towards the CP property only if the conformal class of D agrees with the conformal class stored in the combinatorics of the mesh. Except for topological disks and topological spheres, it is unlikely that this equality happens. As to surfaces, we state
Problem Show that the natural correspondence between combinatorially equivalent CP meshes approximates a conformal mapping of surfaces (and make this statement precise by using an appropriate notion of refinement).
Unfortunately this problem  the only one in this paper which involves mathematics and not applications  is currently unsolved. There is, however, strong numerical evidence for an affirmative answer, and there is the known analogous planar case.
4 The paneling problem
4.1 Global panel optimization
A simple decomposition of a freeform facade typically leads to individual panels with no two of them being identical: in the worst case, their manufacturing is possible only by first manufacturing a mold for each. [22] presents a procedure which combines both combinatorial and continuous optimization in an effort to reduce the total cost, and which is based on the concept of mold reuse. The idea is that a guiding curve network is given. Such guiding curves are highly visible on the finished building so it is safe to assume the architect has firm ideas on their shape! We seek a decomposition of the facade into pieces which are easily manufacturable.
The production processes employed here may be of different kinds: Flat pieces are easily made by cutting them out from readily available panels; cylindershaped pieces have to be bent by a machine which is not cheap; truly freeform pieces have to be shaped by hot bending, using a mold which has to be specially made and which is far from cheap. Note that once a mold is available, we can use it to manufacture any surface which by a Euclidean congruence transformation can be moved so as to be a subset of the mold surface. Using the word ‘mold’ for all kinds of production processes, we state:
Problem Find out how the given panels may be replaced by other panels which can be produced by a small number of molds, thereby minimizing production cost under the side condition that the overall surface does not change visibly.
 (1)
specify a set M of admissible molds. Each mold m has an integer type $i(m)$ and a shape $\sigma (m)$ which typically is some ntuple of reals. There are costs ${\alpha}_{i}$ of providing a mold of type i, and costs ${\beta}_{i}$ of producing a panel from such a mold;
 (2)find an assignment $\mu :P\to M$ of molds to panels, such that the total cost$\sum _{m\in \mu (P)}{\alpha}_{i(m)}+\sum _{p\in P}{\beta}_{i(\mu (p))}$
is minimal, under the sideconditions of bounded deviation from the curve network and bounded kink angles.
The high complexity of this optimization task is caused by the sheer number of panels (thousands) and the coupling of different panels if they are assigned to the same mold. It turns out that it contains an NP hard subproblem. Several devices for acceleration are employed, for example, fast estimates from above of the distance of molds in shape space. For details we refer to [22, 23].
4.2 Wooden panels: level set methods
If a bendable rectangle is forced to lie on a surface, it roughly follows a geodesic curve on that surface (see, for example, [25]). These geodesics are defined by having zero geodesic curvature, and at the same time they are the shortest paths on the surface. The covering of a surface by such panels requires the solution of the following geometric problem:
Problem Find a layout of a pattern of geodesics which are (within tolerance) at equal distance from each other.
(see [26], p. 142]). We accordingly minimize a target functional which combines the competing ${L}^{2}$ norms of both (7) and (8) together with $\parallel \Delta \varphi {\parallel}_{{L}^{2}}$ and additional terms which penalize deviation from other desired properties like prescribed directions, and so on.
This is numerically done by describing the underlying surface as a triangle mesh with typically <10^{6} vertices, and considering ϕ as function on the vertices, with piecewiselinear interpolation in the faces of the mesh. The gradient of such a function is then piecewise constant, and for any vector field X and vertex v, we evaluate $(divX)(v)$ from the flux of X through the boundary of v’s intrinsic Voronoi cell. The resulting optimization problem is solved by standard GaussNewton methods (similar to the other problems of numerical optimization we considered above), augmented by Cholmod for sparse Cholesky factorization [29].
4.3 Segmentation: image processing methods
In general it is not possible to cover a surface by a smooth pattern of panels which in their unbent state are rectangular or at least cut from rectangles. It is necessary to perform segmentation into panelizable parts. This can be formulated as follows:
Problem Decompose a given surface Φ into a finite number of domains with piecewisesmooth boundaries each of which may be covered by a (within tolerance) constantdistance pattern of geodesics.
The dependence on the point $p\in \Phi $ is indicated by the subscript to the scalar products. The norm $\parallel {\gamma}_{V,p}\parallel $ measures how far V deviates from the geodesic property locally around the point p.
Notes
Acknowledgements
This article surveys results obtained within the framework oft the National Research Network ‘Industrial Geometry’ (grants No. S9206, S9209, Austrian Science Fund). It was partly supported by the 7th framework programme of EU (grant agreement ‘ARC’ No. 230520) and by grant No. 813391 of the Austrian Science Promotion Agency (FFG). We are grateful to coauthors of all papers referenced in the paper for the use of figures, and to Waagner Biro Stahlbau, Vienna, to Zaha Hadid Architects, London, to RFR, Paris, and to Evolute GmbH, Vienna, for making geometry data available to us.
Supplementary material
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