Geometric computing for freeform architecture

2.1 Torsion-free nodes

In order to realize a designer’s intended shape as a steel-glass construction, it is in principle easy to find a triangle mesh which approximates that shape, and let beams follow the edges of this mesh, with glass panels covering the faces. This is in fact a very common method. Experience shows that here often the manufacturing of nodes is more complex than one would wish (see Figures 1 and 2), which is caused by the phenomenon that the symmetry planes of beams which run into a vertex do not intersect nicely in a common node axis. The basic underlying geometric question is phrased in the following terms:

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?

The answer for triangle meshes is no, there are not enough degrees of freedom available. For a quadrilateral mesh this is different, and we demonstrate an example which has actually been built: The outer skin of the Yas Island hotel in Abu Dhabi which was completed in 2009 exhibits a quadrilateral mesh with non-planar faces which are not covered by glass in a watertight way. Figure 3 illustrates the tools from Geometric Modeling (subdivision) employed in generating the mesh, the final result is illustrated by Figure 4.

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 torsion-free 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 (quad-dominant 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).

Since (2) - like any other equivalent target functional whose minimization expresses planarity - is highly nonlinear and non-convex, proper initialization is important. Essential information on how to initialize is provided by (1): A mesh covering a given surface Φ can be successfully optimized to become PQ only if the mesh polylines follow the parameter lines of a conjugate parametrization of the surface Φ. One example of such a curve network is the network of principal curvature lines, as demonstrated by Figure 5.

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 steel-glass 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).

There are several nice relations and characterizations of the various properties of meshes mentioned above. We use the term ‘polyhedral surface’ to emphasize that the faces of a mesh are planar.

• A polyhedral surface is capable of torsion-free 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 face-offset, is done by augmenting (2) further by the functional (see Figure 7):

  $\sum _{\mathrm{angles}{\omega }_1,\dots ,{\omega }_4\mathrm{a}\mathrm{t}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{x}}{({\omega }_1-{\omega }_2+{\omega }_3-{\omega }_4)}^2.$

  (3)

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].

3.1 Circle-packing 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 regular-hexagonal 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.

The distances are measured to the tangent plane in the closest-point projection onto Φ; and similarly for the boundary curve’s tangent.

We can further see from Figure 9 that a vertex has the same distance from all incircle contact points on adjacent edges: It follows that the CP property is equivalent to the existence a packing of vertex-centered balls: balls touch each other if and only if the corresponding vertices are connected by an edge (see Figure 10).

In [19] we discuss the relevance of these meshes for freeform architecture which is mainly due to the fact that we can cover surfaces with approximate circle patterns of hexagonal combinatorics, with hybrid tri-hex structures with excellent statics, and other derived constructions (see Figures 11 and 12).

Numerical experiments show that optimizing a triangle mesh towards the CP property works in exactly those cases where topological equivalence implies conformal equivalence, but does not work otherwise. In the case of a surface topologically equivalent to an annulus (Figure 13) optimization works only if one of the two boundary curves is allowed to move freely.

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 discrete-conformal 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.

4.1 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; cylinder-shaped 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.

is minimal, under the side-conditions of bounded deviation from the curve network and bounded kink angles.

An implication of this simple cost model is that one should favour cheap production processes/molds, and if an expensive one is necessary it should be used to produce more than just one panel. Optimization contains a discrete part similar to set cover (the mold type assignment) and a continuous part (choosing the mold shapes). An example is shown by Figures 14–16.

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.

The literature contains suggestions for experimental solutions of this problem (see Figure 17). Firstly it must be said that very few surfaces possess patterns of geodesics which run parallel at constant distance: They exist precisely on the intrinsically flat surfaces with vanishing Gaussian curvature. For infinitesimally close geodesics, the distance has the form $ϵ\cdot w(s)$, where $ϵ\ll 1$s is an arc length parameter, and $w(s)$ obeys the Jacobi differential equation $w^{″}+Kw=0$, where K is the Gaussian curvature. If $K>0$ it is not difficult to show that every interval longer than $\pi /\sqrt{K}$ contains a zero of $w(s)$, so it is not even possible that two geodesics run side by side without intersection [26].

(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 <10⁶ vertices, and considering ϕ as function on the vertices, with piecewise-linear 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 Gauss-Newton 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 un-bent 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 piecewise-smooth boundaries each of which may be covered by a (within tolerance) constant-distance 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.

together with regularizing terms. For details we refer to [27]. The function ρ could be any of the heavy-tailed functions used in image sharpening (cf. [30]), we used the Geman-McClure estimator [31]. This causes high values of $\parallel {\rho }_{p,V}\parallel$ to be concentrated along curve-like regions which become the boundaries of domains. For the actual segmentation we employed the method of [32]. An example is shown by Figures 18 and 19.