Geometric Computing for Freeform Architecture

Johannes Wallner1,2, Helmut Pottmann2,3

1 TU Graz. Kopernikusgasse 24, 8010 Graz, Austria.2 TU Wien. Wiedner Hauptstr. 810, 1040 Wien, Austria3 King Abdullah University of Science and Technology, Saudi Arabia

Email: Johannes Wallner - j.wallner@tugraz.at; Helmut Pottmann - helmut.pottmann@kaust.edu.sa;

Corresponding author

Abstract

Geometric computing has recently found a new field of applications, namely the various geometric problems whichlie at the heart of rationalization and construction-aware design processes of freeform architecture. We reporton 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 circle-packing properties (which isrelated to conformal uniformization), and with the paneling problem. We emphasize the combination of numericaloptimization and geometric knowledge.

1 BackgroundThe time of writing this survey paper coincides withthe summing up of a six-year so-called national re-search network entitled Industrial geometry whichwas funded by the Austrian Science Fund (FWF).By serendipity at the same point in time where thefirst Ph.D. students started their work in this project,a whole new direction of research in applied geom-etry turned up: meshes and three-dimensional geo-metric structures which are relevant for rationaliza-tion and construction-aware design in freeform archi-tecture. It turned out to be fruitful and rewarding,and of course it is also a topic which perfectly fitsthe heading of Industrial Geometry.

It seems that everybody who is in the businessof actually realizing freeform architectural designsas a steel-glass construction, or in concrete, or bymeans of a wooden paneling, quickly encounters thelimits 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 aconsequence there is great demand for a systematicapproach and, most importantly, a full understand-ing of the geometric possibilities and obstructionsinherent in obstacles which present themselves.

We were able to expand knowledge in this di-rection by applying geometry, differential geometry,and geometric algorithms to some of those prob-lems. Cooperation with industry was essential here.We were fortunate to work with with Waagner-BiroStahlbau (Vienna), RFR (Paris), and Evolute (Vi-enna), who provided much-need validation, informa-tion on actual problems, and real-world data. Re-markably the process of applying he known theoryto practical problems also worked in reverse: appli-cations have directly led to research in pure mathe-matics. In this paper we survey some developmentswhich we see as significant:

The discrete differential geometry of quadri-lateral meshes and the sphere geometries of Mobius,

1

Figure 1: Node with Torsion. Manufacturing a vertexwhere symmetry planes of incoming beams do not inter-sect properly is demanding, especially the central part(image courtesy Waagner-Biro Stahlbau).

Figure 2: Nodes without Torsion. If we can align symme-try planes of beams along edges such that they intersectin a common axis, node construction is much simplified.

Laguerre, and Lie, are closely related to the realiza-tion of freeform shapes as steel-glass constructionswith so-called torsion free nodes.

Conformal uniformization appears in connec-tion with circle-packing meshes and derived triangleand hexagonal meshes.

Optimization using various ideas ranging fromcombinatorial optimization to image processing is in-strumental in solving the paneling problem, i.e., therationalization of freeform shapes via decompositioninto simple and repetitive elements.

Our research is part of the emerging interdisci-plinary field of architectural geometry. The inter-ested reader is referred to the proceedings volume[1] which collects recent contributions from differ-ent areas (mathematics, engineering, architecture),

the textbook [2], and the articles [35]. Our aim isto convince the reader that many issues in freeformarchitecture can be dealt with by meshes or othergeometric structures with certain local properties.Further, that we are capable of formulating targetfunctionals for optimization which if successful achieve these properties. It is however important toknow that in many cases optimization without addi-tional geometric knowledge (utilized e.g. by way ofinitialization) does not succeed.

2 Multilayer StructuresIn this section we deal with the remarkable inter-relation between the discrete differential geometryof polyhedral surfaces on the one hand, and prob-lems regarding multilayer structures and torsion-freenodes in steel-glass constructions on the other hand.

2.1 Torsion-Free NodesIn order to realize a designers intended shape as asteel-glass construction, it is in principle easy to finda triangle mesh which approximates that shape, andlet beams follow the edges of this mesh, with glasspanels covering the faces. This is in fact a very com-mon method. Experience shows that here often themanufacturing of nodes is more complex than onewould wish (see Figures 1 and 2), which is caused bythe phenomenon that the symmetry planes of beamswhich run into a vertex do not intersect nicely in acommon node axis. The basic underlying geometricquestion is phrased in the following terms:

Definition. Assume that all edges of a mesh areequipped with a plane which contains that edge. Avertex where the intersection of planes associated withadjacent edges is a straight line is called a node with-out torsion, and that line is called the node axis.

Problem. Is it possible to find meshes (and associ-ated planes) such that vertices do not exhibit torsion,possibly by minimally changing an existing mesh?

The answer for triangle meshes is no, there arenot enough degrees of freedom available. For a quadri-lateral mesh this is different, and we demonstrate anexample which has actually been built: The outerskin of the Yas Island hotel in Abu Dhabi which wascompleted in 2009 exhibits a quadrilateral mesh withnon-planar faces which are not covered by glass in a

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Figure 3: Mesh Design. Top: In order to achieve a meshwhich follows the architects design one can employ aniterative procedure which consists of a subdivision pro-cess [6], a switch to diagonals, and mesh optimizationsuch that the resulting mesh can be equipped with beamswithout node torsion (image courtesy Evolute GmbH).Bottom: final mesh corresponding to Figure 4 (imagecourtesy Waagner-Biro Stahlbau, cf. [7]).

watertight way. Figure 3 illustrates the tools fromGeometric Modeling (subdivision) employed in gen-erating the mesh, the final result is illustrated byFigure 4.

2.2 Meshes with Planar FacesVery often a steel construction is required to haveplanar faces for the simple reason that its faces haveto be covered by planar glass panels. The planarityis of course easy to fulfill in case of triangle meshes(which do not admit torsion-free nodes), but this isnot the case for quad meshes. From the architectsside quad meshes have therefore become attractive(see e.g. [8, 9]), but actual designs relied on simpleconstructions of meshes, such as parallel translationof one polyline along another polyline. The followingproblem turned out to be not so easy:

Problem. Approximate a given surface by a quadmesh v : Z2 R3 with planar faces. The samequestion is asked for slightly more general combina-torics (quad-dominant meshes).

We call such meshes PQ meshes. R. Sauer (see the

Figure 4: Yas Island Hotel during construction. At bot-tom left one can see a detail of the outer skin which ex-hibits torsion-free nodes (images courtesy Waagner-BiroStahlbau).

monograph [10]) has already remarked that a dis-crete surfaces PQ property is analogous to the con-jugate property of a smooth surface x(u, v), whichreads

det(ux, vx, uvx) = 0. (1)

In [11] the convergence of PQ meshes towards smoothconjugate surfaces is treated in a rigorous way. Nu-merical optimization of a mesh towards the PQ prop-erty has been done by [12]. Meanwhile it has turnedout that from the viewpoint of numerics, planarityof quads is best achieved if we employ a target func-tional which penalizes non-intersecting diagonals ofquadrilaterals:

faces v1v2v3v4dist(v1 v3, v1 v4), (2)

where the symbol means the straight line spannedby two points. In practice this target functional hasto be augmented by terms which penalize deviationfrom the reference surface and by a regularizationterm (e.g. one which penalizes deviation of 2nd or-der differences from their previous values).

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Figure 5: Surface Analysis Islamic Art Museum inthe Louvre, Paris (Bellini Architects). Top: A quad-dominant mesh which follows a so-called network of con-jugate curves can be made such that faces are planar.Unfortunately this surface geometry does not leave ussufficient degrees of freedom to achieve a satisfactoryquad mesh [14]. Bottom: Hybrid tri/quad mesh solu-tions with planar faces posses more degrees of freedom(images courtesy A. Schiftner). The one at bottom righthas been realized.

Since (2) like any other equivalent target func-tional whose minimization expresses planarity ishighly nonlinear and non-convex, proper initializa-tion is important. Essential information on how toinitialize is provided by (1): A mesh covering a givensurface can be successfully optimized to becomePQ only if the mesh polylines follow the parameterlines of a conjugate parametrization of the surface. One example of such a curve network is the net-work of principal curvature lines, as demonstratedby Figure 5.

Caveat. Design of freeform architecture does notwork such that an amorphous shape is created, andthis shape is subsequently approximated by a PQmesh for the purpose of making a steel-glass struc-ture. The edges of such a decomposition into planarparts are highly visible and therefore must be partof the original design process. Nowadays it is pos-sible to incorporate the PQ property already in thedesign phase, for instance by a plugin for the widelyused software Rhino (see [13]).

Figure 6: Relation Torsion-free Nodes MultilayerStructures. This image shows an outer layer in frontand an inner layer behind it; these two layes are basedon parallel meshes. One can clearly observe that theplanes which connect corresponding edges serve as thesymmetry planes of beams, and for each vertex theseplanes intersect in a node axis, which connects corre-sponding vertices (image courtesy B. Schneider).

2.3 Meshes with OffsetsFor multilayer constructions the following questionis relevant:

Problem. Find an offset pair M,M of PQ mesheswhich approximate a given surface (meaning thesemeshes are at constant distance from each other).

The distance referred to here can be measured be-tween planes (which are then parallel), leading to aface offset pair of meshes; or it can be measuredbetween edges (an edge offset pair, implying thesame parallelity) or between vertices (if correspond-ing edges are parallel, we call this a vertex offsetpair). For a systematic treatment of this topic werefer to [15]. A weaker requirement is the existenceof a parallel mesh M which is combinatorially equiv-alent to M but whose edges are parallel to their re-spective corresponding edge in M (here translatedand scaled copies of M do not count).

There are several nice relations and characteriza-tions of the various properties of meshes mentionedabove. We use the term polyhedral surface to em-phasize that the faces of a mesh are planar.

A polyhedral surface is capable of torsion-freenodes essentially if and only if it has a nontrivialparallel mesh. This is illustrated by Figure 6.

A polyhedral surface has a face/edge/vertexoffset if and only if there is a parallel mesh whosefaces/edges/vertices are tangent to the unit sphere.

4

Figure 7: This mesh which possesses a face-face offsetat constant distance has been created by an iterativedesign process which employs subdivision and optimiza-tion using both (2) and (3) in an alternating way (imagecourtesy B. Schneider).

A PQ mesh has a face offset if and only ifin each vertex the two sums of opposite angles be-tween edges are equal. Optimization of a quadri-lateral mesh such that its faces become planar, andsuch that in addition it has a face-offset, is done byaugmenting (2) further by the functional

angles 1,...,4 at vertex

(1 2 + 3 4)2. (3)

Similarly, a PQ mesh with convex faces has avertex offset if and only if in each face the sums ofopposite angles are equal (these sums then equal pi).

Any surface can be approximated by a PQmesh which has vertex offsets, and the same for faceoffsets: initialize optimization from the network ofprincipal curves. The class of meshes with edge off-sets is more restricted. For more details see [12,15].

2.4 Curvatures of Polyhedral SurfacesA pair of parallel meshes M,M which are thoughtto be at distance d can be used to define curvaturesof the faces of M . Note that the set of meshes com-binatorially equivalent to M is a linear space, andthe meshes parallel to M constitute a linear sub-space. Consider the vertex-wise linear combinationMr = (1 rd )M + rdM and the area A(fr) of a faceof Mr as r changes: it is not difficult to see that wehave

A(fd) = A(f)(1 2dHf + d2Kf ). (4)

The coefficients Hf ,Kf are expressible via areas andso-called mixed areas of corresponding faces inM,M .This expression is analogous to the well-known Stei-ners formula: The area of an offset surface r atdistance r of a smooth surface is given by the sur-face integral

A(d) =

(1 2dH(x) + d2K(x))d(x), (5)

where H,K are Gaussian and mean curvatures, re-spectively. It therefore makes sense to call Hf ,Kfin (4) the mean curvature and Gaussian curvatureof the face f (w.r.t. to the offset M ).

This definition is remarkable in so far as notableconstructions of discrete minimal surfaces such as[16] turn out to have zero mean curvature in thissense. For details and further developments we referto [11,17,18].

3 Conformal UniformizationUniformization in general refers to finding a list ofmodel domains and model surfaces such that alldomains/surfaces under consideration can be con-formally mapped to one of the models. The unitdisk and the unit sphere serve this purpose for thesimply connected surfaces with boundary and for thesimply connected closed surfaces without boundary,respectively.

Surfaces which are topologically equivalent to anannulus are conformally equivalent to a special an-nulus of the form {z C | r1 < |z| < r2}, but theratio r1 : r2 is a conformal invariant, and there is acontinuum of annuli which are mutually non-equiv-alent via conformal mappings. A classical theoremstates that planar domains with n holes are confor-mally equivalent to a circular domain with n circu-lar holes, and that domain is unique up to Mobiustransformations. A similar result, whose statementrequires the concept of Riemann surface, is true forsurfaces of higher genus with finitely many boundarycomponents.

3.1 Circle-packing MeshesIt is very interesting how the previous paragraph isrelated to the following question, which for designersof freeform architecture is interesting to know theanswer to:

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Problem. Find a covering (within tolerance) of asurface by a circle pattern of mainly regular-hexago-nal combinatorics.

It turns out that this and similar questions can beanswered if one can solve the following:

Problem. Given is a triangle mesh M . Find a tri-angle mesh M such that all pairs of neighbouringincircles of M have a common point, but such thatM approximates M .

The property involving incircles is called the circle-packing (CP) property, and it is not difficult tosee that such meshes are characterized by certainedge length equalities as shown by Figure 9. We cantherefore set up a target functional for optimizationof a mesh M which approximates a surface :

F (M) =

lengths l1,...,l4of triangle pair

(l1 + l3 l2 l4)2

+

vertices vt-dist(v,)2

+

bdry vertices vt-dist(v, )2. (6)

The distances are measured to the tangent plane inthe closest-point projection onto ; and similarly forthe boundary curves tangent.

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

In [19] we discuss the relevance of these meshesfor freeform architecture which is mainly due to thefact that we can cover surfaces with approximate cir-cle patterns of hexagonal combinatorics, with hybridtri-hex structures with excellent statics, and otherderived constructions (see Figures 11 and 12).

=

Figure 8: Optimization of an irregular triangle mesh to-wards the circle-packing (CP) property.

l1

l2

l3l4

Figure 9: A triangle pair as shown has the incircle-pack-ing property l1 + l3 = l2 + l4.

Figure 10: The CP property is equivalent to the existenceof a ball packing with centers in the vertices. Here redballs correspond to vertices with valence 6= 6.

Numerical experiments show that optimizing atriangle mesh towards the CP property works in ex-actly those cases where topological equivalence im-plies conformal equivalence, but does not work oth-erwise. In the case of a surface topologically equiv-alent to an annulus (Figure 13) optimization worksonly if one of the two boundary curves is allowed tomove freely.

3.2 Discrete Conformal MappingsIf we had optimized triangular subdivisions of pla-nar domains instead of spatial triangle meshes, thereason for the behaviour of numerical optimizationmentioned in the previous paragraph would be clear.This is because in the planar case the CP meshesand associated ball packings are the same as the cir-cle packings as studied by [20], and for these much isknown: Two combinatorially equivalent circle pack-ings constitute a discrete-conformal mapping of do-mains, and one can show convergence to the classicalconformal mappings when packings are refined (infact, this approach to conformal mappings was usedin the proof of an extension of the Koebe normal

6

Figure 11: Structures derived from CP meshes: Approxi-mate circle packing with hexagonal combinatorics whichcovers a freeform design.

Figure 12: Structures derived from CP meshes: Hybridtri-hex structure with planar facets and support struc-ture derived from a CP mesh [19].

form theorem to more general domains by He andSchramm in [21]).

To sum up, if a planar CP mesh M covers a do-main D, then the conformal equivalence class of Dis stored in M s combinatorics. We can optimize ageneral triangle mesh which overs D towards the CPproperty only if the conformal class of D agrees withthe conformal class stored in the combinatorics of themesh. Except for topological disks and topologicalspheres, it is unlikely that this equality happens. Asto surfaces, we state

Problem. Show that the natural correspondence be-tween combinatorially equivalent CP meshes approx-imates a conformal mapping of surfaces (and makethis statement precise by using an appropriate notionof refinement).

Figure 13: CP-optimization. Top: Great Court Roof,British Museum, London. Triangle mesh by Chris Wil-liams (as built). Bottom: Combinatorially equivalentmesh with the CP property which approximates the samesurface and its outer boundary. The inner boundarycould move freely during optimization.

Unfortunately this problem the only one in this pa-per which involves mathematics and not applications is currently unsolved. There is, however, strongnumerical evidence for an affirmative answer, andthere is the known analogous planar case.

4 The paneling problemAn important part of the realization process of free-form skins is their decomposition into smaller parts(called panels) such that the entire cost of manu-facturing and handling is as small as possible, andsuch that the numerous side-conditions concerningdimensions, overall smoothness, etc. are satisfied. Inaddition any resolution of the given design into pan-els must not visibly deviate from the original archi-tects design.

7

Figure 14: A freeform design (National Holding Head-quarters, Abu Dhabi, by Zaha Hadid Architects).

(a)(b)

(c)

planarcylindricalparaboloid

toruscubiccustom

gap kink cost(a) 6 mm 1 54173(b) 6 mm 3 27418(c) 6 mm 9 18672

Figure 15: Optimal Decomposition into Panels. Top:the design of Figure 14 is cut into panels along a givennetwork of curves. Panels are of various types (planar,cylindrical, etc.). The total cost depends on the requiredquality.

4.1 Global panel optimizationA simple decomposition of a freeform facade typi-cally leads to individual panels with no two of thembeing identical: in the worst case, their manufactur-ing is possible only by first manufacturing a mold foreach. [22] presents a procedure which combines bothcombinatorial and continuous optimization in an ef-fort to reduce the total cost, and which is based onthe concept of mold reuse. The idea is that a guid-ing curve network is given. Such guiding curves arehighly visible on the finished building so it is safe toassume 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 cut-ting them out from readily available panels; cylinder-shaped pieces have to be bent by a machine which isnot cheap; truly freeform pieces have to be shapedby hot bending, using a mold which has to be spe-cially made and which is far from cheap. Note thatonce a mold is available, we can use it to manufac-ture any surface which by a Euclidean congruencetransformation can be moved so as to be a subsetof the mold surface. Using the word mold for allkinds of production processes, we state:

Problem. Find out how the given panels may bereplaced by other panels which can be produced by asmall number of molds, thereby minimizing produc-tion cost under the side condition that the overallsurface does not change visibly.

A more precise problem statement is the following:Given a network of curves on a freeform surfacewhich is thereby dissected into a collection P of pan-els,

(1) specify a set M of admissible molds. Eachmold m has an integer type i(m) and a shape (m)which typically is some n-tuple of reals. There arecosts i of providing a mold of type i, and costs iof producing a panel from such a mold;

(2) Find an assignment : P M of molds topanels, such that the total cost

m(P )i(m) +

pP

i((p))

is minimal, under the side-conditions of bounded de-viation from the curve network and bounded kinkangles.

orig. (a)

(b) (c)

Figure 16: Reflection lines help in visually inspecting theachieved surface quality for the three examples (a)(c)shown by Figure 15.

8

An implication of this simple cost model is thatone should favour cheap production processes/molds,and if an expensive one is necessary it should beused to produce more than just one panel. Opti-mization contains a discrete part similar to set cover(the mold type assignment) and a continuous part(choosing the mold shapes). An example is shownby Figures 1416.

The high complexity of this optimization task iscaused by the sheer number of panels (thousands)and the coupling of different panels if they are as-signed to the same mold. It turns out that it con-tains an NP hard subproblem. Several devices foracceleration are employed, e.g., fast estimates fromabove of the distance of molds in shape space. Fordetails we refer to [22,23].

Figure 17: Experimental Geodesic Pattern [24].

4.2 Wooden Panels: Level Set MethodsIf a bendable rectangle is forced to lie on a surface, itroughly follows a geodesic curve on that surface (seee.g. [25]). These geodesics are defined by having zerogeodesic curvature, and at the same time they arethe shortest paths on the surface. The covering ofa surface by such panels requires the solution of thefollowing geometric problem:

Problem. Find a layout of a pattern of geodesicswhich are (within tolerance) at equal distance fromeach other.

The literature contains suggestions for experi-mental solutions of this problem (see Figure 17).Firstly it must be said that very few surfaces possesspatterns of geodesics which run parallel at constantdistance: They exist precisely on the intrinsicallyflat surfaces with vanishing Gaussian curvature. For

infinitesimally close geodesics, the distance has theform w(s), where 1, s is an arc length param-eter, and w(s) obeys the Jacobi differential equationw+Kw = 0, where K is the Gaussian curvature. IfK > 0 it is not difficult to show that every intervallonger than pi/

K contains a zero of w(s), so it is

not even possible that two geodesics run side by sidewithout intersection [26].

In [27,28] we have shown how to algorithmicallyapproach the problem of laying out a pattern ofnear-geodesics which have approximately constantdistance from each other. A level set method turnsout to be useful: It is well known that equidistantcurves may be seen as level sets of a function forwhich

1 (7)vanishes (i.e., fulfills the eikonal equation). Thegeodesic property of level sets is expressed by van-ishing of

div(

)(8)

(see [26, p. 142]). We accordingly minimize a targetfunctional which combines the competing L2 normsof both (7) and (8) together with L2 and ad-ditional terms which penalize deviation from otherdesired properties like prescribed directions, etc.

This is numerically done by describing the un-derlying surface as a triangle mesh with typically< 106 vertices, and considering as function on thevertices, with piecewise-linear interpolation in thefaces of the mesh. The gradient of such a function isthen piecewise constant, and for any vector field Xand vertex v, we evaluate (divX)(v) from the fluxof X through the boundary of vs intrinsic Voronoicell. The resulting optimization problem is solvedby standard Gauss-Newton methods (similar to theother problems of numerical optimization we con-sidered above), augmented by Cholmod for sparseCholesky factorization [29].

4.3 Segmentation: Image Processing MethodsIn general it is not possible to cover a surface bya smooth pattern of panels which in their un-bentstate are rectangular or at least cut from rectangles.It is necessary to perform segmentation into panel-izable parts. This can be formulated as follows:

Problem. Decompose a given surface into a fi-nite number of domains with piecewise-smooth bound-

9

Figure 18: Piecewise-Geodesic Vector Fields. Left: A de-sign vector field W . Right: A piecewise-geodesic vectorfield V which approximates W and whose discontinuitieslead to segmentation of the underlying surface [27].

aries each of which may be covered by a (within tol-erance) constant-distance pattern of geodesics.

For that purpose we describe families of curves asintegral curves of a unit vector field. It turns out thatthe geodesic property can be characterized by thesymmetry of the covariant derivative mapping X 7XV , which is linear within each tangent space:

V,p(X,Y ) = XV, Y p X,Y V p = 0. (9)

The dependence on the point p is indicatedby the subscript to the scalar products. The normV,pmeasures how far V deviates from the geodesicproperty locally around the point p.

Any unit vector field W , and in particular onewhich has been created interactively by a designer,can now be approximated by a piecewise geodesicvector field V which occurs as a minimizer of a targetfunctional suitable constructed from weighted inte-grals of the functions

(V,p), where (x) = x21 + x2 ,V W2

together with regularizing terms. For details we referto [27]. The function could be any of the heavy-tailed functions used in image sharpening (cf. [30]),we used the Geman-McClure estimator [31]. Thiscauses high values of p,V to be concentrated alongcurve-like regions which become the boundaries ofdomains. For the actual segmentation we employedthe method of [32]. An example is shown by Figures18 and 19.

Figure 19: A design with bent rectangular panels basedon segmentation of the surface of Figure 18. The under-lying patterns of geodesics have been found by a furthermethod not described in this paper (see [27]).

AcknowledgmentsThis article surveys results obtained within the frame-work oft the National Research Network Industrial Ge-ometry (grant No. S9206, S9209, Austrian Science Fund).It was partly supported by the 7th framework programmeof EU (grant agreement ARC No. 230520) and by grantNo. 813391 of the Austrian Science Promotion Agency(FFG). We are grateful to coauthors of all papers refer-enced in the paper for the use of figures, and to WaagnerBiro Stahlbau, Vienna, to Zaha Hadid Architects, Lon-don, to RFR, Paris, and to Evolute GmbH, Vienna, formaking geometry data available to us.

References1. Ceccato C, Hesselgren P, Pauly M, Pottmann H, Wall-

ner J: Advances in Architectural Geometry 2010. Springer2010.

2. Pottmann H, Asperl A, Hofer M, Kilian A: ArchitecturalGeometry. Bentley Institute Press 2007, [http://www.architecturalgeometry.at].

3. Pottmann H: Geometry and new and future spatial pat-terns. Architectural Design 2009, 79(6):6065.

4. Pottmann H: Architectural geometry as design knowl-edge. Architectural Design 2010, 80(4):7277.

5. Pottmann H, Wallner J: Freiformarchitektur und Mathe-matik. Mitt. DMV 2010, 18(2):8895.

6. Warren J, Weimer H: Subdivision Methods for GeometricDesign: A Constructive Approach. Morgan Kaufmann2001.

7. Frey H, Reiser B, Ziegler R: Yas Island Marina-Hotel Pole position in Abu Dhabi. Stahlbau 2009, 78(10):758753.

8. Glymph J, Shelden D, Ceccato C, Mussel J, Schober H:A parametric strategy for freeform glass structures us-ing quadrilateral planar facets. In Proc. Acadia 2002:303321.

10

9. Schober H: Freeform Glass Structures. In Glass Process-ing Days 2003, Tampere (Fin.): Glass Processing Days2003:4650.

10. Sauer R: Differenzengeometrie. Springer 1970.

11. Bobenko A, Suris Yu: Discrete differential geometry: In-tegrable Structure. American Math. Soc. 2008.

12. Liu Y, Pottmann H, Wallner J, Yang YL, Wang W: Ge-ometric modeling with conical meshes and developablesurfaces. ACM Trans. Graphics 2006, 25(3):681689.

13. Evolute GmbH: Evolutetools for Rhino 2010, [http://www.evolute.at].

14. Zadravec M, Schiftner A, Wallner J: Designing Quad-dominant Meshes with Planar Faces. Computer GraphicsForum 2010, 29(5):16711679. [Proc. Symp. GeometryProcessing].

15. Pottmann H, Liu Y, Wallner J, Bobenko A, Wang W:Geometry of Multi-layer Freeform Structures for Archi-tecture. ACM Trans. Graphics 2007, 26(3):#65, 111.

16. Bobenko A, Hoffmann T, Springborn B: Minimal surfacesfrom circle patterns: Geometry from combinatorics. Ann.Math. 2006, 164:231264.

17. Bobenko A, Pottmann H, Wallner J: A curvature the-ory for discrete surfaces based on mesh parallelity. Math.Annalen 2010, 348:124.

18. Muller C, Wallner J: Oriented mixed area and discreteminimal surfaces. Discrete Comput. Geom. 2010, 43:303320.

19. Schiftner A, Hobinger M, Wallner J, Pottmann H: Pack-ing circles and spheres on surfaces. ACM Trans. Graphics2009, 28(5):#139, 18.

20. Stephenson K: Introduction to Circle Packing. Cam-bridge Univ. Press 2005.

21. He ZX, Schramm O: Fixed points, Koebe uniformizationand circle packings. Ann. Math. 1993, 137:369406.

22. Eigensatz M, Kilian M, Schiftner A, Mitra N, PottmannH, Pauly M: Paneling Architectural Freeform Surfaces.ACM Trans. Graphics 2010, 29(4):#45,110.

23. Eigensatz M, Deuss M, Schiftner A, Kilian M, Mitra NJ,Pottmann H, Pauly M: Case Studies in Cost-OptimizedPaneling of Architectural Freeform Surfaces. In [1]:4972.

24. Spuybroek L: NOX: Machining Architecture. Thames &Hudson 2004.

25. Pirazzi C, Weinand Y: Geodesic lines on free-form sur-faces: optimized grids for timber rib shells. In Proc.World Conference on Timber Engineering 2006. [7pp.].

26. do Carmo MP: Riemannian Geometry. Boston, Basel,Berlin: Birkauser, 2nd edition 1992.

27. Pottmann H, Huang Q, Deng B, Schiftner A, Kilian M,Guibas L, Wallner J: Geodesic Patterns. ACM Trans.Graphics 2010, 29(4):#43, 110.

28. Wallner J, Schiftner A, Kilian M, Flory S, Hobinger M,Deng B, Huang Q, Pottmann H: Tiling Freeform ShapesWith Straight Panels: Algorithmic Methods. In [1]:7386.

29. Chen Y, Davis TA, Hager WW, Rajamanickam S: Algo-rithm 887: Cholmod, Supernodal Sparse Cholesky Fac-torization and Update/Downdate. ACM Trans. Math.Softw. 2008, 35(3):#22, 114.

30. Levin A, Fergus R, Durand F, Freeman WT: Image anddepth from a conventional camera with a coded aperture.ACM Trans. Graphics 2007, 26(3):#70, 19.

31. Geman S, McClure DE: Statistical methods for to-mographic image reconstruction. Bull. Inst. Internat.Statist. 1987, 52(4):521.

32. Pauly M, Keiser R, Gross M: Multi-scale Feature Extrac-tion on Point-sampled Surfaces. Comput. Graph. Forum2003, 22(3):281290.

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