An important part of the realization process of freeform skins is their decomposition into smaller parts (called panels) such that the entire cost of manufacturing and handling is as small as possible, and such that the numerous sideconditions concerning dimensions, overall smoothness, and so on are satisfied. In addition any resolution of the given design into panels must not visibly deviate from the original architect’s design (see Figures 14 and 15).
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.
A more precise problem statement is the following: Given a network of curves on a freeform surface which is thereby dissected into a collection P of panels,

(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.
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 \u03f5\cdot w(s), where \u03f5\ll 1s is an arc length parameter, and w(s) obeys the Jacobi differential equation {w}^{\u2033}+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].
In [27, 28] we have shown how to algorithmically approach the problem of laying out a pattern of neargeodesics which have approximately constant distance from each other. A level set method turns out to be useful: It is well known that equidistant curves may be seen as level sets of a function ϕ for which
\parallel \nabla \varphi \parallel 1
(7)
vanishes (that is, ϕ fulfills the eikonal equation). The geodesic property of level sets is expressed by vanishing of
div\left(\frac{\nabla \varphi}{\parallel \nabla \varphi \parallel}\right)
(8)
(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.
For that purpose we describe families of curves as integral curves of a unit vector field. It turns out that the geodesic property can be characterized by the symmetry of the covariant derivative mapping X\mapsto {\nabla}_{X}V, which is linear within each tangent space:
{\gamma}_{V,p}(X,Y)={\u3008{\nabla}_{X}V,Y\u3009}_{p}{\u3008X,{\nabla}_{Y}V\u3009}_{p}=0.
(9)
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.
Any unit vector field W, and in particular one which has been created interactively by a designer, can now be approximated by a piecewise geodesic vector field V which occurs as a minimizer of a target functional suitable constructed from weighted integrals of the functions
\begin{array}{r}\rho (\parallel {\gamma}_{V,p}\parallel ),\phantom{\rule{1em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}\rho (x)=\frac{{x}^{2}}{1+\alpha {x}^{2}},\\ \parallel VW{\parallel}^{2}\end{array}
together with regularizing terms. For details we refer to [27]. The function ρ could be any of the heavytailed functions used in image sharpening (cf. [30]), we used the GemanMcClure estimator [31]. This causes high values of \parallel {\rho}_{p,V}\parallel to be concentrated along curvelike 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.