Wire Mesh Design
Akash Garg1∗
1

Andrew O Sageman-Furnas2
Eitan Grinspun1
Mark Pauly3

Columbia University

2

Bailin Deng3
Yonghao Yue1
2
Max Wardetzky

University of Göttingen

3

EPFL

Figure 1: Wire mesh design allows creating physical realizations (1st and 5th images) of a given design surface (2nd and 4th images)
composed of interwoven material (middle image) in an interactive, optimization-supported design process. Both the torso and the Igea face areconstructed from a single sheet of regular wire mesh.

Abstract1

We present a computational approach for designing wire meshes, i.e.,
freeform surfaces composed of woven wires arranged in a regulargrid. To facilitate shape exploration, we map material propertiesof wire meshes to the geometric model of Chebyshev nets. This
abstraction is exploited to build an efficient optimization scheme.
While the theory of Chebyshev nets suggests a highly constrained
design space, we show that allowing controlled deviations from theunderlying surface provides a rich shape space for design exploration. Our algorithm balances globally coupled material constraints
with aesthetic and geometric design objectives that can be specified
by the user in an interactive design session. In addition to sculpturalart, wire meshes represent an innovative medium for industrial applications including composite materials and architectural façades.
We demonstrate the effectiveness of our approach using a variety
of digital and physical prototypes with a level of shape complexity
unobtainable using previous methods.

Wire meshes enjoy broad application in art, architecture, and engineering, including handmade sculptures, filters, support structures incomposite materials, and architectural façades (see Fig. 3). Despite
their widespread use, a systematic design methodology for freeform
wire meshes is lacking. While physical exploration helps build
intuition in early concept design, rationalizing a surface entails numerous constraints that are often globally coupled. Artists currently
use an incremental, trial-and-error approach, where an initially flatpiece of wire mesh is gradually bent by hand to conform to a desired
surface. Likewise, in architecture wire meshes are restricted to very
simple shapes, such as planes, cylinders, cones, or half-spheres, despite great interest in freeform façades. We show that a much richer
space of wire meshes can be more effectively explored using digital
tools, which automatically account for the strong global coupling ofphysical and geometric constraints.

CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Physically based modeling; J.2 [Physicalsciences and engineering]: Engineering—Computer-aided design
Keywords: Wire mesh, interactive shape modeling, Chebyshev
nets, discrete differential geometry, design, global optimization
Links:DL

PDF∗ e-mail: akg2110@columbia.edu

ACM Reference Format
Garg, A., Sageman-Furnas, A., Deng, B., Yue, Y., Grinspun, E., Pauly, M., Wardetzky, M. 2014. Wire Mesh
Design. ACM Trans. Graph. 33, 4, Article 66 (July 2014), 12 pages. DOI = 10.1145/2601097.2601106
http://doi.acm.org/10.1145/2601097.2601106.
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DOI: http://doi.acm.org/10.1145/2601097.2601106

Introduction

While in our fabrication examples (but not for our design tool), we
have focused on wire mesh made of steel, wire mesh encompasses amuch broader range of materials, such as fishnet stockings, woven
reinforcements in composite materials, or even onion nets. Indeed,even something as prosaic as a simple onion net reveals some of thecore structural properties of wire mesh: inextensible fibers that are
woven in a criss-cross pattern such that the warp and weft directions
cannot stretch but may significantly shear towards (or away from)one another (see Fig. 5). In order to gain intuition for designing withwire mesh, one may try to “dress” a given target shape, such as avase, a bust, or a ball with an onion net. Soon one then discovers
that due to shearing some features cannot be captured, that morematerial may be required in certain areas, or that it is difficult topreserve the fine details of the given target shape.
Such difficulties are ubiquitous when working and designing withwire mesh: If a wire mesh is required to lie exactly on a given target
design surface, incremental layout often fails to adequately represent
the desired shape. We substantiate this observation by modelingwire meshes as discrete Chebyshev nets (§3), revealing fundamental
limitations in the kind of shapes that can be equipped with a single
wire mesh. Further insights from the theory of Chebyshev nets allow
us to formulate an optimization scheme where the mesh can deviate
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Choose Coarse
Wire Mesh Resolution

Load Target Shape

Explore Initial Conditions§4.1

Phase I: Partially cover target w/ interpolating wire mesh

Edit Wire Mesh

Add material §4.2.1
Cut material §4.2.2
Fix overlaps §4.2.3

Subdivide

Figure 3: Wire meshes are a popular medium in abstract (left) andfigurative (middle) sculptural art. These freeform shapes are createdby manually bending a single, flat piece in an incremental, trial-anderror process. Lacking an effective digital design process, the use ofwire meshes in architectural façades (right) is currently limited tosimple geometries.

§4.2.5

Potential Design

Global Optimization

Paint weights §4.2.4
Adjust parameters §4.2.4

Phase II: Improve coverage w/ approximating wire mesh

Figure 2: The wire mesh design process.
from the target surface in a controlled way. This scheme balances
inextensibility of wires and limits on shearing angles with design
objectives imposed by the user. Optimization is then incorporated
into an interactive design tool that leverages the user’s high-level
understanding of shape (§4). The tool interleaves user input on
artistic decisions with global optimization to explore the design
space. This facilitates an effective pipeline from surface modelingto physical realization.
In contrast to previous approaches, our methodology (i) lifts the
restriction of a priori curvature bounds for a given target shape(which is omnipresent in the mathematical literature and previous
computer-aided tools) and (ii) works without insertion of darts (i.e.,folds stitched into the material). Indeed, unlike garments, where
the inclusion of darts are considered a premium and a signature ofthoughtful design, darts generally introduce a point of failure, overengineering, or manufacturing complexity in wire mesh structures.
Instead, our design tool allows the user to interactively and iteratively
grow a wire mesh on a target shape, thus allowing fabrication with asingle piece of material.
Our contribution is an example of how geometric modeling andoptimization-based shape exploration can lead to new material-aware
design solutions that enable creative works not feasible before. The
design process (Fig. 2) has two phases: a setup phase that directly
extends previous work on interpolating Chebyshev nets which partially cover the target and a novel design loop where the designer
interactively explores approximating Chebyshev nets to increase
coverage.

2

Related Work

Computational tools for material- and fabrication-aware design haverecently become a prominent topic in computer graphics research.
A typical example are algorithms for the design and optimizationof discrete freeform surfaces with planar polygons [Liu et al. 2006;
Wang et al. 2008; Bouaziz et al. 2012; Poranne et al. 2013]. Ensuring planarity of mesh elements facilitates the use of cost-effective
materials and construction technologies, for example in stone or
ACM Transactions on Graphics, Vol. 33, No. 4, Article 66, Publication Date: July 2014

glass façades. Additional constraints such as torsion-free nodes
can be incorporated to improve structural performance and further
simplify the fabrication process [Liu et al. 2006]. Related examples include rationalization and shape exploration for developablesurfaces [Pottmann et al. 2008], geodesic patterns [Pottmann et al.
2010], curved panelings [Eigensatz et al. 2010], or circular arc
structures [Bo et al. 2011]. A series of papers exploits geometricabstractions of compression-only surfaces to facilitate the design ofself-supporting structures [Vouga et al. 2012; Panozzo et al. 2013;de Goes et al. 2013; Liu et al. 2013]. The construction of planar intersecting pieces has been investigated by Schwartzburg andPauly [2013] who map assembly and fabrication restrictions to geometric constraints of a mixed discrete/continuous optimization. The
common theme of these and other related methods is that material
behavior or physical restrictions are mapped to geometric properties
or constraints of the design surface—a methodology that we also
follow here. This strategy avoids the complexities of a full physical simulation and enables efficient computations for interactive
form-finding and design exploration.
As explained in §3, wire meshes are best modeled by Chebyshev
nets—a geometric model of woven materials using a two dimensional net composed of inextensible yarns, first proposed by Chebyshev in 1878 [Tschebyscheff 1878]. The same model is used in themechanical theory of pure networks, i.e., grids of inextensible yarns
with no shear resistance [Rivlin 1958; Rivlin 1964; Rivlin 1997].
When shearing is incorporated, the model is known as a reinforced
network [Adkins 1956]; investigations into bending resistant inextensible networks have been considered in [Wang and Pipkin 1986].
Pipkin analyzed stress in reinforced networks on arbitrary curved
surfaces [Pipkin 1984] and the distribution of wrinkles of the solution [Pipkin 1986], but assumes—different from our approach—that
the reinforced network already lies on the surface. A purely discrete
version of Chebyshev nets has been used in computer aided design
to model the forming of woven reinforced composite materials tosurfaces. However, many studies focus only on simple geometries
such as a hemisphere [Robertson et al. 1981; Ye and Daghyani 1997],
polyhedral and rounded cones [Robertson et al. 2000; Baillargeon
and Vu-Khanh 2001] or translated sine curves [Wang et al. 1999].
In terms of application domain, virtual design with Chebyshev nets
was considered by Aono et al. [1994; 1996; 2001]. Building upon
initial computational investigations [van West et al. 1990], Aono etal. presented a method for finding Chebyshev nets interpolating agiven surface via automatic dart insertions. Their approach mimics
the process by which a tailor drapes garments over the human form,
pinning initial lines of material onto a dress form, then working
outwards from these constraints, making cuts or inserting darts asWire Mesh Design

required to fit the underlying form. Our work is inspired by andbuilds upon the work of Aono and coworkers, while (i) lifting the
restriction that total Gaussian curvature be bounded by 2π (thereby
expanding the scope of possible designs), (ii) working with a single
piece of material without the need of inserting darts, (iii) expanding
the range of admissible target shapes (including cylindrical topologies), and (iv) extending modality by deeply integrating the user into
the design loop with a varied tool set.

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where K(u, v) denotes the Gaussian curvature at the point r(u, v).
This equation reveals that changes in the shearing angle directly
correspond to the encoding of curvature. Indeed, regions where
the magnitude of Gaussian curvature is high correspond to large
magnitude (close to π/2) shearing angles or a high rate of changeof shearing angles (which then leads to large magnitude shearing
angles nearby). For wire mesh design, where large magnitude shear
angles are prohibitive (see Fig. 5), this results in difficulties forcovering regions of high curvature.

Chebyshev Nets
An important obstruction that
results from the coupling between shear angle and curvature isprovided by the formula of Hazzidakis [1879]: Consider an axisaligned (with respect to the u and v parameter directions) rectangulardomain D ⊂ R2 , then it follows from Equation (1) that
Global obstructions to existence

As prototypical wire meshes we consider metal (steel)wires woven in a plain weave. In these most ubiquitouswire meshes, longitudinal warp and transversal weft
wires are interwoven (but not welded) in a criss-crosspattern (see inset figure). For the typical forces anddeformations applied to wire meshes, the stretching ofeach metal wire may be reasonably neglected, thus eachwire is adequately modeled as an inextensible elastic curve. The
elastic response of the ensemble is then governed by the bending ofeach curve and the interactions induced by the weave pattern.
The weave induces a “soft” interlocking of wire: while each wire
may slightly slide over the crossing wires, significant sliding is uncharacteristic because it occurs only under exceedingly large forces.
Consequently, adjacent contact points maintain their intrinsic distance, even for large extrinsic deformations, resulting in a structure
where warp and weft directions cannot stretch but significantly shear
towards or away from one another. The corresponding mathematicalmodel is the theory of Chebyshev nets.
Chebyshev nets are akin to the conformal parameterizations commonly used for texture mapping. While conformal maps exactly
preserve angles but allow for uniform stretching, Chebyshev nets preserve lengths along two parameter (warp and weft) directions but allow for shearing of angles between warp and weft. Let r : R2 → R3
be a parameterization of a smooth surface describing a patch of a surface in space. Then r(u, v) is Chebyshev if |ru | = |rv | = 1, where(u, v) is an orthonormal coordinate system for R2 . A collection ofsuch patches describing a whole surface, such that coordinate transitions are given by translations only, is called a smooth Chebyshev
net.

3.1

Smooth Chebyshev Net Theory

One of the intricate mathematical difficulties for constructing smooth
Chebyshev nets results from the fact that while one can locally equip
every smooth surface in 3-space with a Chebyshev net [Bieberbach
1926], this is no longer the case globally without producing singularities. This mathematical difficulty translates into very concrete
challenges in the design process. While it is locally possible to fit
a wire mesh to a given shape, there are strong global obstructions;moreover, small local changes might have drastic global effects—
making manual design cumbersome, time consuming, or intractable.
The coupling between shearing of the wire
mesh and curvature makes global existence of Chebyshev nets
on an arbitrary smooth surface a delicate matter. Let γ(u, v) be
the shear angle of a Chebyshev net r(u, v), i.e., sin γ(u, v) =ru (u, v) · rv (u, v). Simply, γ measures the signed angle deviationof the originally orthogonal warp and weft directions under the deformation of the surface. The so-called Gauß equation for the local
parameterization reads [Pipkin 1984]
Curvature and shear

K(u, v) cos γ(u, v) = γuv (u, v) ,(1)

Z
K(u, v)dA = 2π −
D

3
X

αi ,

(2)

i=0

where dA is the area element on the surface and the αi are theinterior angles of the quadrangle given by the image of the axisaligned rectangle D under the Chebyshev net r.
As a consequence of Hazzidakis’ formula, if one requires the boundary of a rectangular patch D to coincide with parameter lines, thenit is impossible to cover the image of D with a Chebyshev net if
this image has total Gauß curvature greater than 2π. Perhaps surprisingly, despite the Hazzidakis obstruction, Voss [1882] showedthat there exists a global Chebyshev net on any surface of revolution
which does not meet the rotation axis—even if total Gauß curvature
exceeds 2π. Recently, Ghys [2011] proposed a Chebyshev net on
the sphere (with singularities along two spherical arc segments at thesouth pole) that is different from the solution of Voss (with singularities at the poles where the profile curve meets the rotation axis).
These results show that Hazzidakis’ obstruction ceases to be valid if
one gives up on insisting on axis-aligned parameter domains.
Inspired by these observations, our design tool introduced in §4,
allows the user to both: construct non axis-aligned domains and to
add or remove material, thus changing the shape of the domain.
Sufficient conditions for existence Hazzidakis’ formula provides necessary conditions for the existence of Chebyshev nets. The
search for optimal results about sufficient conditions for the existenceof global Chebyshev nets on surfaces is still ongoing [Bakelman
1965; Samelson 1991; Samelson and Dayawansa 1995; Burago et al.
2007]. Although some of these works offer constructive proofs, they
do not immediately lead to a computationally feasible algorithm.
More importantly, these works assume that total negative and positive Gauß curvature does not exceed 2π in magnitude—a bound too
restrictive for real-world design—and that the resulting Chebyshev
net lies exactly on the given surface—a requirement that is neither
practical nor strictly necessary in design.

The above mathematical properties translate into practical difficulties when designing shapes with wire mesh. We tackle
these challenges in our design tool by (i) allowing the parameterdomain to be changed by the user by adding or removing material,(ii) working with Chebyshev nets nearby (but not exactly on) a given
target shape, and (iii) accommodating for the global nature of the
problem with the help of an optimizer.
Summary

3.2

Discrete Chebyshev Nets and the Role of Shear

We model wire mesh by discrete Chebyshev nets, rhombic nets
comprised of inextensible equilateral edges such that each interior
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mesh vertex has valence four. Notice that we do not require therhombi to be planar. Akin to the smooth case, discrete Chebyshev
nets have a long history in mathematics. Originally introduced forthe special case of constant Gauß curvature surfaces [Wunderlich
1951; Sauer 1970; Bobenko and Pinkall 1996; Hoffmann 1999;
Pinkall 2008], their more general theory is still an active area of
research.

60000

Ratio: 7.937
Ratio: 6.944
Ratio: 6.173
Ratio: 5.555

50000

Force [N/m2]

66:4

40000
30000
20000
10000

-20

0

0

20

40

60

80

Magnitude of shear angle [deg]Force vs. magnitude of shear angle

Figure 4: Local construction of a discrete Chebyshev net.
As in the smooth case, there exists a discrete Chebyshev net locallyaround any point p on a target surface S ⊂ R3 . Indeed, given asmall neighborhood U of p on S, choose an edge length ` and twounit vectors v, w ∈ R3 such that p1 = p + `v and p2 = p + `w
are in U . Generically the three points p, p1 , p2 determine a uniquefourth point p3 ∈ S such that the quadrilateral (p, p1 , p2 , p3 ) is a(non-planar) rhombus (see Fig. 4). To obtain p3 , consider the two
spheres of radius l around p1 and p2 , respectively, which intersect
in a circle (shown in blue). Both p and p3 lie on the intersection ofthis circle with S. Generically, p3 is unique and distinct from p.
While the wires of a wire mesh do not stretch
and offer some resistance to bending of warp or weft directions,the quadrilateral structure of the weave allows for a considerable
amount of shear, which is ultimately responsible for the rich set of
possible wire mesh deformations. We measure the discrete signed
shear angle, γ, as the deviation from 90◦ of the interior angles of
the quadrilaterals. Wire meshes exhibit little resistance to in-plane
shearing as long as the magnitude of the shear between warp andweft lines does not exceed a certain threshold. Beyond this point therequired shear force increases drastically.

Instron measurement setup

Figure 5: Measurements of shear resistance for four different wire
meshes. While very little force is required to shear the mesh initially,an exponential increase in force can be observed starting at a shear
angle magnitude of about 45◦ . In the measurements shown above,
we have tested four wire meshes with different mesh opening / wire
thickness ratios. The diameter of the wires in all meshes is 0.009
mm and the meshes have per inch 14 cells (ratio: 7.937), 16 cells(ratio: 6.944), 18 (ratio: 6.173) cells and 20 cells (ratio: 5.555),
respectively.

The role of shear

This claim is validated by several experiments that we conducted
on real materials using an Instron machine, a device that measurestensile (or compressive) forces under a prescribed deformation. As
shown in Fig. 5, the energetic cost of shearing wire mesh samplesof varying gauge is negligible for shears with magnitude below a(consistent) threshold of approximately 45◦ . To deform a wire meshbeyond this threshold requires excessive force.
These experiments validate a bounded-shear model. When optimizing for a discrete Chebyshev net, we thus restrict the magnitude of the allowable shearing to a user-defined bound; our examples set the shear limit or shear bound to γmax = π/4. Eachrhombus is thus individually restricted to interior angles between[π/2 − γmax , π/2 + γmax ]. We point out that the motivation for our
specific choice of γmax arises from experiments—other choices areindeed possible. We use the adjective realizable to refer to discrete,bounded-shear Chebyshev nets.

3.3

Building Discrete Chebyshev Nets

A prevalent way for constructing discrete Chebyshev nets is througha process called integration from appropriate initial data or initialconditions. This approach plays a central role in our implementation
for initializing a discrete Chebyshev net on a target surface.
The observation that three points of arhombus in a discrete Chebyshev net on a (smooth or triangulated)surface S determine the fourth point leads to a construction ofInterpolating integration

ACM Transactions on Graphics, Vol. 33, No. 4, Article 66, Publication Date: July 2014

Figure 6: Top: integration using Cauchy initial data (black polygonal curve). Bottom: diagonal initial data given by a curve (yellowdots) and desired angles (blue, at black dots in 2nd picture fromleft); diagonal data determine two zig-zag curves along the diagonal. Left-to-right: orange dots denote new data that are computedfrom previously known ones. Rightmost figure: bounding box shows
region in the parameter domain that can be covered.

Chebyshev nets from certain initial condition curves. To this end,
consider a curve on S that is equidistantly sampled (with respect tothe extrinsic metric of R3 ). We refer to such a curve as Cauchy initial
data. Any vertex of this curve whose adjacent edges (with respect
to its adjacent sample points) form an angle that obeys the shear
limit constraint can be used as a seed for integration, see Fig. 6 (top)for a schematic illustration of this process. We additionally allow
for specifying what we call diagonal initial data, which, different
from Cauchy data, are given by a (discrete) curve on S together with
angles (obeying the shear limit) for each curve segment. In this case,curve segments serve as diagonals of rhombi and angles specify
the two (necessarily equal) angles opposite to the diagonal in eachrhombus, see Fig. 6 (bottom). Notice that on a topological cylinder,
diagonal initial data are required since they allow for covering theentire cylinder, while Cauchy data do not—see Fig. 7 for a schematic
illustration.
While there exists a discrete
Chebyshev net of some resolution ` around every point p on S given
by initial data, it is impossible to know in advance how far these data
will propagate the net. This makes it difficult to know if particular
initial data are sufficient to construct a global discrete Chebyshev
net. There are three ways in which initial data can fail to propagate arealizable Chebyshev net to globally cover the entire target surface:(i) the generic construction of Fig. 4 fails to find a new point p3
because the circle of intersection of the constraint spheres only meetsthe surface at the original point p; (ii) the three points p, p1 , p2already determine an angle that violates the shear limit; (iii) the
Limits to interpolating integration

Wire Mesh Design

III

IV

8

I

8

II

Figure 7: Influence of initial data for a topological cylinder—in both
figures, left and right side of the square are identified (glued) to forma cylinder. Left: axis-aligned Cauchy data allow for covering a finite
region only, i.e., the depicted parts of quadrants I to IV; additionally,in general there is no guarantee that the results of integration matchon left and right—possibly leading to discontinuities. Right: zigzag (from diagonal data) allows for covering an infinite cylinder—
without further restrictions except for the shear limit.
prescribed initial data were not sufficient to cover the target surface
because shearing in one place pulls material from another part, so
more material is actually required. These three failure modes are
dependent on the initial data and on the initial resolution, `, at which
the Chebyshev net was formed. Choosing ` too large in comparison
to the target surface might produce an extreme approximation suchas a single large rhombus for the entire target surface, while choosing` too small might introduce unnecessary curvature detail that is either
not of artistic interest or stems from artifacts of a triangulation.
We employ an alternative
method to construct discrete Chebyshev nets when interpolating
integration fails. This translation surface integration method relieson the fact that three non collinear points p, p1 , p2 in 3-space suchthat |p − p1 | = |p − p2 | uniquely determine a fourth point p3 suchthat (p, p1 , p2 , p3 ) is a planar rhombus. That is, instead of propagating a Chebyshev net such that the fourth vertex resides on the target
shape, we offer the possibility to propagate, from an initial curve on
S, such that the resulting rhombi are planar, while maintaining thegeneral integration paradigm depicted in Fig. 6 (top). Notice thatin general the resulting net will deviate from the target shape—aproperty that is desirable for initializing an approximate (instead ofinterpolating) Chebyshev net in scenarios where integration fails.
We refer to §4 for details of when we use translation surfaces insteadof interpolating integration. We remark that discrete Chebyshev nets
that are entirely comprised of planar rhombi are discrete translation
surfaces—in analogy to smooth translation surfaces that are definedby r(u, v) := a(u) + b(v), where a, b : R → R3 are smooth curves[Voss 1882; Pipkin 1984].
Translation surface integration

Next, we describe how these insights guide our design of wire meshin practice.

4

Design Tool

We facilitate the creation of a single, contiguous piece of wire meshthat can be cut out of the plane, and bent without inserting darts andwith bounded shear to approximate a desired surface, or guide form.
Theory informs us that the Chebyshev net constraints are globallycoupled; when we additionally constrain the net to interpolate agiven guide form, the design space is intractably small. Fortunately,
considerable additional freedom can be gained by allowing for slightdeviations from the guide form. We therefore seek designs thatapproximate rather than interpolate a given shape.
With a guide form given, the first phase is to create initial wire meshmaterial that interpolates a part of the guide form. Due to the limitations of interpolating integration laid out in the previous section,
to extend coverage a second approximating phase is required. This

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second phase interweaves adding or removing wire mesh material,
weight-painting, and globally optimization.
We find that a coarse-to-fine design process is effective. The designer first situates the wire mesh andresolves the coarsest features, before refining to focus on details.
The design session begins by establishing a coarse (large cell size)
wire mesh, and proceeds by iteratively subdividing the wire mesh,revising the shape, and repeating, until the finest details are resolved.
The revisions employ several types of tools, categorized as either
local or global in effect.
Coarse-to-fine design

Tools with local effect alter only the selected region of the wire mesh, e.g., adding or removing mesh material; these tools do not allow the deforming of the wire mesh: theChebyshev constraints are inherently global in nature, prohibiting
such a local deformation. To deform the mesh, we employ a global
optimizer; it necessarily alters almost the entire wire mesh shape.
This optimizer is incorporated into the interactive workflow, andthe user controls the optimization by painting weights to prioritize
approximation of some target regions over others.
Local vs. global tools

We now survey these tools and refer the reader to the accompanying
demonstrations in the supplementary video.

4.1

Phase I - Interpolating the Guide Form

We present two interpolating integration tools to quickly and easily
lay out the initial patch of material. The first is a novel zig-zag
tool for both cylindrical and disk topologies, while the second is
the Cauchy integration tool for disk topology similar to the work of
Aono et al. [1994; 1996; 2001].
The designer draws a single “diagonal” curve segment on the surface, and specifies the (possibly varying) shear along
the diagonal; using these data, the computer generates two curves
that zig-zag on and off the diagonal, one on each side of the diagonal(see Fig. 6-bottom); the computer then integrates an interpolating
wire mesh patch outward, using the diagonal initial data. To specify
cylindrical topology a closed loop on the surface is specified as theinitial diagonal curve instead of a curve segment.
Zig-zag tool

The designer draws two curve segments that intersect
at a point; the computer then integrates an interpolating wire mesh
patch from each quadrant of Cauchy initial data (see Fig. 6-top).
Using the Cauchy tool with a geodesic curve, for instance, ensures aconstant shear along the geodesic. Indeed, the geodesic curvatureof the u-parameter lines and v-parameter lines are given by γu and−γv , respectively [Pipkin 1984]. Therefore, a Chebyshev net’s
parameter line lies along a geodesic if and only if the shearing γ isconstant along that line. Using geodesics as initial conditions makes
the Cauchy approach appealing if a geodesic is made to pass overmultiple hills and valleys. We found this useful when designing
the Igea head and for capturing the face and neck of the bunny(see Fig. 10).
Cauchy toolDiscussion A cylindrical Chebyshev net requires diagonal initialconditions and thus the zig-zag tool. In the case of a disk topology,
either type of initial conditions may be used, and the choice is oneof aesthetics, versatility, and convenience.

The zig-zag approach is otherwise preferred because of its versatility. One example of a recurring strategy with this tool is to pass a
diagonal through a high curvature region: Knowing that curvature
will decrease in magnitude away from the diagonal, the designer
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specifies a high shear along the diagonal, thus allowing the mesh to
reduce in shear when integrated outward.
With either of the tools, the user selects the resolution of integration,and specifies whether the shear bound should be respected. Integration then continues as far as possible subject to the initial conditions,boundary of the guide form, and (optionally) the shear bound.
Both integration methods have been optimized using spatial hashingfor accurate and fast surface intersection tests. This allows interactive exploration of various integration options in order to find a goodwire mesh patch that can be used in Phase II of the design process.
These tools require the user to draw curves on
the guide surface. In our implementation we allow for three simple
approaches: (i) the designer positions a plane, and the algorithm
computes the plane-surface intersection; (ii) the designer picks asurface point, a direction, and a distance, and the algorithm integrates
out a geodesic; (iii) the designer picks a sequence of points forminga polyline, and the algorithm projects the polyline to the guide form.
Drawing curves

4.2

Phase II - Approximating the Guide Form

The second part of the design process allows for modeling toolsthat approximate instead of interpolate the guide form. There are
multiple reasons why interpolating integration alone falls short:First, interpolating integration typically does not create as muchmaterial as desired, e.g., due to exhaustion of initial data. Second,
interpolation may be too strong a request (§3.3). After other editing
tools are used, the wire mesh will approximate rather than interpolate
the guide form.
4.2.1

Adding Material

New material must be added to a given wire mesh patch when interpolating integration can no longer proceed. The tools described here
allow for the creation of new material that respect the Chebyshev
conditions at the expense of interpolating the guide form.
This tool extends an existing wire mesh by propagating parallel to a profile curve drawn on
the guide form, but does not additionallyseek proximity to the guide form. The designer selects a wire mesh boundary anddraws a profile curve on the guide form.
The computer then creates additional wire
mesh material following the translation surface integration described in §3 using the profile guide form andthe wire mesh boundary as the two translation curves. This tool isparticularly useful for adding wire mesh material along regions ofhigh or oscillatory curvature.
Translational surface toolThis tool extends an existing wire mesh (blue quads in inset figure)by a small, local addition. After selecting
a wire mesh region, the user taps the tool’s
hot key, and the computer extends the wire
mesh by one cell. At corner inclusions,the remaining fourth vertex is uniquely determined by the Chebyshev and planarity
conditions (green quad), i.e., just as forthe translation surface tool. At boundary edges, the two remaining vertices are uniquely determined by reflecting the face acrossthe boundary edge, trivially ensuring compatibility with adjacent
Reflection tool

ACM Transactions on Graphics, Vol. 33, No. 4, Article 66, Publication Date: July 2014

extensions (orange quads). If reflection is repeated without an optimization pass (discussed below), material can be constructed that
protrudes far from the guide form.
There are two hotkeys for each of these tools. Both create new
Chebyshev material using the specified tool, but one guarantees theshear limit is not violated while the other allows a user specified
amount of violation, usually ten percent. Therefore adding new approximate material using these tools either strictly satisfies or nearlysatisfies the wire mesh realizability constraints, thereby drasticallyfacilitating the optimizer’s task (described below). By allowing theuser to create new material which neither satisfies the shear constraint nor interpolates the surface, new material may always be
added.
4.2.2

Removing Material

The designer uses the cutting tool to eliminate excess material that mayotherwise buckle, to trim
boundaries, or to punchholes. We do not allow Figure 8: From left to right: (1) cutting a
cut edges to be stitched to- boundary cell removes one, two, or three wire
gether into a dart as this edges; (2) cutting an interior cell marks thewould introduce valence- cell as deleted, but does not remove wires; (3)when two adjacent cells are missing, the septhree vertices which canarating wire is removed; (4) when the mesh is
not be fabricated.
subdivided, wires are not inserted in deleted
The cutting tool is simple cells.
to use: the designer selects wire mesh cells, and presses the cutting
hot key. The selected cells are then marked as deleted. Wires thatbound a live (not deleted) cell are retained, and the remaining wires
are discarded.
4.2.3

Visualizing and Correcting Parametric Overlap

Repeated cutting and material addition can unintentionally create
a design that cannot be cut out of a planar wire mesh, by adding
excess material that overlaps in the parametric domain. Throughout
the design process, overlapping regions of the parametric domain
are brought to the designer’s attention via highlighted wire meshcells, as depicted in the inset figure of the bunny.
Detecting cells that overlap in the parametric domain is a straightforward exercise inreference counting. Because the wire meshhas strictly regular grid connectivity, eachcell is easily indexed by integer Cartesian
coordinates; when two or more cells haveidentical coordinates, they overlap.
Our interface automatically detects andhighlights, but does not prevent parameteric overlap. We have found the ability
to temporarily create overlaps to be an indispensable discretion during the design
process. We benefited from freely extending material, temporarily ignoring the highlighted overlaps, and laterchoosing whether to correct them by cutting from the new material,or from the older overlapping region. The inset shows that the wire
mesh on the left ear of the bunny and on its back refer to the same
region in the parameter domain, so the artist must make a choice.
Similar choices were made in many of our examples (see Fig. 10),where the designer traded: the entire left arm of the armadillo man
for the shell on its back; the bump on the Stanford bunny’s back forthe exterior of its left ear; and the back of the Igea head for the front.

Wire Mesh Design

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folding onto itself. Fclose is the weighted sum of squared distancefrom the mesh vertex to the guide form
Fclose (x) =nX

W (P (xi ))kxi − P (xi )k2 ,i=1

Figure 9: Unconstrained global optimization finds a wire meshclose to the target (left), while fixing the wire mesh along two curves(in red) as hard constraints during global optimization produces
large deviations from the target (right). Deviation from the target iscolored from blue to red.
4.2.4

Optimization and Form Shaping

Due to the global nature of Chebyshev nets, our design workflow
uses a global optimization approach to improve the shape quality while ensuring the realizability of a design. The optimizer isintended to make a quick calculation that does not impede the interactive, multi-faceted system of tools afforded for design. To use
the optimizer, the designer paints weights onto the guide surface toindicate where a close fit should be prioritized. The optimizer seeks
to balance the quality of the fit against the fairness of the wire mesh.
Global optimization with no hard constraints is essential to find asatisfactory result as illustrated in Figure 9.
The constraints are enforced using a generic geometricframework [Bouaziz et al. 2012]. The objective function, together
with these constraints, are solved using the augmented Lagrangian
technique of [Deng et al. 2013] with one modification: the closenessterm is based on distance measured to the guide form, as opposed
to displacement from original vertex positions. Briefly, this solver
works by introducing auxiliary variables to transform the original
problem into separable subproblems with closed-form solutions.
This separable structure allows subproblems to be solved in parallel
leading to significant speedups on multi-core systems [Bouaziz et al.
2012] and rapid convergence to approximate solutions [Deng et al.
2014].
SolverGiven a design represented as a Chebyshev
net, we write the optimization problem in terms of vertex positions
x = (x1 , . . . , xn ):Problem statement

min wfair Ffair (x) + Fclose (x)xs.t. kxi − xj k = ` ∀(i, j) ∈ E, (Chebyshev net)π| − ∠xi xj xk | ≤ γmax ∀(i, j, k) ∈ A, (Shear limit)
2
where the function Fclose penalizes the deviation between the meshand guide surface, Ffair measures the fairness of the mesh, E is theindex set of vertices that lie on a common edge, A is the index setof vertices that form a corner of a quad face; ` is the constant edge
length, γmax is the maximum amount of shear in radians, and wfair
is a user-specified positive weight to control the tradeoff between
closeness and fairness. Ffair is a quadratic energy defined using thesecond order difference of vertices
X
Ffair (x) =kxi − 2xj + xk k2 ,(i,j,k)∈F

where F is the index set of three consecutive vertices that lie on
a common wire. Such a fairing term inhibits the wire mesh from

where P (xi ) is the closest projection of xi onto the guide form, andW is a local weight function painted onto the guide form by the user,
or a global constant when no local weights are specified.
Closeness term The closest projection P (xi ) of a vertex of thewire mesh is approximated using a signed distance field. This provides significant speed improvements to the optimization. The field
is precomputed on a very high resolution grid (with 20% padding)to capture all the details of the guide form. To emphasize important features of the guide form during global optimization, the local
weights W (P (xi )), are painted onto the guide form vertices andlinearly interpolated across the faces.

The optimization has three parameters which may bechanged at any point during the design loop: (i) the global fairness
weight wfair , (ii) the surface closeness weights W (P (xi )), and (iii)the number of iterations to perform. Recall that the design loop tools
guarantee that the mesh is Chebyshev and in almost every instancethat the shear limit constraint is also satisfied, even though material
is being added and removed. Additionally, any new material roughly
approximates the guide form. Therefore the modified wire mesh is agood initialization for the global optimization.
Discussion

We chose a solver which finds approximate solutions
quickly. The compute time for a single iteration is 27ms on a modern laptop computer for a mesh with 15,000 vertices. This allows
us to achieve interactive performance even for 500-1000 iterations
which satisfies the Chebyshev net constraint usually within 1% andthe shear limit constraint within 3-5%. The designer therefore has
good intuition of the final result at interactive speeds. At the highest subdivided resolution (the edge length is that of the physicalwire mesh) we run a final optimization. As for many non-convex
optimization problems, there is no guarantee that the solver converges. In practice, however, we observed that 10,000 iterations
were sufficient to fabricate the results.
Timings

4.2.5

Enriching detail via subdivision

To resolve finer details, the designer invokes the subdivision tool.
The subdivision scheme globally quadrisects each cell keeping all
the original (“even”) vertices fixed, while introducing new (“odd”)
vertices at the cell centroids and edge midpoints. This subdivision
automatically preserves the Chebyshev constraints.

5

Results

Our design tool is implemented as a plugin of O PEN F LIPPER[Möbius and Kobbelt 2012]. The accompanying video provides
a didactic, visual explanation of the design and interaction process.
Figures 10 & 13 showcase designs created with this process, andTable 1 summarizes the associated statistics. These examples demonstrate coverage of large parts of intricate geometries with a single
sheet of wire mesh. In all designs, the Chebyshev nets are restricted
to a shear limit of π/4. The optimization automatically distributes
shear non-uniformly (see shear distributions, Fig. 10, fourth column)
so as to simultaneouly satisfy the wire mesh constraints and adhere
closely to the guide surface (see deviations plotted in Fig. 10, thirdcolumn).
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66:8

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A. Garg et al.

% vertices
vertices
%
1.0
0.8
0.6
0.4
0.2

% vertices
0.4
0.3
0.2
0.1

1

2

3

4

5

deviation (mm)% vertices

0.8
0.6
0.4
0.2

1

2

3

4

5

deviation (mm)% vertices
0.8
0.6
0.4
0.2
1

2

3

4

5

deviation (mm)% vertices

1.0
0.8
0.6
0.4
0.2

1

2

3

4

5

deviation (mm)% vertices
1.0
0.8
0.6
0.4
0.2
1

2

3

4

5

deviation (mm)

20

40

60

80

40

60

80

40

60

80

20

40

60

80

20

40

60

80

% vertices
0.25
0.20
0.15
0.10
0.05
20

% vertices
0.25
0.20
0.15
0.10
0.05
20

% vertices
0.30
0.25
0.20
0.15
0.10
0.05

% vertices
0.30
0.25
0.20
0.15
0.10
0.05

shear (degrees)

shear (degrees)

shear (degrees)

shear (degrees)

shear (degrees)

Figure 10: Three computer graphics classics, a male torso, and a freeform facade modeled as wire meshes. From left to right: guide surface
and final flattened wire mesh, overlay of wire mesh and guide surface, deviation from guide surface, shear distribution, final wire mesh.
ACM Transactions on Graphics, Vol. 33, No. 4, Article 66, Publication Date: July 2014

Wire Mesh Design

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variation on the facade model, making the design process moretime consuming as the exact amount of material required for a fullcovering has to be found via interaction. As one bump on the facadeis captured better through optimization, material is pulled awayfrom other regions, necessitating addition of material and further
optimization. After three levels of subdivision, the optimized meshes
fit well to the target surface (see Fig. 10, third column). The flatback of the facade model was trimmed away to reveal the final disk
topology.

Figure 11: Physical fabrication workflow: A 3D scaffold is createdby laser cutting intersecting planar pieces (top-left); the planar wire
mesh material is labeled and cut according to the flattened Chebyshev design net (bottom); the mesh is bend into place according
to the labelled curves and pinned to the support (top-right); after
removing the support, a free standing sculpture of the torso is obtained. The slight tilt of the model results from the non-horizontal
lower boundary curve.

Figure 12: The fabricated facade (center), with a comparison between renderings of the designed Chebyshev net (top and bottom
left) and photographs of the physical model (top and bottom right).

For the Igea, bunny, and Armadillo models, thedomain of each Chebyshev net has been designed interactively using the tools describedin §4. Through the combination of interaction
and optimization, we can capture not only geometrically delicate features such as the bunnyears, but also the global surface structure of
the guide surfaces. The Armadillo model isparticularly challenging due to the high total
curvature resulting from the geometric complexity of its salient features. In the design ofthis wire mesh, clear tradeoffs have to be madebetween surface coverage and guide surface adherence. For example,
as seen on the inset figure to the left, when trying to retain the bumpon the back of the Armadillo the left arm cannot be covered without
overlap in the parametric domain.
The male torso, female torso, Moai statue, and facade models havebeen designed starting from a cylindrical topology. After finding an
initial layout of a coarse cylindrical mesh, the design is refined byinterleaving mesh edits, subdivision and optimization, to gradually
capture prominent features of increasing geometric frequency. While
the facade may seem simpler than the torso, there is more curvature

Figure 13: Wire meshes of a Moai statue (top) and a female torso
sculpture (bottom), designed using zig-zag initial conditions to account for their cylindrical topologies.
ACM Transactions on Graphics, Vol. 33, No. 4, Article 66, Publication Date: July 2014

66:10•A. Garg et al.

Model
I GEA
A RMADILLOB UNNY
FACADET ORSONK

Time

41,70066,01998,23966,351
230,880

66.21
175.9665.72
107.04124.40

10 min2 hr
2.5 hr45 min10 minTable 1: Statistics for our design studies. N denotes the numberof vertices, K measures the total discrete Gaussian curvature as
the sum of all vertex angle defects. Time is the total design time
including exploration. All numbers refer to the final wire mesh.
Observe the choice of a diagonal orientation of the wires in manycases and the non axis-aligned domains, thus circumnavigating the
restrictions enshrined by the Hazzidakis formula. Indeed, the total
discrete Gaussian curvature in all examples significantly exceeds thefundamental 2π limit that is imposed on axis aligned rectangular domains by the Hazzidakis formula. This illustrates that the choice ofthe right domain is essential when aiming for single-sheet coverageof curved surfaces.
While numerous artists have manually created compelling wire mesh
sculptures of the human body, to our knowledge, no example existsthat has cylindrical topology like our torso model, i.e., represents acomplete section of a body. Previous examples like the one shownin Fig. 3 only show the front part with a wire mesh of disk topology.
The facade model illustrates the potential for architectural applications. We cover a complex facade with a single sheet of wire mesh,avoiding patch boundaries with their attendant inconsistencies andvisible seems; such contiguous designs improve the visual quality ofwire mesh claddings and freeform facades.

Second, we color the scaffold, referring to an intersection map:
We intersect the digital wire mesh against the digital scaffold model;
the intersecting faces typically form a network of curves, to which
we assign three colors. We then color the contours of the fabricatedscaffold according to this color convention. Correspondingly, we
digitally map the colored curve network to the parametric plane
using the parametric plane construction of §4.2.3. The flattened
network of colored curves forms our intersection map, which we
print on paper and transfer onto a large piece of planar shear-freewire mesh (see Fig. 11-bottom).
Third, the planar piece of wire mesh is manually bent: We bend
the mesh so as to bring the guide curves into alignment with thecontours of the scaffold, using push pins to fix the mesh in place.
We chose pins with heads large enough to prevent too much slippage
from occurring, but small enough to allow the wire mesh to shear.
We leave the mesh affixed to the scaffold for 48–72h, allowing time
for plastic flow under the applied strain (see Fig. 11-top-right), at
which time we remove the pins, and the wire mesh, from the scaffold,yielding a freestanding wire mesh (see Figs. 1 & 12).

6

Conclusion & Future Work

Computational wire mesh design is a new approach for creatingcompelling 3D models composed of woven materials. We employ
results from the theory of Chebyshev nets to shed light on the intrinsic difficulties of designing with wire meshes. Our analysis calls fora global approach with local control. We leverage and coordinate
the human ability to understand shape, and the computer’s ability to
optimize shape subject to thousands of constraints.
A fundamental challenge,
both theoretically and practically, of
Chebyshev nets is whether a given guide
form can be covered in its entirety. Consequently, in our digital design process it
is difficult to anticipate the amount of material required to cover the target surface.
Our user therefore iteratively adds material,
guided by intuition. However, if too much Figure 15: Left: before
material is supplied at a coarse resolution, optimization. Right: after
subdivision introduces buckles and folds, optimization, causing selfwhich are most easily corrected by back- intersections.
tracking the design process. Conversely, if too little material isavailable near a cut, the optimizer may attempt to close the cut, producing non-local self-intersections; while the fairing energy helps
reduce local self-intersections and buckling, it does not prevent moregeneral self-intersections (see Fig. 15). Providing more powerfultools to address buckles and self-intersections would accelerate thedesign process.
Limitations

To validate the agreement of physical wire mesheswith digitally designed Chebyshev nets, we fabricated four of thedesigns. We use 0.34 mm gauge stainless steel wires, woven witha plain weave into a wire mesh with 1 mm square openings; correspondingly, our digital designs have at their finest resolution a 1.34
mm centerline spacing. We fabricate in three stages (see Fig. 11):Fabrication

First, we fabricate a scaffold: The wire mesh design is triangulated; to avoid the bias introduced by cutting a quadrilateral by adiagonal, we add a vertex at the center of each quadrilateral. We employ Autodesk’s 123D Make to transform the triangulated mesh into
two orthogonal families of planar cross sections, which we laser-cut
from 4mm softwood, glue together, and sand at each contour planeintersection (see Fig. 11-top-left).

Figure 14: Physical realizations (middle & right) of the female
torso (left).
ACM Transactions on Graphics, Vol. 33, No. 4, Article 66, Publication Date: July 2014

Our fabrication process also poses challenges when dealing with the“spring-back” of the physical wire mesh material. A real wire meshmust be “over bent” in certain regions so that it springs back into thedesired form. It would be interesting to investigate how to accountfor this over bending in future work. Even without over bending,
producing scaffolds for arbitrary shapes is not easy; 123D Make
constructs strong scaffolds for closed, nearly convex surfaces such as
the torso or facade. Generating such scaffolds for highly non-convex
shapes, such as the bunny or armadillo, is more difficult. While
123D Make generates two families of planar contours orthogonal toeach other, the works of Cignoni et al. [2014] and Schwartzburg &Pauly [2013] construct scaffolds from planar pieces cut at arbitrary
angles to one another; these works can hopefully be extended toprovide sufficiently strong scaffolds for our fabrication process.

Wire Mesh Design

While we make no attempt to directly advance thetheory of Chebyshev nets, we hope that by exposing empirical evidence for the rich space of Chebyshev nets that abound under relaxed
constraints on surface adherence, our work might inspire new theoretical investigations on approximative nets.
Future work

We look forward to extensions of the optimization step that incorporate additional design objectives, such as accounting for the influenceof gravity, or optimizing shadows and shade. Indeed, wire meshes
are popular in facade applications to reduce solar radiation, where
wire thickness, spacing, and shearing all affect shadowing capacity.
Extending our software to the design of wire meshes with desirable
spatially varying shading capacity could therefore be a powerful tool
in architectural form finding.

•

66:11

B OBENKO , A. I., AND P INKALL , U. 1996. Discrete surfaces withconstant negative Gaussian curvature and the Hirota equation.
Journal of Differential Geometry 43, 3, 527–611.
B OUAZIZ , S., D EUSS , M., S CHWARTZBURG , Y., W EISE , T., AND
PAULY, M. 2012. Shape-up: Shaping discrete geometry withprojections. Comp. Graph. Forum (SGP ’12) 31, 5, 1657–1667.
B URAGO , Y. D., I VANOV, S. V., AND M ALEV, S. G. 2007. Remarks on Chebyshev Coordinates. Journal of Mathematical Sciences 140, 4, 497–501.
C IGNONI , P., P IETRONI , N., M ALOMO , L., AND S COPIGNO , R.
2014. Field-aligned mesh joinery. ACM Trans. Graph. 33, 1,
11:1–11:12.

Acknowledgements

DE G OES , F., A LLIEZ , P., OWHADI , H., AND D ESBRUN , M. 2013.

We thank Cosmo Wenman for providing the digital models of the torsos from their physical counterparts in The Louvre; Zohreh Sasanianfor helping with the fabrication process; Keenan Crane, Henrique
Maia and Nora Wixom for helping prepare the final version of thispaper; Keith Yeager for allowing us to use the Instron machine; andMario Botsch for initial discussions on wire mesh design.

D ENG , B., B OUAZIZ , S., D EUSS , M., Z HANG , J.,
S CHWARTZBURG , Y., AND PAULY, M. 2013. Exploring
local modifications for constrained meshes. Computer Graphics
Forum (EUROGRAPHICS ’13) 32, 2, 11–20.

This work was supported in part by the JSPS Postdoctoral
Fellowships for Research Abroad, NSF (Grants IIS-1319483,
CMMI-1331499, IIS-1217904, IIS-1117257, CMMI-1129917, IIS0916129), Israel-US BSF, Niedersachsen-Israel (grant “Spectral
Methods in Geometry Processing: Theory and Applications”), SNSF
(grant 200021 137626), Intel, The Walt Disney Company, Autodesk,
Side Effects, and NVIDIA. This research has received funding fromthe European Research Council under the European Unions Seventh
Framework Programme (FP/2007-2013) / ERC Grant Agreement257453, ERC Starting Grant COSYM.

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