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Abstract and Figures

Geodesic curves in surfaces are not only minimizers of distance, but they are also the curves of zero geodesic (sideways) curvature. It turns out that this property makes patterns of geodesics the basic geometric entity when dealing with the cladding of a freeform surface with wooden panels which do not bend sideways. Likewise a geodesic is the favored shape of timber support elements in freeform architecture, for reasons of manufacturing and statics. Both problem areas are fundamental in freeform architecture, but so far only experimental solutions have been available. This paper provides a systematic treatment and shows how to design geodesic patterns in different ways: The evolution of geodesic curves is good for local studies and simple patterns; the level set formulation can deal with the global layout of multiple patterns of geodesics; finally geodesic vector fields allow us to interactively model geodesic patterns and perform surface segmentation into panelizable parts.
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Geodesic Patterns
Helmut Pottmann
KAUST, TU Wien
Qixing Huang
Stanford University
Bailin Deng
TU Wien
Alexander Schiftner
Evolute GmbH, TU Wien
Martin Kilian
TU Wien
Leonidas Guibas
Stanford University
Johannes Wallner
TU Graz, TU Wien
Figure 1: Geodesic patterns on freeform surfaces. Left: For the cladding of a surface by wooden panels bendable only about their weak axis,
we perform segmentation into parts which can be covered by geodesic strips of roughly constant width. Right: A timber construction derived
from a hexagonal geodesic web has good manufacturing and statics properties.
Abstract
Geodesic curves in surfaces are not only minimizers of distance, but
they are also the curves of zero geodesic (sideways) curvature. It
turns out that this property makes patterns of geodesics the basic ge-
ometric entity when dealing with the cladding of a freeform surface
with wooden panels which do not bend sideways. Likewise a geo-
desic is the favored shape of timber support elements in freeform
architecture, for reasons of manufacturing and statics. Both prob-
lem areas are fundamental in freeform architecture, but so far only
experimental solutions have been available. This paper provides a
systematic treatment and shows how to design geodesic patterns in
different ways: The evolution of geodesic curves is good for local
studies and simple patterns; the level set formulation can deal with
the global layout of multiple patterns of geodesics; finally geodesic
vector fields allow us to interactively model geodesic patterns and
perform surface segmentation into panelizable parts.
CR Categories: I.3.5 [Computer Graphics]: Computational Ge-
ometry and Object Modeling—Geometric algorithms, languages,
and systems; I.3.5 [Computer Graphics]: Computational Geometry
and Object Modeling—Curve, surface, solid, and object represen-
tations
Keywords: computational differential geometry, architectural ge-
ometry, geometry of webs, timber rib shell, cladding, freeform sur-
face, pattern, geodesic, Jacobi field.
1 Introduction
In recent years it has become apparent that methods from Geomet-
ric Computing bear a great potential to advance the field of freeform
architecture. This fact has created the new research area architec-
tural geometry, which draws from various branches of geometry
and which is motivated by problems originating in architectural de-
sign and engineering – see for instance the proceedings volume
[Pottmann et al. 2008a]. The topics studied in the present paper
belong to this line of research. They have as a common theme the
design of a pattern of geodesics on a freeform surface.
One problem concerns the cladding of a general double curved sur-
face with wooden panels. Such claddings will be mainly applied
to interior spaces (see Fig. 2). Even if the material may be differ-
ent from wood, the panels are assumed to be close to developable
and their development should fit well into a rectangle whose length
is much larger than its width. Hence, each panel should follow a
geodesic curve. The cladding problem can be approached in an ex-
perimental way as illustrated by Fig. 2. Computationally it means
decomposing a given surface into regions, each of which can be
covered by a sequence of nearly equidistant geodesic curves (see
Fig. 1).
Other applications of geodesic patterns lie in wooden construc-
tions where the geodesics are used for the supporting structure of
a curved shell. Extending pioneering technologies by J. Natterer
[2002], ongoing research at the EPF Lausanne aims at the design
of freeform timber rib shells, which are composed of a grid of geo-
desic curves (see Fig. 3). Other innovative timber constructions, as
seen in recent projects by Shigeru Ban (Fig. 4), would also benefit
from an efficient computational approach to the layout of geodesic
patterns on surfaces. One reason why geodesic curves are a pre-
ferred shape is statics: Geodesics – being minimizers of distance
– are the equilibrium shapes of elastic curves constrained to the
surface. Another reason is the manufacturing of laminated beams,
which are much easier to make if the individual boards can simply
be twisted and bent and along the weak axis [Pirazzi and Weinand
2006].
Figure 2: Experimental cladding using paper strips (left) results in
an office space design by NOX Architects [Spuybroek 2004].
Figure 3: Assem-
bling screw-laminated
beams for a timber rib
shell prototype based
on a 2-pattern of geo-
desics. Image cour-
tesy IBOIS, the tim-
ber construction lab at
EPF Lausanne.
Related work. Let us briefly address the literature in geometric
computing, as far as it is obviously related to our problem formula-
tion. We will later encounter more connections to previous work.
A geodesic curve gis a locally shortest path on a surface S. The
computation of geodesics is a classical topic: For smooth surfaces,
pursuing a geodesic curve emanating from a point with a given tan-
gent vector is equivalent to solving an initial value problem for
a 2nd order ODE, while the boundary value problem (connecting
two points on a surface by a geodesic path c(t)) can be converted
into the constrained minimization of the quadratic energy Rk˙
ck2
[do Carmo 1976]. For triangle meshes, shortest polylines cross
edges at equal angles, and ambiguities at vertices may be resolved
by the concept of “straightest geodesics” [Polthier and Schmies
1998]. Finding the truly shortest geodesic paths requires the com-
putation of distance fields, for which several efficient algorithms
have been developed, see for instance [Chen and Han 1996] or
[Kimmel and Sethian 1998].
Early research on the cladding of freeform surfaces with devel-
opable panels evolved from the architecture of F. Gehry [Shelden
2002]. One contribution to the present cladding problem is pro-
vided by the geodesic strip models of Pottmann et al. [2008b].
This name is used for continuous surfaces composed of devel-
opable strips which have nearly straight development. One can
view them as semi-discrete versions of smooth families of geode-
sics on a smooth surface. In general, several geodesic strip models
with different directions are required to cover a freeform surface
(see Fig. 2). The choice of these directions and the initialization of
the optimization in [Pottmann et al. 2008b] has not been systemati-
cally investigated so far.
Figure 4: Timber construction for
the Yeoju Golf Club by Shigeru
Ban; CAD/CAM by designtoproduc-
tion. The beams are no geodesics and
manufacturing thus requires CNC ma-
chining. We address the design of 3-
patterns of geodesics and so contribute
to their simplified manufacturing.
Our approach to geodesic patterns is closely related to classical
results on the geometry of webs which was developed mainly by
W. Blaschke and his school. We refer to the survey article [Chern
1982] and the monograph [Blaschke and Bol 1938].
Contributions and overview. Inspired by practical problems in
architecture, we study geodesic N-patterns on surfaces. These are
formed by Ndiscrete families of geodesics which are subject to
additional constraints arising from the specific application. We put
particular emphasis on the cases N= 1 (Fig. 2), N= 2 (Fig. 3)
and N= 3 (Fig. 4). The overall goal is to provide an efficient
computational framework to support the user’s navigation through
design space. Our results include:
*ways to control the strip width variation in a geodesic 1-pattern
based on geometric concepts such as Jacobi fields and striction
curves (Sec. 2);
*two computational approaches for designing geodesic 1-patterns,
namely an evolution algorithm guided by Jacobi fields (Sec. 3)
and a level set approach which is also used for the design of gen-
eral N-patterns (Sec. 4);
*the design of geodesic webs (special 3-patterns and 4-patterns)
and the extraction of further patterns from such webs, for instance
in the style of Islamic art (Sec. 5);
*a user-friendly design tool for the solution of the cladding prob-
lem which results in an aesthetically pleasing segmentation of the
design surface into regions covered by geodesic 1-patterns. This
tool is based on the concept of piecewise-geodesic vector fields
(Sec. 6).
Each of these approaches solves different instances of the problem
of covering a surface with geodesic curves.
2 Distances between geodesics
gi+1
gi
ww
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
In this section, we discuss general se-
quences (1-patterns){gi}of successive
geodesic lines in a surface and the distances
between them. It is not usually possible to
express by formulas such a distance – say
between the geodesic curve giand its neigh-
bor gi+1. Not even an exact definition of
such a distance is straightforward. Only in
some highly symmetric cases, like evenly
distributed meridian curves of a rotational surface (see the figure)
the measurement of distance is elementary. However, a first order
approximation of that distance is well known: Start at time t= 0
with a geodesic curve g(s), which is parametrized by arc length s,
and let it move smoothly with time. A snapshot at time t=εyields
a geodesic g+near g(see Fig. 5):
g+(s) = g(s) + εv(s) + ε2(...).
The derivative vector field vis called a Jacobi field. It is known
that without loss of generality we may assume that it is orthogonal
to the curve g(s), and it is expressed in terms of the geodesic’s
g+(s)g+(s)
g+(s)
g+(s)
g+(s)
g+(s)
g+(s)
g+(s)
g+(s)
g+(s)
g+(s)
g+(s)
g+(s)
g+(s)
g+(s)
g+(s)
g+(s)
g(s)
g0(s)
v(s)
g(0)
Figure 5: A geodesic g(s)with
Jacobi field v(s), and a neigh-
boring geodesic g+(s)which is
at distance εkv(s)k. The pa-
rameter sis the arc length along
the geodesic g, and vobeys the
Jacobi differential equation (1).
Here v(0) = 0.
tangent vector g0as
v(s) = w(s)·Rπ/2(g0(s)),where w00 +Kw = 0.(1)
Here Rαis the rotation by the angle αin the tangent plane of the
surface, and Kis the Gaussian curvature [do Carmo 1976].
Since distances between infinitesimally close geodesics are gov-
erned by (1), that equation also approximately governs the width of
a strip bounded by two geodesic curves of small finite distance.
Note that the Jacobi equation can also be used to fabricate sur-
faces of given Gaussian curvature, by gluing together strips of paper
whose width obeys the Jacobi differential equation (see Fig. 6).
w(s) = αcosh(sp|K|)
Figure 6: Do it yourself K-sur-
face. We glue together strips
whose width w(s)fulfills the
Jacobi equation for some con-
stant value K < 0. This results
in a surface of approximately
constant Gaussian curvature.
A first design method for geodesic 1-patterns. For the design
of a geodesic 1-pattern {gi}it is important to control the positions
of points at which the distance of the curve gi+1 from the curve gi
assumes a minimum or maximum. It turns out that it is not difficult
to design 1-patterns of geodesics where the locus of these points is
prescribed. The Jacobi relation (1) is the key to understanding the
local behavior of the strip width.
Assuming w > 0, we have w00 <0whenever the Gaussian curva-
ture is positive. Therefore in an area where K > 0, the strip width
can have only maxima. If the Gaussian curvature is negative, then
the strip width can have only local minima. The case of constant
strip width is only possible if K= 0, which means developable
surfaces.
Setting aside the special case of developable surfaces, we get a pic-
ture of the locus of extremal strip width by first looking at the case
of a smooth family {gt}of curves on a surface. The striction curve
sis the locus of extremal width of the infinitesimal strips defined
by two neighboring curves of the family (generically sis curve-like,
but it may also degenerate). H. R. M ¨
uller [1941] showed the follow-
ing: If the curves gtare geodesics, their initial tangent vectors are
geodesically parallel along s. Conversely, if we compute a geodesi-
cally parallel vector field v(t)along a curve s(t)and then trace the
s
ss
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
Figure 7: The striction curve sof a geodesic 1-pattern. If the initial
tangent vectors of geodesics constitute a geodesically parallel vec-
tor field along s, then extremal distances between successive curves
occur near s.Left: the piecewise-developable surface of Fig. 6.
Right: The areas of extremal distance of a geodesic curve from its
neighbors are indicated by white color. This surface is taken from
the Cagliari museum project by Zaha Hadid architects.
geodesics with start at points s(t)with initial tangent vector v(t),
then sis (part of) the striction curve of this family of geodesics.
Knowing this, the design of geodesic patterns with prescribed stric-
tion curve s(t)is simple: Once the striction curve sis chosen, there
is only one remaining degree of freedom in the choice of the par-
allel vector field. From this 1-parameter family of patterns we can
take the one which best fits the design intent (see Fig. 7).
3 Geodesic 1-patterns by evolution
This section presents a strategy for the design and computation
of geodesic 1-patterns on triangle meshes which represent general
freeform surfaces. It is based on an evolution of curves, where the
transfer from a current geodesic curve gto the next one (denoted by
g+) considers only the local neighborhood of gand can thus nicely
be governed by Jacobi vector fields.
Solution of the Jacobi differential equation. The computation
of all possible Jacobi vector fields v(s)orthogonal to a geode-
sic curve which is given by an arc length parametrization g(s)
(0sL) is easy. We only have to compute the function w(s)
which gives the length of the Jacobi field, and which satisfies the
differential equation w00 +Kw = 0. This is a linear ODE with
coefficient function K=K(g(s)). Any solution w(s)is a linear
combination
w(s) = λ1w(1)(s) + λ2w(2) (s)
of two linearly independent fundamental solutions w(1) and w(2).
We have implemented the solution of the Jacobi differential equa-
tion as follows: The geodesic under consideration is represented as
a polyline p0, p1,...,pMwith edge lengths Li=kpi+1 pik.
The unknown function wis given by its values wi=w(pi)in the
vertices pi. Now the second derivative w00 is approximated by
w00
i2
Li1+Li1
Li
(wi+1 wi)1
Li1
(wiwi1).
In this way the Jacobi equation turns into a sparse linear system,
involving M1equations for w00
1,...,w00
M1in M+1 unknowns
w0,...,wM. If we prescribe the pair (w0, w1)of values, the sys-
tem reduces to a triangular one. For the computation of the funda-
mental solutions w(1) and w(2) we simply choose these pairs to be
(0,1) and (1,0), respectively.
Selecting a Jacobi field. The selection of the Jacobi field which
is to guide the next geodesic g+depends on the design intent. If we
are interested in constant strip width wW=const., we compute
λ1, λ2by minimizing
Fw=[0,L](λ1w(1)(s) + λ2w(2) (s)W)2ds, (2)
which amounts to a linear 2×2system. Since we have only two
degrees of freedom, we cannot expect λ1w(1)(s) + λ2w(2) (s)
const. in all cases. It is therefore advisable to check afterwards if
we really have w(s)[Wε1, W +ε2], for certain pre-assigned
tolerances ε1, ε2. See Figures 8a,d for an example.
Replacing the constant Wby a prescribed function W(s)is compu-
tationally the same, but opens up many possibilities: The decompo-
sition of a freeform surface into strips might require a denser sam-
pling if the curvature across the strips is high. We therefore guide
strip width by minimizing Fwwith W(s) = φ(κn(s)), where φ
is a strictly decreasing function and κnis the surface’s normal cur-
vature in the direction orthogonal to the given geodesic. For an
example, see Fig. 8b. Prescribed patterns of geodesic curves can be
achieved by appropriate choices of W(s)(see Fig. 8c).
(a) (b) (c) (d) (e)
Figure 8: Evolution of geodesics, starting from a source curve (dark blue). (a) Regular equidistant evolution. Thin blue pieces of curves
show where the distance constraint is violated. (b) Aesthetic reasons might require denser strips if the normal curvature across the strip is
high. (c) Pattern transfer. The small image shows some intended width functions W(s). (d) Because of positive Gaussian curvature, in some
areas evolution is possible only if we allow intersections of adjacent geodesics. (e) Evolution of the same geodesic curve as in (d), but by
introducing breakpoints if the strip width deviates too much from a constant value. The breakpoints are guided along the red curves.
Computing the next geodesic. The function w(s)computed
above approximately describes the distance between the geodesic
curve gand the ‘next geodesic’ g+. We could now find g+by
simply moving the endpoints g(0), g(L)sideways by the amounts
w(0),w(L), respectively, and connecting them by a geodesic g+.
The following method takes more information into account:
h0h1
X0X1
g(s0)g(s1)
g+
g
YV
We sample the original geodesic at parameter values si=i
NL
(i= 0,...,N) and move the points g(si)sideways on geodesics
hiorthogonal to g. This results in points Xi. The next geodesic g+
is represented by a point Ywhich slides along a curve orthogonal to
the current geodesic gand by an initial tangent vector V. These two
degrees of freedom are determined such that Pidist(g+hi, Xi)2
(each distance measured along hi) becomes minimal. Being close
to the solution already, this can be done by a Levenberg-Marquardt
method which avoids second derivatives.
Limitations of the evolution method. The method above which
finds a smooth geodesic at a certain nonzero distance from a given
one works only if one can find a width function wwhich solves
the Jacobi equation and which has no zeros in the considered in-
terval. It turns out that we can tell the existence of such ‘useful’
solutions simply by testing if the fundamental solution w(1) has an-
other zero in the considered interval. The proposition below, proved
in [do Carmo 1992], characterizes the two possible cases (good and
bad) and sums up some of their geometric properties.
PROP OS IT IO N 1. Consider a geodesic curve g(s), where s[0, L]
is an arc length parameter. Assume a fundamental solution w(1) (s)
of the Jacobi equation with w(1)(0) = 0. Then there are the fol-
lowing two cases:
case 1 case 2
# zeros of w(1)(s)for 0< s L:01
# zeros of any solution in the interval [0, L]:11
existence of solution nonzero in [0, L]yes no
glocally minimizes distance of g(0),g(L)yes no
The inequality K(s)>(π/L)2for s[0, L]implies case 2. Anal-
ogously K(s)<(π/L)2implies case 1 (surely true if K0).
If the geodesic curve under consideration is a case 2 curve we have
two choices (illustrated by Figures 8d,e): Either to put up with the
fact that we cannot have a proper non-intersecting next geodesic, or
to consider broken geodesics, which are the topic of the following
paragraph.
(a) (b)
Figure 9: Evolution of a 1-pattern of broken geodesics, which starts
with an unbroken one (blue, at extreme left). The threshold for
the introduction of breakpoints in (a) and (b) is that the strip width
deviates more than 7.5% or 5%, resp., from the desired value (this
surface is taken from the top of the Lilium tower, Warsaw, by Zaha
Hadid Architects).
Extension to broken geodesics Evolving a geodesic curve into
a 1-pattern frequently runs into obstacles, due to the few degrees
of freedom available. We therefore relax the geodesic condition
and consider broken geodesic curves. Such a curve gconsists of
geodesic arcs gidefined in arbitrary subintervals [si, si+1]. These
arcs fit together at points Pi=gi1(si) = gi(si) = g(si). We
evolve each arc giseparately, with its own width function wi(s)
which satisfies the Jacobi equation. Obviously, the single functions
wiare not independent. We draw the following picture:
wi1(si1)
wi1(si)
P+
i=g+(si)
Pi=g(si)
αi
αi
αi
αi
αi
αi
αi
αi
αi
αi
αi
αi
αi
αi
αi
αi
αiβi
βi
βi
βi
βi
βi
βi
βi
βi
βi
βi
βi
βi
βi
βi
βi
βi
wi(si+1)
wi(si)
g(si+1)
g+(si+1)
gi1
g+
i1
With the angles αi, βibetween curve normals and the line connect-
ing breakpoints, we read off the approximate relation
wi1(si)
cos αi
=wi(si)
cos βi
,(3)
which assumes that the derivatives of wi1,wiare small. We can
now find the single width functions wiby first choosing the di-
(a) (b) (c) (d)
κg= 10 1 0.1 0.01
Figure 10: 1-patterns of geodesic curves which are found as level sets of a real-valued function defined on the surface. (a) Optimization with
a low weight on Fwyields almost true geodesics. Three geodesics are shown in orange for comparison. (b) A higher weight on Fwgenerates
strips of even width, but lets level sets deviate from true geodesics and creates higher geodesic curvatures. (c) and (d): Deviation from the
geodesic property for subfigures (a) and (b). We show the geodesic curvature κgon a logarithmic scale; bounding box diagonal equals 4.
rection of movement of the breakpoints, reading off the angles αi,
βi, and subsequently minimizing the sum of all Fwias defined by
(2) with (3) as side condition. The ‘next’ geodesics g+
iare sub-
sequently fitted to the width functions in a manner similar to the
computation of the next geodesic which is described above. With
the path of breakpoints already prescribed, the number of degrees
of freedom for the broken geodesic g+is the number of segments
plus one.
In the examples of Figures 8e and 9 breakpoints are automatically
inserted whenever a strip would violate the distance constraint, and
the paths of breakpoints are guided such that they bisect the angle
of their adjacent geodesic segments.
4 Geodesic N-patterns from level sets
The Jacobi field approach is well suited if, in a 1-pattern, we want
to move from one geodesic to the next, in a way which allows us
to control the distance between these two geodesics. For global
tasks such as an optimal alignment of a 1-pattern of geodesics with
a vector field, or design problems involving several 1-patterns, we
prefer to represent the geodesics of a 1-pattern as selected level sets
of a real valued function φwhich is defined on the given surface S.
Geodesic level sets are not new, in fact they represent the main idea
in the geodesic active contour method of Caselles et al. [1997]. The
difference to our setting is that we consider global patterns of geo-
desics. For level set methods in general we refer to [Osher and
Fedkiw 2002].
Geodesic curves are characterized by vanishing geodesic curvature
κg[do Carmo 1976]. If a curve is given in implicit form as a level
set φ=const., then its geodesic curvature can be computed by
κg=div φ
k∇φk,(4)
where div,are understood in the intrinsic geometry of S(see e.g.
[do Carmo 1992], p. 142).
We implement construction of a function φas follows. The surface
Sis represented by a triangle mesh (V, E , F ). The function φis
determined by its values on vertices and is considered to be linear
within each triangular face. The level sets under consideration are
polylines on the mesh. The vector field φis constant in each face.
For any vector field Xwe evaluate div(X)at a vertex vby com-
puting the flux of Xthrough the boundary of v’s intrinsic Voronoi
cell, divided by the area of that cell.
For regularization, we wish to keep φ=div(φ)small. Further,
applications might require the distance between level sets φ=c
and φ=c+hto equal some value w; for that distance we have
whk∇φk.(5)
We thus optimize φby solving a nonlinear least squares problem
which is governed by a linear combination of three functionals:
Fκpenalizes deviation from zero geodesic curvature, Fis for
smoothing/regularization, and Fwpenalizes deviation from the de-
sired strip width w. Definitions are:
Fκ=XvVA(v)`div φ
k∇φk(v)´2
,
F=area(S)XvVA(v)∆φ(v)2,
Fw=XfFarea(f)k∇φ(f)k − h
w2.
Here A(v)means the area of the Voronoi cell of v. All three func-
tionals are invariant w.r.t. scaling of the geometry. We therefore
formulate our optimization as
Fκ+λF+µFwmin .(6)
We initialize optimization by a function which minimizes Falone,
under the side condition of 3 arbitrary function values. Results of
this method can be seen in Fig. 10. As is to be expected we cannot
make make Fκand Fwvanish at the same time: Only for devel-
opable surfaces (K= 0) the geodesic property is compatible with
constant width between curves.
For an N-pattern of geodesics, each of the Nfamilies of curves in-
volved is represented as level sets of a function φi(i= 1,...,N).
We now minimize the sum of the single target functions analogous
to (6). We can also incorporate additional requirements, for in-
stance constant intersection angle αbetween families iand j. To
this end we augment the target functional by
ν·Fangle =νX
fF
area(f)
area(S)Dφi
k∇φik, R π
2α`φj
k∇φjk´E2.
Here Rπ
2αmeans a rotation by the angle π
2αin the respective
face; νis the weight given to Fangle. For an example see Fig. 16.
The level set approach is an integral part of the examples and im-
ages in Sections 5 and 6.
Remark 1. Our level set approach works only for simply connected
surfaces. It is possible to extend it to arbitrary surfaces by using
branched coverings, such as in [K¨
alberer et al. 2007].
Implementation details. To compute the function φwhich min-
imizes the combined functional (6), we employ a classical Gauss-
Newton method with line search for optimization [Madsen et al.
2004]. The variables of this optimization are the values of the un-
known function φon the vertices of a mesh. All required first order
derivatives are calculated analytically (i.e., are not approximated).
(a) (b) (c) (d) (e)
Figure 11: A hexagonal web of geodesics imposed on the surface of Fig. 6. The level sets of functions φ1, φ2, φ3are geodesics, if they obey
div(φi/k∇φik) = 0. If they also obey φ1+φ2+φ3= 0, they form a hexagonal web. (a) Integer level sets for each of φ1, φ2, φ3and true
geodesics for comparison. (b) Flattening of the surface by the mapping ψ(x) = (φ1(x),(φ1(x)+2φ2(x))/3). (c)–(d): Using ψas texture
mapping, patterns with hexagonal combinatorics can be transferred to the surface. (e) Detail of a geometric pattern inspired by Islamic art.
Long and thin components follow geodesic curves on the surface and can therefore be manufactured by bending of wooden panels.
The linear systems to be solved in each round of iteration are sparse,
since the single terms which contribute to (6) involve only local dif-
ferential operators which are defined in terms of small vertex neigh-
borhoods. For sparse Cholesky factorization we employ CHOLMOD
[Chen et al. 2008].
5 Geodesic webs
Timber constructions like the one of Figure 4 follow a curve pattern
with regular combinatorics. For manufacturing and statics reasons,
one would like these guiding curves to be geodesics. Questions
of this kind immediately lead us to the investigation of systems
of curves on surfaces with both geometric and topological proper-
ties. The most important concept here is the hexagonal web, which
means 3 families of curves which can be continuously mapped to 3
families of parallel straight lines in the plane, as shown by Fig 11b.
By selecting a discrete sample of curves from the web we can gener-
ate a variety of patterns – see Figures 1 (right), 4, and 11. The con-
dition that families of curves form a hexagonal web is of a topolog-
ical nature and belongs to the so-called geometry of webs [Blaschke
and Bol 1938; Chern 1982]. Neglecting some details concerning
the domain where a web is defined, we quote a result:
THE OR EM 2. A complete hexagonal web of straight lines in the
plane consists of the tangents of an algebraic curve of class 3 (pos-
sibly reducible); any class 3 curve yields a hexagonal web [Graf
and Sauer 1924]. The variety of geodesic hexagonal webs in a sur-
face is the same as in the plane the surface has constant
Gaussian curvature [Mayrhofer 1931].
In summary this means that there is a 9-parameter family of pos-
sible hexagonal webs made from geodesics, if the surface under
consideration is the plane, or its Gaussian curvature Kis constant
(see Fig. 12). Note that such surfaces possess mappings into the
plane where geodesics become straight lines; such surfaces carry
one of the non-Euclidean geometries if K6= 0.
Little seems to be known about surfaces which carry a smaller va-
riety of hexagonal webs made from geodesics. Anyway, for our
applications it is more important to have a computational tool for
generating a hexagonal web whose curves are as geodesic as possi-
ble. The level set approach is well suited for that. We describe the
three families of curves as level sets of functions φ1,φ2,φ3with
φ1+φ2+φ3= 0.(7)
This equation results in a hexagonal web: Mapping the surface S
into the plane via x7→ (φ1(x), φ2(x)) maps the level sets of φ1
and φ2to lines x=const. and y=const., resp., while the level
sets of φ3are mapped to the lines x+y=const. Equation (7) is
known in the theory of webs; in fact every hexagonal web can be
found in this way.
We implement geodesic webs as follows: We consider two func-
tions φ1, φ2on the given surface S, and optimize them such that
the sum of target functionals according to (6), evaluated for φ1,φ2,
and φ3:= φ1φ2becomes minimal. For results, see Figure 1
(right) and Figure 11. A different kind of web is shown by Fig. 19.
(a) (b)
Figure 12: Hexagonal webs and class 3 curves. (a) The condi-
tion that a straight line u0+u1X+u2Y= 0 is in the web reads
Pi+j+k=3
i,j,k0
aijk ui
0uj
1uk
2= 0. This equation characterizes the tan-
gents of a class 3 curve, and is determined, up to a factor, by 9
given lines in general position. The colors given to the tangents
identify the 3 families of a hexagonal web. (b) A geodesic web on
the sphere. It is transformed to a planar one by projection from the
center.
6 Global solution of the cladding problem
When the input surface is more complicated, it is harder to cover it
with a single geodesic pattern. We have already encountered such
difficulties in Section 3 where we discussed the evolution of a geo-
desic curve. We saw that the Gaussian curvature of the input surface
is responsible for the maximal length of a strip which is bounded by
geodesic curves. We are thus led to the question of segmentation of
the input surface such that each piece can be covered by a geodesic
1-pattern.
For this purpose we employ a device not used in previous sec-
tions, namely geodesic vector fields and piecewise-geodesic vec-
tor fields. This section describes a general setup related to geode-
sic vector fields, shows how to interactively design a near-geodesic
vector field by means of a certain reduced eigenbasis of fields, and
demonstrates how to subsequently modify (sharpen) it to make it
(a) (b) (c) (d)
Figure 13: Processing pipeline for the global cladding problem. (a) The first two elements of the reduced basis which spans the vector field
design space. (b) User’s selection v
f1,v
f2, . . . indicated by arrows, and blue design vector field vadapted to this selection. (c) Sharpened
vector field vwhich is now piecewise geodesic together with the boundaries of macro patches which lie where the vector field is sharp. (d)
Segmentation into finitely many geodesic 1-patterns which are aligned with the user’s selection. This surface is taken from the interior facade
of the Heydar Aliev Merkezi Project by Zaha Hadid Architects.
piecewise geodesic. Finally, we perform segmentation of the input
surface along the curves where the resulting vector field is sharp.
To find the actual geodesic curves defined by the vector field, we
refer back to the level set method of Section 4.
Geodesic vector fields. A vector field von a surface Sis geode-
sic if it consists of tangent vectors of a 1-parameter family of geode-
sic curves covering S. Later we also encounter piecewise-geodesic
fields which fulfill the geodesic property in the inside of certain
patches. The following characterization of the geodesic property
employs the notation x(v), which means the covariant derivative
of a vector field vin direction of the tangent vector x, and which
is defined as the tangential component of the ordinary directional
derivative [do Carmo 1976].
PROP OS IT IO N 3. A vector field vof constant length is geodesic
for all points of the surface, the linear mapping x7→ ∇x(v)
in the tangent plane is symmetric.
Proof: kvk=const. implies that x(v)is orthogonal to v, for
all x. The geodesic condition is v(v) = 0 everywhere. Among
the linear mappings which map the entire tangent plane to v, the
symmetric ones are exactly those which map vto zero.
Implementing geodesic vector fields. In our implementation,
the surface Sis represented as a triangle mesh (V, E , F). The vec-
tor field is represented by unit vectors v
fattached to the incenters
mfof faces fF. Consider two adjacent faces f1,f2, such that
the face f2has been unfolded into the plane of f1(in the following
text, the unfolded positions of items associated with f2are marked
with a hat). In order to capture the condition of Prop. 3, we endow
each face with a local coordinate system and a Jacobi matrix Jf
such that
v
f1b
v
f2=v
f1+Jf1·(b
mf2mf1) + re,
e
mf1b
mf2
where Jf=g1,f g2,f
g2,f g3,f .(8)
Here reis a remainder term associated with the edge e=f1f2.
This manner of discretizing the symmetry of the covariant deriva-
tive – as postulated by Prop. 3 – comes from two facts: (i) covariant
differentiation is invariant if an isometric deformation is applied, so
we may unfold the neighboring triangles into the plane of the trian-
gle under consideration; (ii) in a plane, x(v)equals the ordinary
directional derivative of vw.r.t. the vector x, i.e., it equals multi-
plication of a Jacobi matrix with x. The condition of Prop. 3 now
means symmetry of the Jacobi matrix.
In summary, the vector field v= (v
f)fFis a geodesic vec-
tor field, if its length kv
fkis constant for all faces f, and we
can find a collection of coefficients g= (gf)fFwith gf=
(g1,f , g2,f , g3,f ), such that (8) holds with the r’s below some
threshold.
The following functional attempts to quantify how well vsatisfies
the geodesic property. It will turn out to be very useful for the next
task (interactive editing of geodesic vector fields). We let
Q(v) = min
gR3|F|X
eE
wekre(v,g)k2+λrkgk2.(9)
The term including λris for regularization; we choose the factor
λrproportional to a characteristic edge length in the mesh. The
edge weights we0are for downweighting areas where we do
not care about the geodesic property and where we (later) want to
encourage formation of a patch boundary. We chose to downweight
areas of high curvature; so we let we= exp(κ2
e/2µ2),where κe
is the normal curvature across the edge e, and µis computed as a
median of all absolute values |κe|. We say that these weights are
median-weights w.r.t. the mapping e7→ |κe|.
Since the re’s which contribute to the value Q(v)depend only on
v, we can also write re(v). The following turns out to be important:
PROP OS IT IO N 4. There is a positive semidefinite symmetric matrix
H, which depends on the given mesh, such that Q(v) = vTHv.
Proof: The bracket expression in (9) has the general form gTAg+
2gTBv+vTCvwith symmetric A, B, C =its minimum is
attained for g=A1Bv, so Q(v) = vT(CBA1B)v. Pos-
itivity is obvious from Q(v)0.
Interactive vector field selection. We show how the user can in-
teractively design a piecewise geodesic vector field, by prescribing
the values
v
f1,...,v
fk(10)
of that field in user-selected faces f1,...,fk. This procedure,
which is described in the following paragraphs and which is mo-
tivated by [Huang et al. 2009], first amounts to choosing a vector
field vwhich is not exactly geodesic but has a reasonably small
value of Q(v). In a subsequent step vis sharpened so as to be-
come a piecewise geodesic vector field v.
In order to achieve interactive editing rates, we precompute a re-
duced basis v(1),...,v(n)of vector fields and and try to realize
the user’s selection by a linear combination
v=x1v(1) +···+xnv(n).
of these eigenvectors alone. Our choice is to take the first neigen-
vectors of the quadratic form Q(i.e., eigenvectors of the matrix
Has defined in Prop. 4). In order to determine x1,...,xnand
to reinforce the condition kv
fk= 1 which is not satisfied by the
eigenvectors, we employ two steps in an alternating way: (i) we let
¯
v
f=v
f/kv
fk, and (ii) we determine x1,...,xnsuch that
Fprox =
k
X
i=1 kv
fiv
fik2+λrkvk2+λnX
fFkv
f¯
v
fk2min .
The weights λn,λrgovern the influence of normalization and
regularization terms. The choice of these values is not critical;
we used λn= 0.1and λr= 0.01. We initialize this itera-
tion by letting λn= 0 in the first round. Each round amounts
to solving a linear system and a matrix-vector multiplication, and
takes O(n3) + O(n· |F|)time. For instance, with n= 60 and
|F|= 40 000 we experienced 10 ms per round on a 2 GHz PC.
Further, we found 3–5 rounds to be sufficient. Thus, vector field
selection can be performed in real time.
Remark 2. The number ndetermines the degrees of freedom of-
fered to the user; we found n50 to work well. Guidance is given
by the magnitude of the first neigenvalues which should be small,
and by the complexity of the object under consideration. Choos-
ing the first eigenvectors of Qto span our design space amounts to
choosing a design space where Q(v)has small values under the
‘wrong’ and geometrically meaningless normalization constraint
kvk2= 1. As all vector fields undergo further treatment any-
way, this wrong normalization does not matter. Our purpose was to
find a design space which contains enough degrees of freedom, and
which is taken from the low frequency end of the spectrum (thus
avoiding unreasonably small patch sizes).
Figure 14: Sharpening a vector field
(left) such that it becomes piecewise
geodesic (right). This is a detail of
Fig. 13.
Vector field sharpening. Segmentation of the given surface S
into patches is based on piecewise geodesic vector fields, which
fulfill the geodesic property in the inside of certain patches (i.e., re
is small there), but we allow high values of krekalong the patch
boundaries. We assume that a vector field vis given, and we wish
to find another vector field vwhich is close to v, but is piecewise
geodesic. We set up an optimization problem as follows:
F(v,g) = λvFdist(v) + λrkgk2+Fgeod (v,g)min,(11)
Fgeod(v,g) = X
eE
weρ(kre(v,g)k), ρ(x) = x2
1 + αx2,(12)
Fdist(v) = X
fF
wfkv
fv
fk2.(13)
The function ρcould be any of the heavy-tailed functions used for
image sharpening (see for instance [Levin et al. 2007]). Its purpose
is to push the deviations to accumulation areas. We used the robust
estimator of Geman and McClure [1987], with α= 100 in all ex-
amples. Global weights λr,λvwhich govern the influence of the
regularization term and the proximity of vto vhave to be set ac-
cording to the application. We maintain the condition kv
fk= 1,
so the variables in this optimization are, besides gR3|F|, the
collection of angles θfwhich define the vector v
fin the local coor-
dinate system of the face f. The optimization problem is solved in
the same way as that of Section 4.
The face weights wfoffer the possibility to keep vclose to v
where valready is a geodesic vector field – indicated by smallness
of re(v). We employ median-weighting w.r.t. the mapping f7→
Pe∂f kre(v)k.
Surface segmentation and pattern layout. Having found a
piecewise geodesic vector field v, we now define patches by cut-
ting along the edges where vis sharp. We first collect all edges
e=f1f2where the angle between vf1and vf2is greater than a
threshold value α; such edges indicate patch boundary curves (we
use α= 20). We then use the method of Pauly et al. [2003] to
polish these curves. We omit details since we do not consider seg-
mentation a new result. Segmentation being completed, we end
up with patches P1, . . . , PMwhich we know can be covered by a
smooth geodesic vector field. The layout of evenly spaced geode-
sics within each Pjis done according to Section 4, augmenting the
target functional (6) by
F(j)
align =λuser X
i:fiPjh∇φ(f),v
fii2+λsharp X
fPjh∇φ(f),vfi2.
Minimizing F(j)
align means that the geodesics occurring as level sets
of φare aligned with the sharpened vector field vand/or with the
user’s selection {v
fi}. The corresponding weights λsharp and λuser
have to be set accordingly. Results are shown in Figs. 1, 13, and 15.
Figure 15: Instead of properly
segmenting a surface into parts
which can be covered by smooth
geodesic 1-patterns, we may look
for a weaker solution: cutting it
open along curves such that it can
be covered by 1-patterns which
are smooth everywhere except at
the cuts. The present example is
almost a true segmentation with
only 1 dangling edge.
General remarks. There are several reasons why the interactive
procedure of Section 6 is not only nice to have in applications, but
is actually necessary: First, the highly nonlinear minimization of
the function F(v,g)in Equation (11) will typically get stuck in
local minima, so we cannot expect that unguided minimization of
F(v,g)succeeds. Second, there is a great variety of local minima
which cannot clearly be distinguished by the magnitude of F(v,g)
alone. For this reason it is necessary to let the user choose.
7 Discussion
Architectural applications are not limited to timber structures, as the
static properties of geodesics apply to any material. An additional
bonus is that beams which follow geodesics, being shortest, need
less material than other shapes. Fig. 16 shows an example, featuring
a 2-pattern of geodesics which is designed for realization in steel
rather than in wood.
Comparing the vector field method with other approaches.
The segmentation procedure of Section 6 is governed by both user
interaction and Gaussian curvature, since it produces patches inside
which the Gaussian curvature has a nice behaviour — ‘nice’ being
defined in terms of the capability of being covered with a 1-pattern
of geodesic curves. Segmentation of surfaces driven by Gaussian
curvature as proposed by [Yamauchi et al. 2005] for even distribu-
tion of Gaussian curvature or by [Julius et al. 2005] for near-de-
velopable patches probably produces patches which can be covered
Figure 16: Relations between fami-
lies of geodesics. The level set ap-
proach is capable of accommodating
additional requirements, such as a de-
sired intersection angle αbetween the
level sets of functions φ1, φ2. This 2-
pattern of geodesic curves is imposed
on the Warsaw Lilium Tower by Zaha
Hadid Architects. True geodesics in
orange are shown for comparison.
by geodesic 1-patterns. The method of the present paper, however,
directly works with geodesics in a user-controlled way; Gaussian
curvature is present only implicitly.
This paper contains another way of segmenting a surface into
patches which can be covered by almost equidistant geodesics,
namely the evolution of broken geodesics as illustrated by Fig. 9 (it
even yields a consistent spacing of curves along the patch bound-
aries). The method of Section 6 is much more flexible, however,
and treats each part of the surface in the same way, independent of
an initial choice of geodesic to evolve from.
Computation details and timings. For the level set approach,
details on the choice of weights, the number of variables, etc., as
well as timings are given by Table 1. For the geodesic vector field
approach, the choice of weights is detailed by Section 6. Timings
are given by Table 2 for three examples, each of which have 40k
faces. Preprocessing (eigenspace computation with Arpack) needs
30–40 seconds. One round of sharpening takes 1.1 seconds, and
the final segmentation needs about 1 second. That table also shows
times for the evolution of broken geodesics according to Section 3.
Here about 70% of the time is used for the piecewise fitting of
curves, the rest is postprocessing like merge, trim, etc.
Fig. var iter. sec FκFFwFangle λ µ ν
1 33k 4 32 .25 20 10200
10a 10k 10 16 .0006 2470 25 1081040
10b 10k 11 17 .71 392 .0007 10410+4 0
11 21k 12 77 .005 145 10600
16 159k 10 419 .66 1050 .26 3·10402.8
Table 1: Details for the level set approach. We show the number of
variables, number of iterations, total time for a 3GHz PC, the values
of functionals and the weights used in optimization.
Figure 9a 9b 1 (left ) 13 17 (left)
seconds 86 113 102 85 73
Method Evolution Vector fields
Table 2: Representative timings for evolution of geodesics, and for
the geodesic vector field method. Data apply to a 2GHz PC.
Limitations. Since in a fixed surface the geodesics are only a two-
parameter family of curves, often the designer’s request cannot be
met and one has to compromise (see Fig. 10b for an example of a
decision for equal spacing rather than for the geodesic property).
This phenomenon can also cause the segmentation process of Sec-
tion 6 to produce unsatisfactory results, in which case it has to be
iterated (see Fig. 17).
Geodesic webs pose many constraints on the involved curves and
in fact we cannot expect them even to exist in a mathematically
exact way on arbitrary surfaces. Our level set approach produces
webs of curves which are as geodesic as possible, but deviations
of level curves from true geodesics are inevitable (see Fig. 18). As
mentioned in Theorem 2, on constant Gaussian curvature surfaces
we have the same variety of geodesic webs as for straight line webs
in the plane. This is verified by the fact that the level curves in
Figures 11 and 19 are indistinguishable from true geodesics.
Figure 17: A situation where two rounds of segmentation are nec-
essary because of the unsatisfactory result of the first round.
Figure 18: A hexagonal web of
near-geodesics. We show some
curves of the web Fig. 1 is based
on. Due to the strong variation
in Gaussian curvature they de-
viate from true geodesics (thick
curves).
One major motivation for geodesic patterns is the cladding of
freeform surfaces by thin wooden panels. This cladding problem
has been considered previously by Pottmann et al. [2008b], in the
context of developable strips which certainly are a shape which thin
panels can assume. In that paper the importance of curve’s tangents
staying away from asymptotic directions was mentioned. We could
incorporate this condition in our level set approach, but we have
completely neglected it in this paper. In the negatively curved areas
of surfaces, it would very much obstruct our available degrees of
freedom. The reason why we chose not to aim at developable strips
is that they are not the shape that bent wooden panels assume in
general; such panels can be twisted (for other materials, however,
this additional aspect has to be observed to a higher degree).
Conclusion and future research. Motivated by problems in
freeform architecture we have described three different approaches
to the layout and interactive design of geodesic patterns on surfaces.
Each of the methods treated in this paper has its specific strength:
The Jacobi field approach to the evolution of geodesic curves is best
suited to deal with local issues. The level set approach can deal with
the global layout of patterns very well, and is efficient in dealing
with multiple patterns of geodesic curves which are in some rela-
tion, such as the condition that they form a hexagonal 3-web. The
vector field approach is capable of solving the global layout and
segmentation problem in an interactive way.
This paper seems to be the first where the classical geometry of
webs is employed in a geometry processing context. It is likely that
this theory has yet more potential and applications, in particular in
architecture.
Acknowledgments. This research has been supported by the
Austrian Science Fund (FWF) under grants No. S92-06 and S92-
09 (National Science Network Industrial Geometry), and by the
European Community’s 7th Framework Programme under grant
agreement 230520 (ARC). The authors gratefully acknowledge the
support of NSF grants 0808515 and 0914833, of NIH grant GM-
072970, and of a joint Stanford-KAUST collaborative grant. We
want to express our thanks to Zaha Hadid Architects, London, to
be able to work on data which come from some of their current
projects.
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(a) (b)
Figure 19: Geodesic 4-pattern. For a combination of two hexag-
onal webs, functions φ1,φ2are optimized such that level sets of
φ1, φ2, φ1±φ2are geodesic. (a) Level sets (thin) and true geode-
sics (thick). Level sets of all four functions are almost truly geo-
desic, because the Gaussian curvature is nearly constant for this
particular surface. (b) Transfer of an Islamic art pattern according
to [Sutton 2007] to the surface by the simple texture coordinates
u=φ1,v=φ2. The four principal directions in the pattern follow
geodesic curves.
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