Content uploaded by Romain Mesnil

Author content

All content in this area was uploaded by Romain Mesnil on Feb 12, 2018

Content may be subject to copyright.

https://doi.org/10.1177/0266351117738379

International Journal of Space Structures

1 –15

© The Author(s) 2017

Reprints and permissions:

sagepub.co.uk/journalsPermissions.nav

DOI: 10.1177/0266351117738379

journals.sagepub.com/home/sps

Introduction

The design of complex architectural shapes has bene-

fited from great advances from the computer graphics

community in the last decade. For instance, significant

efforts were made to develop numerical methods for the

covering of free-form surfaces with planar panels. These

methods differ from the common knowledge of archi-

tects and engineers. Although properly implemented

methods are efficient and relatively simple to use, they

remain black boxes providing little insight on the nature

of the design constraints imposed by facet planarity. The

technique proposed in this article aims thus at modelling

efficiently meshes with planar facets and at providing

the designers with an understanding of the design space.

The proposed method takes inspiration from descriptive

geometry, a tool used by architects for centuries, and

turns it into a real-time design tool for planar quadrilat-

eral (PQ)-meshes.

Prior works

Geometrically constrained approach. PQs have been identi-

fied by practicians as an important optimisation target for

the construction of double curved surfaces, as they avoid

using curved panels.1 Several techniques for generating

exact PQ-meshes were proposed, mostly relying on affine

transformations, which preserve planarity, a notion illus-

trated in Pottmann et al.2 For example, Scale-trans sur-

faces, introduced in Schober,3 use composition of two

affine transformations: translation and homothetic transfor-

mations. The designer controls the shape with two curves,

making the process highly intuitive. Despite formal limita-

tions, these shapes have been used in many projects.

The term of geometrically constrained design approach

has been introduced by Bagneris et al.4 Constrained geo-

metric approaches use shapes that are well known and can

Marionette Meshes: Modelling free-form

architecture with planar facets

Romain Mesnil1,2, Cyril Douthe2, Olivier Baverel2

and Bruno Leger1

Abstract

We introduce an intuitive method, called Marionette, for the modelling of free-form architecture with planar facets. The

method takes inspiration from descriptive geometry and allows to design complex shapes with one projection and the

control of elevation curves. The proposed framework achieves exact facet planarity in real time and considerably enriches

previous geometrically constrained methods for free-form architecture. A discussion on the design of quadrilateral

meshes with a fixed horizontal projection is first proposed, and the method is then extended to various projections and

patterns. The method used is a discrete solution of a continuous problem. This relation between smooth and continuous

problem is discussed and shows how to combine the marionette method with modelling tools for smooth surfaces, like

non-uniform rational basis spline or T-splines. The result is a versatile tool for shape modelling, suited to engineering

problems related to free-form architecture.

Keywords

architecture, descriptive geometry, fabrication-aware design, Marionette Meshes, mesh planarisation, structural morphology

1Bouygues Construction, France

2 Laboratoire NAVIER, Ecole des Ponts, IFSTTAR, CNRS, (UMR 8205),

Université Paris-Est (UPE), France

Corresponding author:

Romain Mesnil, Laboratoire NAVIER, École des Ponts, IFSTTAR, CNRS

(UMR 8205), Université Paris-Est (UPE), Avenue Blaise-Pascal, 77420

Champs-sur-Marne, France.

Email: romain.mesnil@enpc.fr

738379SPS0010.1177/0266351117738379International Journal of Space StructuresMesnil et al.

research-article2017

Article

2 International Journal of Space Structures 00(0)

be rationalised efficiently, for example, towards a high rep-

etition of nodes or panels.5 More recently, other design

strategies exploring more complex shapes based on Möbius

transformations and inspired by Bobenko and Huhnen-

Venedey6 have been proposed.7 The idea to use groups of

transformations to study geometrical properties of surfaces

is not new,8 but recent applications to architecture show

that this has a great potential. Generally speaking, these

methods suffer, however, from a lack of flexibility and

form a restricted design space. New geometrically con-

strained techniques merging fabrication and structural con-

siderations have been proposed recently and extend the

potential of classical methods. For example, an elegant

design methodology based on planar pre-stressed networks

through Airy stress function allowed Adriaenssens et al.9 to

design a funicular irregular gridshell with planar facets.

The lack of flexibility of usual modelling techniques led to

the introduction of post-rationalisation strategies in order to

cover arbitrary shapes with PQs.10

Optimisation-based shape exploration. Most recent methods

propose hence to explore design space of feasible solutions

for a given mesh topology with the help of optimisation

techniques.11,12 The mesh is interactively deformed by the

user with the help of control handles. The overall smooth-

ness is checked with discrete functions of the vertices. To

go further, an efficient solver handling quadratic constraints

was presented in Tang et al.13 and used in Jiang et al.14

Projections and subspace exploration are efficiently used

for constrained-based optimisation in Bouaziz et al.15 and

Deng et al.11,16 These methods provide a great design free-

dom, but illustrations shown in the cited references are

local, handle-based, deformations of meshes. Local defor-

mations of shapes are of particular interest in the computer

graphics community, but architectural modelling paradigms

are generally thought as ways to steer a shape as a whole.

The idea here is to use the notion of projection, which is

commonly used in architecture, especially with plane view

and elevations, and to link subspace exploration tech-

niques with representation techniques based on projections

in architecture.

Descriptive geometry. Descriptive geometry is a technique

of shape representation invented by French mathematician

Gaspard Monge.17,18 It is based on planar orthogonal pro-

jections of a solid. The planes, in which the projections are

done, are usually the horizontal and vertical planes. Figure

1 is a typical drawing of descriptive geometry: it describes

an ellipsoid with a plane view, displayed with some eleva-

tions. The curve network corresponds to the horizontal

projection of lines of curvature.19

Because architectural objects have to deal mainly with

gravity and vertical forces, it makes naturally sense to sep-

arate projections in vertical and horizontal planes. The

idea to use these projections to guide structural design was

used recently in computational frameworks based on the

Thrust Network Analysis20 where compression-only struc-

tures are found from a planar network at equilibrium.21,22

The objective of this article is to show that descriptive

geometry can be turned into a general tool for the design of

PQ-meshes and their structural optimisation. The method,

called Marionette method is presented in section

‘Marionette Meshes’, where the relation between smooth

and discrete geometry for PQ-meshes is discussed. Section

‘Architectural design with Marionette Meshes’ explores

then some applications in architecture. Section

‘Generalisation of the method’ shows finally the generality

of the proposed method, which can be extended to meshes

other than regular quadrilateral meshes and therefore con-

stitutes a promising versatile tool to intuitively integrate

fabrication constraints into architectural design.

Marionette Meshes

Marionette Quad

The principles of descriptive geometry can be transposed

to architectural shape modelling. The use of appropriate

projections provides a simple interpretation of the problem

of meshing with flat quadrilaterals. For simplification, we

discuss the case of a projection in the

()

XY plane in this

section: the generalisation to other projections is illustrated

in section ‘Generalisation of the method’.

Consider first Figure 2, four points have a prescribed

plane view ABCD in the horizontal plane

()

P

1. Three

points ′

A, ′

B

and ′

D have prescribed heights zA, zB

and zD. In general, there is only one point ′

C with the

imposed projection C, so that

′′′′

ABCD is planar.

The planarity constraint reads

det ′′ ′′ ′′

()

=AB AC AD,, 0 (1)

Figure 1. Lines of curvatures of an ellipsoid with descriptive

geometry.19

Mesnil et al. 3

Expressing coordinates in a Cartesian frame of

()P

1

and writing dBC =det2D

()

AB AC,, dBD =det2D

()

AB AD,

and dDC =det 2D

()

AD AC,, if the points A, B and D

are not aligned, then, one gets

zz d

dzz d

d

zz

CA

BC

BD

DA

DC

BD

BA

−

()

=

⋅−

()

+

⋅−

()

(2)

Figure 2 shows vertical lines used for construction, recall-

ing the strings of a marionette, which gives the name

Marionette Quad. Note that the system is under-constrained

if the points A, B and D are aligned: in that case, the vec-

tors AB and AD are colinear and dBD is equal to zero. This

configuration corresponds to vertical a quad. A projection in

the horizontal plane thus allows only for the modelling of

height fields. This limitation can be overcome using other

projections (see section ‘Generalisation of the method’).

Regular Marionette Meshes

Consider now a quadrangular mesh without singularity as

depicted in Figure 3. The plane view in the horizontal

plane is fixed, and two intersecting curves are defined on

the projection planes P

2 and P

3. Then, provided that the

planar projection admits no degenerated quad (i.e. quad

where three points are colinear), equation (2) can be propa-

gated through a strip, and by there, through the whole

mesh. Indeed, on the highlighted strip of Figure 3, the first

quad (top left) has three prescribed z-values, and equation

(2) can be used and so forth. The same applies for all the

quads of the strip.

For a

NM×

mesh, the propagation requires NM

applications of equation (2), the memory needed is 3NM .

The marionette technique guarantees hence that the num-

ber of operations varies linearly with the number of nodes

within a structure. The method performs thus in real time

even for meshes with thousands of nodes, as discussed in

section ‘Computational setup’.

Link with smooth geometry

Partial differential equation. The proposed method has some

interesting relations with smooth geometry. The problem

of covering curved shapes with planar panels is linked

with the integration of conjugate curves networks.10,23

Conjugate networks correspond to parameterisations

()

uv, satisfying the following equation23

det ∂∂∂

()

=

uvuv

2

,, 0 (3)

Consider now that the components in

x

and

y

are

fixed, like in the problem solved by the Marionette tech-

nique. We are looking for the height functions fz satisfy-

ing equation (3). Adopting the notation fu to denote

differentiation of f with respect to

u

, equation (3) is

reformulated into

det

fff

fff

fff

u

x

v

x

uv

x

u

y

v

y

uv

y

u

z

v

z

uv

z

=0 (4)

Equation (4) is defined if the parameterisation in the plane

()

XY is regular, which means if the study is restricted to

height fields. We expand equation (4) using adjugate

matrices

ff

ff

fff

ff

fff

ff

f

u

x

v

x

u

y

v

yuv

zv

x

uv

x

v

y

uv

yu

zu

x

uv

x

u

y

uv

yv

z

+−=0 (5)

Equation (5) is a second-order linear equation in fuv

z

()

,.

The only term of second-order is fuv

z: the equation is thus

hyperbolic. Hyperbolic equations often correspond to the

propagation of information in a system (think of the wave

equation). It is thus no surprise that the marionette method

corresponds to a propagation algorithm. Loosely speaking,

it can be shown that solutions of hyperbolic equations

Figure 2. Creation of a Marionette Quad with a plane view and

two elevations.

Figure 3. Two elevations and a planar projection define a

unique Marionette Mesh.

4 International Journal of Space Structures 00(0)

retain discontinuities of initial conditions. The smoothness

of the shape obtained by the marionette method is thus

dependent on the smoothness of the input data (plane view

and elevation curves).24

Boundary conditions. With the marionette method, we pre-

scribe the values of fz on two boundaries. Mathemati-

cally, we choose two functions

fv

1

()

and

fu

2

()

which

correspond to the height of the two guide curves

fuuv fv

fuvv

fu

fu fv

z

z

=

()

=

()

=

()

=

()

()

=

()

01

02

20

10

,

, (6)

The last equality corresponds to a compatibility condition

between equation

fv

1

()

and

fu

2

()

, so that the height of

fuv

z

()

00

, is known without ambiguity. This equation

corresponds to an integration of the second member of

equation (7). Writing fu fuu

32

=() (/)( )∂∂ , we have

fuuv

fv

f

uuv vfu

z

z

=

()

=

()

∂

∂=

()

=

()

01

03

,

,

(7)

We see now that we specify the height of a guide curve

and the slope on the second curve. This kind of boundary

condition based on both values and derivatives is called

Cauchy boundary condition and is particularly suited for

hyperbolic equations.24 The smooth problem solved by the

marionette method is thus a classical problem in the theory

of partial differential equations. Classical results on the

existence, uniqueness and regularity of solution can be

applied, even though it is not the purpose of this article.

Marionette Meshes with singularities

The modelling of complex shapes requires the introduc-

tion of internal vertices with a valence other than four,

called singularities in the following. For example, the

mesh displayed in Figure 4(a) has one singularity: the cen-

tral node has a valence of six. The mesh can be subdivided

into six patches with no inner singularity (in blue and

white). This kind of procedure can be applied to any quad

mesh. Each patch is a regular mesh, and the marionette

technique can be applied. There are, however, restrictions

on the curves used as guide curves due to compatibility

between patches. For example, in Figure 4(a), it is clear

that the six curves attached to the singularity can be used

as guides for the six patches, whereas choosing the 12

curves on the perimeter over-constrain the problem.

For an arbitrary quad mesh, it is possible to compute the

number of guide curves that can be used to generate a

Marionette Mesh. The mesh can be decomposed into sim-

ple quad domains without any singularity, for example,

using the methods described in Tarini et al.25 or Takayama

et al.26 For example, Figure 4(a) has six domains and the

mesh in Figure 5(a) has nine domains. These domains are

four sided, and it is possible to extract independent fami-

lies of strip-domains, like displayed in Figure 5. Depending

on the n-colourability of the mesh, the number of families

varies. The example showed is two-colourable. As a result,

two families of strips can be found and are shown in Figure

5(b) and (c). Exactly one curve can be chosen across each

strip-domain. Since strips are independent, the height of

these nine curves can be chosen independently and will not

over-constrain the problem.

Closed Marionette Meshes

Closed strips. Marionette Meshes create PQ-meshes by

propagation of a planarity constraint along strips. One can

easily figure that if the strip is closed, the problem becomes

overconstrained. Indeed, consider Figure 6, the plane view

of a closed strip and the height of the points

()

P

i of one

polyline are prescribed. If the height of the first point used

for the propagation P

0

* is chosen, the planarity constraint

can be propagated along the strip. The points of the outer

line are, therefore, imposed by the method, and the designer

Figure 4. A Marionette Meshes with a singularity: (a)

decomposition of a complex mesh into simple patches and (b)

the corresponding lifted mesh.

Mesnil et al. 5

has no control on them. The last point P

N

*

is therefore gen-

erally different from the initial point P

0

*

, leading to a geomet-

rical incompatibility of PQ-meshes.

In the following, we develop a method to deal with the

geometrical compatibility of closed strips. The results,

however, can then be extended to general Marionette

Mesh with closed strips. Suppose that the two prescribed

curves are defined as the inner closed curve and one

radial curve (see Figure 6). By propagation of equation

(2), we easily see that the height of the last point zN

*

depends linearly on the height of the first point z0

* and on

the heights of the points on the inner curve Z. It also

depends on the in-plane projection of the strip. Formally,

there exists a vector V and a scalar

a

, both functions of

the plane view so that

VZ⋅+⋅az zN0

**

= (8)

We are interested in the case where

zz

N0

**

=. There are

two possibilities:

1. a

=

1: in this case, the condition restricts to

VZ

⋅=

0 and does not depend on z0

*. The vector

Z is in the hyperplane of V, which leaves N

−

1

degrees of freedom (DOFs).

2. a ≠ 1 : there is only one solution for z

0

*

. This is the

most constrained case: the designer can only con-

trol the inner curve of the strip.

Detailed calculations on closed strips and particular

examples satisfying the condition a

=

1 are given in

Appendix 1.

Closed meshes. The meshes with one solution are less flex-

ible, but they can still generate interesting shapes, like the

one displayed on Figure 7, which recalls the example of

Figure 6. The designer has a total control on the height of

the inner curve and the plane view, but cannot manipulate

freely the outer curve.

The most interesting case occurs when the designer has

potentially the control of two curves. It relies on a condi-

tion on the planar projection explained above. A simple

case where this condition is fulfilled is when it has a sym-

metry. In this case, there is a N

−

1 parameters family of

solutions for the height of the inner curve. The elevation of

a closed guide curve can be chosen arbitrarily and pro-

jected into the hyperplane of normal V, keeping the nota-

tions of equation (8). This operation is straightforward and

allows to control the elevation of a second curve, like for

open meshes. An example of this strategy is displayed in

Figure 8, where all the meshes have the same planar pro-

jection, which has a plane of symmetry. The Marionette

Figure 5. Decomposition of a mesh into two families of strip-domains. Marionette Meshes can be generated by choosing one guide

curve across each strip-domain: (a) initial mesh, (b) family of four strip-domains and (c) family of five strip-domains.

Figure 6. Closed Marionette strip with incompatible closing

condition induced by the prescription of the plane view of the

whole strip (yellow) and the heights of the inner curve (blue).

Figure 7. Architectural design with a closed Marionette Mesh:

the height of the inner curve is prescribed and the designer

does not have control on the outer curve.

6 International Journal of Space Structures 00(0)

Mesh is on the second row, and the column on the right in

Figure 8 has, however, no plane of symmetry.

Another look at the problem. The problems specific to closed

strips or meshes can be understood by the consideration of

the equivalent smooth problem. The partial differential equa-

tion (5) remains unchanged, but the boundary conditions

expressed by equation (7) are not valid anymore. Indeed, a

closed surface imposes a periodicity of the solution. Con-

sider the case where we want the curves

()

u=constant to

be closed, there exists a certain period T so that

fuuv fv

f

uuv vfu

uf uv Tfuv

z

z

zz

=

()

=

()

∂

∂=

()

=

()

∀+

()

=

()

01

03

,

,

,, ,

(9)

This additional boundary condition might over-

constrain the problem and the existence of a solution is not

certain.

Architectural design with Marionette

Meshes

Computational setup

The algorithms described in this article have been imple-

mented in the visual-scripting plug-in Grasshopper™ for

the modelling software Rhino™. This allows interaction

with other numerical tools necessary for architectural

design, like finite element analysis software Karamba™.

An example of interaction between fabrication-aware shape

generation and structural analysis is shown in section ‘Case

study: fabrication-aware structural optimisation’.

Marionette Meshes only require the solution of a sparse

linear system. The computation time is thus low, typically

it takes 3 ms to lift a mesh of 10,000 faces, with no pre-

factorisation involved. Real-time computation provides

great design flexibility, even for large meshes. It also opens

possibilities to perform optimisation with numerous func-

tion calls, for example, with genetic algorithms, as dis-

cussed in section ‘Case study: fabrication-aware structural

optimisation’.

The basic input of the method are a planar projection

and several elevation curves. The marionette framework

allows the architect and the engineer to choose a topology

and some mesh features (like alignment of the mesh to a

boundary) and generates a design space of feasible solu-

tions. In comparison, post-rationalisation techniques fit

perfectly a target geometry (e.g. with conjugate fields

integration), with less control over the grid topology.

Since parameterisation is equally important as shape in

the overall aesthetics and structural behaviour of a grid

structure, the marionette technique, like some optimisa-

tion-based algorithms,11,13 offers an interesting alternative

to post-rationalisation.

In our framework, the planar projections are generated

with non-uniform rational basis spline (NURBS) patches,

and the elevation curves are drawn as Bézier curves. The

smoothness of the final mesh depends thus on the smooth-

ness of the in-plane parameterisation. A C0 projection

yields a C0 solution to the hyperbolic equation (4), so that

shape functions with creases can easily be propagated

through the mesh.27 Figure 9 shows a corrugated shape

generated from a C0 planar projection and smooth guide

curves. Such corrugations can be used in folded plate

structures and could extend the formal possibilities of

methods developed in Robeller et al.28 or discussed in

Lebée.29

Geometrical optimisation

General remarks. The method used in this article constructs

a space of solutions with planar facets. This space is a vec-

tor space, which has some interesting implications for

some optimisation problems. We can indeed see the pla-

narity constraint as a linear constraint on the coordinates of

all the vertices of a mesh. There exists a matrix A and list

of heights

z

and zp so that

Az z

p

⋅= (10)

The marionette method gives an intuitive way to construct

this matrix, as zp corresponds to the heights of vertices on

Figure 8. Some shapes with planar faces and a closed mesh

generated with the method proposed in this article. All the

shapes are built from the same planar projection.

Mesnil et al. 7

the guide curves and

A

depends on the planar projection.

For a mesh with NM faces, A is a matrix of size

(()( ))NM

NM

++ +⋅ +1,

11

It is a well-known fact that minimisation of quadratic

functions under linear constraints is equivalent to the solv-

ing of a linear system.30 An example of such optimisation

problems with useful applications for architectural design

is given in the following.

Surface fitting. A common problem described in the litera-

ture is the approximation of a given shape with a PQ-mesh.

In the following, we consider that the designer prescribes a

planar projection and looks for the closest Marionette

Mesh to a reference surface.

The problem is illustrated in Figure 10, the heights of

the vertices in the Marionette Mesh are written

z

and the

heights of the points on the reference surface are written

z0. The function to minimise is written as follows

JT

zzzzz

00

()

=−

()

−

()

(11)

The design space is the Marionette Meshes, which have

the considered planar projection. This constraint is written

in equation (10). The optimisation problem follows

zAzp

p0

p0

zAzz Az z

=

minminJT

()

=−

()

−

()

(12)

Expanding the equation, one gets

Jp

TT TT

TT

zzAAzzAz

zz

pp pp

00

()

=−+20 (13)

A necessary condition to find a solution is to verify that

∇Jp=0. The system reduces, therefore, to

AAzAz

p0

TT

= (14)

Equation (14) is typical of least square problem. It is clear

that the rank of the matrix A is

()

NM

++

1. It follows

that the rank of

AA

T is also

()

NM

++

1. Since

AA

TNM NM∈++++

()

1, 1 , this matrix is inverti-

ble. Equation (14) has, therefore, one unique solution.

Since

AA

T is clearly definite positive, it follows that the

extremum is in fact a local minimum. Finally, the behav-

iour when zp

→∞

demonstrates that this is a global

minimum.

An application is illustrated in Figure 11, where a target

NURBS is approached by a surface of translation, which is

well known in architectural design.1 This optimal can be

considered poor, but the key information is that it is the

best in the design space chosen by the designer, so that the

designer knows that to improve the solution, he has to

explore other planar projections or mesh topologies. The

surface displayed is indeed the best solution possible for

the planar projection chosen by the designer. As computa-

tion is done in real time, it is easy to generate very quickly

different plane views with different topologies, keeping

control of the aesthetic and layout of the cladding. The

mesh topologies could be generated from a catalogue, like

the ones generated in Takayama et al.,31 and their relevance

for the shape-fitting problem could be efficiently assessed

with the marionette method. The selection of a proper

mesh topology remains, however, open for future work.

Shape exploration with Marionette Meshes

The framework introduced here intrinsically accounts for

planarity of panels. Its mathematical formulation is, how-

ever, suited for many architectural constraints. Hard con-

straints must be fulfilled exactly, whereas soft constraints

are included into the function to minimise.30 Since the

planarity constraint is linear, soft constraints expressed as

linear or quadratic functions can easily be included in the

objective function. In this case, the optimisation problem

will be similar to a classical least square problem and can

be solved efficiently.

Figure 9. A non-smooth mesh with planar facets generated

with the marionette method.

Figure 10. Optimisation problem: approximation of a

reference surface with a given planar projection (dashed lines).

8 International Journal of Space Structures 00(0)

Hard constraints defined by linear equations are treated

effectively within the proposed framework. Examples of

linear constraints are prescribed volume and a maximal

allowable height. The marionette method imposes

NM NM−+−

(1

) out of NM parameters, this means

that another NM

+−

1 linear constraints can be applied

without over-constraining the optimisation problem.

Perhaps, the most interesting application is the pre-

scription of a boundary, as depicted in Figure 12. In this

figure, the planar projection is imposed and the user pre-

scribes the height of some points of the mesh along a curve

(white circles). In this case, the number of prescribed

points is superior to the number of DOFs, and the problem

might be overconstrained. It might hence be preferable to

turn this problem into a soft constrained problem with a

quadratic function to minimise.

Other constraints could be used. For example, in the

manner of NURBS modelling, the user could control the

height of some handle-points, each handle decreasing the

size of the space of solution by 1 DOF. This kind of

approach has been used in optimisation-based shape explo-

ration, but it loses the notion of global shape control.

Case study: fabrication-aware structural

optimisation

The formal possibilities offered by Marionette Meshes are

broad enough to offer an interesting design space for

engineering problems. Among them, structural optimisation

is a particularly relevant. The quick generation of a param-

eterised design space and the coupling with advanced analy-

sis software seems particularly promising.32 Indeed,

non-linear criteria, like the buckling capacity, are of high

importance for practical design of thin shell or grid shells.33

An illustration of the potential of Marionette Meshes

for a structurally informed architectural design is proposed

in Figure 13: the shell is a Marionette Mesh spanning over

an ellipse. The plane view is inspired by Figure 1. The

mesh is constitutes six NURBS patches and has two singu-

larities (white dots in the image), and guide curves are

found with the method proposed in this article. The bound-

ary curve is constrained in the horizontal plane (blue curve

on Figure 13). One curve in the other direction (orange

curve in Figure 13) defines the whole elevation of the

dome. The shell is submitted to gravity load. All

Figure 11. A target surface (left) and the optimal

approximation by a surface of translation (right).

Figure 12. A plane view (thin lines) with a prescribed

boundary (thick lines).

Figure 13. A result of an optimisation procedure: the shell

structure is a Marionette Mesh constructed from six patches

(top) minimising total elastic energy. Middle: guide curves and

top view. Bottom: red areas indicate compression, the shell

works mostly with compression.

Mesnil et al. 9

the translations at the outer boundary are restricted, and

rotations at the supports are allowed (hinges). The model is

computed with finite element software Karamba3D™.

The shape generation of a 1000-face mesh requires less

than 1 ms with the marionette technique, far less than the

assembly and computation of a shell model with finite ele-

ment method (FEM).

The structure is optimised towards a minimum of the

total elastic energy by the means of genetic algorithms. In

Figure 13, the colour scheme is used to represent the maxi-

mal principal stress σ11 on the upper side of the shell. Area

with tensile stress should appear in blue, but is not visible

in the figure. Indeed, tensile stress in Figure 13 is almost

non-existent: the maximal tensile stress is 0.2 MPa for a

dome with a span of 40 m and a shell thickness of 10 cm.

Hence, if defined properly with an accurate number of sin-

gularities, the design space offered by Marionette Meshes

is wide enough to find compression-dominant shapes by

the means of structural optimisation.

Generalisation of the method

General projections

It appeared that prescribing a horizontal view and applying

the propagation technique presented here only allows for the

modelling of height fields. This is a limitation of this method,

although height fields surfaces are commonly used for roof

covering. Other projections can be used for more shape

flexibility. The planarity constraint for a quad can be extended

to the case of non-parallel projections, like in Figure 14.

Some projections are of practical interest for archetypal

projects. Towers and facades can be modelled with cylin-

drical projections. Stadia can be designed using projec-

tions on torus or on moulding surfaces, with the offset

directions corresponding to the normals of the smooth sur-

face. Moulding surfaces fit naturally the geometry of sta-

dia (see Figure 15(a)) and have some interesting features,

discussed in Mesnil et al.:5

• Their natural mesh contains planar curves, which

are geodesics of the surface: the planarity is pre-

served by the marionette transformation.

• They are naturally meshed by their lines of curva-

tures, which gives a torsion-free beam layout on the

initial surface and on the final shape.

Extension to other patterns

This section proposes to extend the marionette method to other

patterns than quads. First, we discuss the estimation of the size

of the design space offered by the marionette technique in the

most general cases. We illustrate then those remarks on the

generality of the method with various patterns.

Size of the design space. The facet planarity constraint is lin-

ear, which means that the space of meshes with planar facets

is a vector space. Deng et al.16 proposed a criterion to evalu-

ate the dimension of this vector space. For each facet, three

points can be chosen independently (3 DOF for each points),

and the remaining points must be chosen in the constructed

plane (1 DOF deleted for these nodes). Writing nF the

number of vertices for each face, the estimation of the size

of the space of meshes with planar facets follows16

NN n

nodes

Faces

F

33−−

()

∑

(15)

For a quad mesh, we get

NN

nodes

2. This number is

high and is difficult to interpret for the designer. The

Figure 14. A Marionette Quad with non-parallel guide lines.

Figure 15. Design of stadia obtained from a projection on a moulding surface: the prescribed curves are the inner ring and a

section curve: (a) reference moulding surface, (b) non-symmetrical design and (c) symmetrical design.

10 International Journal of Space Structures 00(0)

projection technique used in this article reduces the size of

the design space. Since the planar projection is prescribed,

each point looses 2 DOF. Equation (15) writes

NN n

marionette nodes

Faces

F

∼− −

()

∑

3 (16)

The size of the design space is reduced compared to gen-

eral methods, but the smoothness of the final shape is easily

controlled. Note that techniques relying on generation of the

whole vector space have to introduce fairing energies, as the

design space contains both smooth and non-smooth meshes.

For a quadrilateral mesh with

nm×

faces without sin-

gularity, we have Nnm

nodes =+

⋅+

()

()11

and

nm

faces.

The application of equation (16) shows that the size of the

design space is nm

++

1, which is exactly what is found

by the marionette method.

Application to non-standard patterns. Equation (16) can be

applied to meshes composed of triangles and hexagons,

also known as Kagome lattices. It reveals that the number

of DOFs is comparable to the one of the quadrilateral

meshes. There is, therefore, a straightforward way to lift

Kagome lattices with the marionette technique. Figure

16(a) shows the guide curves for the Kagome pattern.

Other isolated points are required to lift the mesh. The

height of these points can, for example, be chosen in order

to optimise mesh fairness, which has been characterised in

numerous works by an energy

defined in equation

(17), where vi is the ith vertex of a polyline

=−+

∑∑ ++

polylines i

vv v

ii1i2

22

(17)

The functional is quadratic and is not difficult to mini-

mise under linear constraints. Figure 16(c) shows a mesh

derived from an hexagonal pattern: three guide curves can

be used to lift the mesh. The number of DOFs of the exam-

ples of Figure 16 are evaluated in Appendix 2.

Illustration. Figure 17 shows a Kagome lattice covered

with planar facets generated with the marionette method.

The design started from a planar projection generated with

a NURBS patch, and a Kagome was then generated fol-

lowing the isoparametric lines and lifted with the mario-

nette technique. One of the guide curve is the parabolic

arch of the entrance, and the other is an undulating curve

following the tunnel. Like for PQ-meshes, the computa-

tion is done in real time.

Conclusion

We have introduced an intuitive technique for interactive

shape modelling with planar facets. It is based on descrip-

tive geometry, which has been used by architects and engi-

neers for centuries. The concept has many applications, in

particular, the modelling of PQ-meshes with or without

singularity. Some examples show the formal potential of

Figure 16. Marionette method applied to several patterns, white dots correspond to prescribed heights: (a) Kagome lattice, (b)

dual Kagome lattice and (c) hex pattern.

Figure 17. Free-form design covered by planar Kagome

lattice.

Mesnil et al. 11

our method. The framework was also extended to Kagome

and dual Kagome lattices. It is likely that other polyhedral

patterns can be treated with the marionette technique. The

generality of the method has also been demonstrated by

changing the projection direction, a method with large

potential if used on mesh with remarkable offset

properties.

Our discussion on Marionette Meshes with singularities

highlights the fact that the choice of mesh topology influ-

ences greatly the size and nature of the design space for

meshes with planar facets. Selecting a proper mesh topol-

ogy is thus crucial in practice. This article dealt with the

definition and implementation of the marionette method as

a parametric design space exploration. The choice of a rel-

evant mesh topology a priori from boundary conditions

should be addressed in future work.

Quadratic optimisation problems, like surface-fitting

problems, can be solved efficiently with the marionette

technique. A simple example where only the heights of the

guide curve are the only parameters was detailed, but con-

trolling the plane view with NURBS patches could allow

for a more general solution of such problems. The separa-

tion of variables in horizontal plane and vertical plane can

potentially give birth to efficient numerical methods for

geometrical optimisation.

Furthermore, we made a comment on the underlying

smooth problem solved by the method, which gives

indications on the smoothness of the shapes arising

from this framework. We have seen that this smoothness

depends on the smoothness of both the planar projection

and the guide curves, which can be generated with any

usual modelling tool based on NURBS, T-spline and

Bézier curves. Moreover, it was shown that Marionette

Meshes give an intuitive illustration on the principle of

subspace exploration, a powerful tool for constrained

optimisation of meshes. It was shown in section ‘Case

study: fabrication-aware structural optimisation’ that

the marionette method could be used as an alternative to

NURBS modelling for the parameterisation of struc-

tural optimisation problems for thin shells or gridshells.

The relative performances of the two modelling tech-

niques in the context of structural optimisation should

be assessed in future work.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with

respect to the research, authorship and/or publication of this

article.

Funding

This work was done during Mr Mesnil’s doctorate within

the framework of an industrial agreement for training through

research (CIFRE number 2013/1266) jointly financed by the

company Bouygues Construction SA and the National

Association for Research and Technology (ANRT) of France.

References

1. Glymph J, Shelden D, Ceccato C, et al. A parametric strat-

egy for free-form glass structures using quadrilateral planar

facets. Automat Constr 2004; 13(2): 187–202.

2. Pottmann H, Asperl A, Hofer M, et al. Architectural geom-

etry. Exton, PA: Bentley Institute Press, 2007.

3. Schober H. Geometrie-prinzipien für wirtschaftliche und

effiziente schalentragwerke (teil 1). Bautechnik 2002;

79(1): 16–24.

4. Bagneris M, Motro R, Maurin B, et al. Structural morphol-

ogy issues in conceptual design of double curved systems.

Int J Space Struct 2008; 23(2): 79–87.

5. Mesnil R, Douthe C, Baverel O, et al. Isogonal moulding

surfaces: a family of shapes for high node congruence in

free-form structures. Automat Constr 2015; 59: 38–47.

6. Bobenko A and Huhnen-Venedey E. Curvature line para-

metrized surfaces and orthogonal coordinate systems:

discretization with Dupin cyclides. Geometriae Dedicata

2012; 159(1): 207–237.

7. Mesnil R, Douthe C, Baverel O, et al. Generalised cyclidic

nets for shape modelling in architecture. Int J Architect

Comput 2017; 15(2): 148–168.

8. Klein F. Vergleichende Betrachtungen über neuere

geometrische Forschungen. Math Ann 1893; 43(1): 63–100.

9. Adriaenssens S, Ney L, Bodarwe E, et al. Finding the form

of an irregular meshed steel and glass shell based on con-

struction constraints. J Architect Eng 2012; 18(3): 206–213.

10. Liu Y, Wang W, Pottmann H, et al. Geometric modeling

with conical meshes and developable surfaces. ACM T

Graphic 2006; 25(3): 681–689.

11. Deng B, Bouaziz S, Deuss M, et al. Interactive design

exploration for constrained meshes. Comput Aided Design

2015; 61: 13–23.

12. Yang YL, Yang YJ, Pottmann H, et al. Shape space explora-

tion of constrained meshes. ACM T Graphic 2011; 30: 124.

13. Tang C, Sun X, Gomes A, et al. Form-finding with polyhe-

dral meshes made simple. ACM T Graphic 2014; 33(4): 70.

14. Jiang C, Tang C, Tomicic M, et al. Interactive modeling of

architectural freeform structures: combining geometry with

fabrication and statics. In: Block P, Wang W and Knippers J

(eds) Advances in architectural geometry. Cham: Springer,

2014, pp. 95–108.

15. Bouaziz S, Deuss M, Schwartzburg Y, et al. Shape-up:

shaping discrete geometry with projections. Comput Graph

Forum 2012; 31(5): 1657–1667.

16. Deng B, Bouaziz S, Deuss M, et al. Exploring local modi-

fications for constrained meshes. Comput Graph Forum

2013; 32(2 part 1): 11–20.

17. Monge G. Géométrie descriptive. Paris: Baudouin, 1798.

18. Javary A. Traité de géométrie descriptive. Delagrave, 1881,

http://gallica.bnf.fr/ark:/12148/bpt6k28506r

19. Leroy C. Traité de stéréotomie, comprenant les applications

de la géométrie descriptive à la théorie des ombres, la per-

spective linéaire la gnomonique, la coupe des pierres et la

charpente, avec un atlas composé de 74 planches in folio.

Paris: Mallet-Bachelier, 1857.

20. Block P and Ochsendorf J. Thrust network analysis: a new

methodology for three-dimensional equilibrium. J Int Ass

Shell Spat Struct 2007; 48: 167–173.

12 International Journal of Space Structures 00(0)

21. Rippmann M, Lachauer L and Block P. Interactive vault

design. Int J Space Struct 2012; 27(4): 219–230.

22. Miki M, Igarashi T and Block P. Parametric self-supporting

surfaces via direct computation of Airy stress functions.

ACM T Graphic 2015; 34(4): 89.

23. Bobenko A and Suris YB. Discrete differential geometry:

integrable structure. American Mathematical Society,

2008, http://bookstore.ams.org/gsm-98

24. Alinhac S. Hyperbolic partial differential equations. New

York: Springer Science+Business Media, 2009.

25. Tarini M, Puppo E, Panozzo D, et al. Simple quad domains

for field aligned mesh parametrization. ACM T Graphic

2011; 30(6): 142.

26. Takayama K, Panozzo D, Sorkine-Hornung A, et al. Sketch-

based generation and editing of quad meshes. ACM T

Graphic 2013; 32(4): 97.

27. Mesnil R, Douthe C, Baverel O, et al. Structural morphol-

ogy and performance of plated structures with planar quad-

rilateral facets. J Int Ass Shell Spat Struct 2017; 58(1): 7–22.

28. Robeller C, Stitic A, Mayencourt P, et al. Interlocking

folded plate: integrated mechanical attachment for structural

wood panels. In: Block P, Knippers J, Mitra N, et al. (eds)

Advances in architectural geometry 2014. Cham: Springer,

2015, pp. 281–294.

29. Lebée A. From folds to structures, a review. Int J Space

Struct 2015; 30(2): 55–74.

30. Nocedal J and Wright S. Numerical optimization. New

York: Springer Science+Business Media, 2006.

31. Takayama K, Panozzo D and Sorkine-Hornung O. Pattern-

based quadrangulation for N-sided patches. Comput Graph

Forum 2014; 33: 177–184.

32. Preisinger C and Heimrath M. Karamba – a toolkit for para-

metric structural design. Struct Eng Int 2014; 24(2): 217–

221.

33. Firl M and Bletzinger KU. Shape optimization of thin walled

structures governed by geometrically nonlinear mechanics.

Comput Method Appl M 2012; 237–240: 107–117.

Appendix 1

Some results on closed strips

The aim of this section is to discuss with more detail the

problem of closed strips. First, we write the propagation

problem on a strip. This step is purely computational, but

is necessary to introduce a quantity of interest. We inter-

pret then the geometrical meaning of the compatibility

condition with respect to the mathematical formalism

introduced. Then, we present some particular cases where

the closing of a strip is possible regardless of the choice of

the height on the outer curve.

Propagation equation. Consider the closed strip discussed

in section ‘Closed Marionette Meshes’. For each facet, we

can apply the planarity constraint of equation (2). Writing

zi the height of the ith point of the inner curve and zi

* the

height of the ith point of the outer curve, we can rewrite

this equation. For the sake of simplicity, we replace the

ratios of the two-dimensional (2D) determinants by scalars

ai and bi. We make following identifications

PA

PB

PC

PD

a

b

i

i

i

i

i

i

⇔

⇔

⇔

⇔

⇔

()

()

⇔

+

+

1

*

1

*

,

,

det

det

det

2D

2D

2D

AB AC

AB AD

ADDAC

AB AD

,

,

()

()

det2D

We get hence following equation

zabz az bz

iiii ii ii

++

=−−

()

++

1

**

1

1 (18)

We make the following hypothesis, which is easily veri-

fied by recurrence

zvzA

zi

i

k

i

kk i

**

,=+∀>

=

∑

0

00 (19)

In fact, we can be even more precise and compute the

value of Ai. We make the hypothesis that

Aai

i

k

i

k

=∀

>

=

−

∏

0

1

0, (20)

Proof. This is true for i

=

1 due to equation (18). Then, we

proceed by recurrence. Assume that equation (20) is true

for i, then we show that this is true for i

+

1. We plug

equation (19) into equation (18) and get

zabz avzA

zb

z

iiii i

k

i

kk

ii

i+

=

+

=−−

()

++

+

∑

1

0

01

1

**

There is only one term in z0

*, and it verifies equation (20).

Geometrical interpretation. The ratios ai can be interpreted

with elementary plane geometry. Consider Figure 18, the

ratio

a

is defined with 2D determinants and can be

expressed with the vectors norms and angles. We have

a=

AB AC

AB AD

sin

sin

β

α

(21)

We recognise the areas of the triangles ABC and ABD,

so that

a

can be rewritten as

Mesnil et al. 13

aABC

ABD

=

(22)

The two triangles used for the computation of ai are

shown in Figure 19.

General solutions for a closed strip. Recall that we are inter-

ested in finding the solutions, so that

zz

N

**

=0, which also

writes

zz vz

az

N

k

N

kk

k

N

k

** *

−= +−

=

==

−

∑∏

0

00

1

0

10

(23)

We also want the space of solutions to be as large as

possible, and therefore, we do not want it to depend on the

choice of the height on the outer curve z0

*. This implies a

new equation

k

N

k

a

=

−

∏=

0

1

1 (24)

In the following, we discuss the invariance of this equa-

tion under some transformations and show some particular

cases where it is satisfied.

Invariance. To have a complete overview on the problem

of closed strips, we provide transformations that map

compatible strips to other compatible strips. The study of

group of transformations that preserve a given quantity is

at the core of modern geometry, and for the sake of com-

pleteness, we show this point of view dating back from

Klein.8

Linear maps. The transformations we are interested

in preserve equation (24). The most straightforward way

to do this is to preserve each ai. It is clear that all linear

transformations in the plane (translation, scaling, shearing)

preserve each individual ratio. Consider indeed transfor-

mations defined by

fxymm

mm

x

y

X

Y

,

()

=

⋅

+

11 12

21 22

0

0

(25)

Consider two points

()

xy

00

, and

()

xy

11

,. We write

their image by f, respectively,

()

′′

xy

00

, and

()

′′

xy

11

,. We

call M the matrix written in equation (25), then we have

detdet det

′′

′′

=

()

xx

yy

xx

yy

01

01

2

01

01

M (26)

It is clear that the linear map preserves the ratio of 2D

determinant, since the factor detM depends only on the

parameters of the transformations. Linear maps preserve

thus the geometrical compatibility of closed strips. This is

not a surprise, since linear maps preserve PQ-meshes.2

Combescure maps. We give now another set of trans-

formations that preserve the geometrical compatibility. We

rewrite now equation (21) using the properties of area of

triangles

ai=

BC

AD

sin

sin

γ

α

(27)

Computing the product of all these values, we notice

that the lengths cancel out (each length is exactly one

time at the numerator and one time at the denominator),

so that

i

N

i

i

N

i

i

a

=0

1

=0

1−−

∏∏

=sin

sin

γ

α (28)

Therefore, a transformation that preserves discrete

angles preserves also the geometrical compatibility. Such

transformations are known as Combescure transforma-

tion. The image of a mesh by a Combescure transforma-

tion has its edges parallel to the initial mesh, but it does not

necessary preserve lengths. Examples of such transforma-

tions are given in Mesnil et al.5

Figure 18. Planar projection of a quadrilateral and angle

notations.

Figure 19. The two triangles used to compute ai.

14 International Journal of Space Structures 00(0)

Particular cases. We give three simple examples where

equation (24) is verified.

Example 1: parallel edges. Equation (24) is verified when

all the ak are equal to one. This condition translates into

AC

AD =sin

sin

β

α

(29)

We write the equation of AB

CD

,,, in the Cartesian

plane where eX is parallel to

AB

, and we have

AB

AB

AC AC

AD AD

=0

0

=

0

=

cos

sin

cos

si

β

β

α

nn

α

0

(30)

We plug then equation (29) into equation (30) and com-

pute the vector CD . We get following result

CD AD=

−

sin

sincoscos

α

β

βα

0

0

(31)

Remarkably, we notice that the vectors CD and

AB

are parallel. Reciprocally, if these two vectors are parallel,

then equation (29) is satisfied. Therefore, a closed strip

where all the projected quads are trapezoids satisfies equa-

tion (24). Such planar projections provide thus a large

design space and the maximal design flexibility for closed

strips.

Example 2: symmetry. Consider the case where the pla-

nar projection of the strip has an axis of symmetry. Con-

sider Figure 19, each ai is defined as the ratio of the area

of the blue to orange triangles. When the curve has a sym-

metry, like the one depicted in Figure 20, the role of orange

and blue triangle is inverted by the symmetry. Two faces

related by a symmetry have, therefore, inverse values of

ai. Their product is naturally equal to

1

, which proves

that strips with an axis of symmetry satisfy equation (24).

Example 3: orthogonal fields. The first two examples

are based on equation (24), where the ai are expressed

as ratios of areas. The propagation rule is applied to each

quadrilateral, but in the case of closed curves, it is also

interesting to look at each vertex.

Consider Figure 21, equation (28) is verified if

γα

01

=,

γα

12

= and so forth. In other terms, if the transverse edge

is the bisecting line of the inner curve, then we have a solu-

tion to the problem of closed strips.

This condition is a discrete counterpart of orthogonality

of vector fields. Examples of such meshes are obtained by

moulding surfaces or Monge’s surfaces.5 Discretisation of

orthogonal parameterisation of the plane will, therefore,

yield strips that are very close to be geometrically compat-

ible. The smooth counterpart of this problem would be

looked at carefully in further work.

Figure 20. A curve with an axis of symmetry and the

inversion of the blue and orange triangles.

Figure 21. A closed curve and the angles used in equation

(28).

Mesnil et al. 15

Appendix 2

Size of the design space for periodic patterns

We propose here to count the number of DOFs for the

meshes drawn in Figure 16. We use equation (16) to esti-

mate the available DOFs and compare this number with

the number of prescribed points drawn in Figure 16. Each

time, it is easy to propagate the heights in the manner of

what has been done with quadrilateral meshes. We illus-

trate here the fact that equation (16) is exact for meshes

with no closed curves.

Kagome pattern. The Kagome pattern shown in Figure

16(a) features 191 vertices, 112 triangles, 48 hexagons

and 8 pentagons. The estimated number of DOFs given by

the marionette method follows

N

N

N

=− ⋅−

()

+⋅−

()

+⋅ −

()

()

=−++

()

=

191 112 33 48 63 853

191 0 144 16

31

This is exactly the number of prescribed nodes in Figure

16(a).

Dual Kagome pattern. The dual Kagome pattern shown in

Figure 16(b) has 185 vertices and 156 quadrilateral facets

N

N

=−⋅−

()

185 156

43

=29

This is the number of vertices with prescribed heights

in Figure 16(b).

Hexagonal pattern. The pattern derived from an hexagonal

mesh has 183 vertices and 162 quadrilateral facets. The

number of DOFs is thus

N

=−=

183 162 21 (32)

Remark. The calculation provided here shows hints for the

number of guide curves to use to lift the mesh. Notice,

however, that the DOF must be uncoupled: for example, it

is not possible to prescribe independently the four heights

of the vertices of a quadrilateral face. Our choice of guide

points does not violate this constraint.

For both the dual Kagome and the hexagonal pattern, the

planarity constraint cannot be propagated throughout the whole

mesh, and additional isolated heights have to be prescribed.