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62

Marionette Mesh

From Descriptive Geometry to Fabrication-Aware Design

Romain Mesnil, Cyril Douthe, Olivier Baverel, and Bruno Léger

R. Mesnil, C. Deute, O. Baverel

Laboratoire Navier, Champs-sur-Marne, France

romain.mesnil@enpc.fr

B. Léger

Bouygues Construction, France

S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016

© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_7, ISBN 978-3-7281-3778-4

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63

Abstract

This paper introduces an intuitive method for the modelling of free-form architec-

ture with planar facets. The method, called Marionette by the authors, takes its

inspiration from descriptive geometry and allows one to design complex shapes

with one projection and the control of elevation curves. The proposed framework

only deals with linear equations and therefore achieves exact planarity, for quad-

rilateral, Kagome, and dual Kagome meshes in real-time. Remarks on how this

framework relates to continuous shape parameterisation and on possible appli-

cations to engineering problems are made.

Keywords:

structural morphology, descriptive geometry, fabrication-aware design

S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016

© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_7, ISBN 978-3-7281-3778-4

http://vdf.ch/advances-in-architectural-geometry-2016.html

64

with a plane view, displayed with some elevations. The curve network corresponds to the

horizontal projection of lines of curvature (Leroy,1857).

Figure 1: Lines of curvatures of an ellipsoid with descriptive geometry (Leroy,1857).

Because architectural objects have to deal mainly with gravity and vertical forces, it

makes naturally sense to separate projections in vertical and horizontal planes. The idea to

use these projections to guide structural design was used recently in the framework of the

Thrust Network Analysis where compression-only structures are found from a planar network

at equilibrium (Rippmann et al.,2012;Miki et al.,2015). The objective of this paper 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 2, where the relation between smooth and discrete geometry for PQ-meshes. Section

3explores then some applications in architecture. Section 4shows ﬁnally the generality of

the proposed method, which can be extended to meshes other than the regular quadrilateral

meshes and therefore constitute a promising versatile tool to integrate intuitively fabrication

constraints into architectural design.

3

Figure 1. Lines of curvatures of an ellipsoid with descriptive geometry (Leroy 1857).

S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016

© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_7, ISBN 978-3-7281-3778-4

http://vdf.ch/advances-in-architectural-geometry-2016.html

65

1. Introduction

The design of complex architectural shapes has beneﬁted from great advances

within the computer graphics community in the last decade. For instance, sig-

niﬁcant 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 architects and engineers, making them hard use for non-specialists

to use. The technique proposed in the present article aims thus at bridging this

gap with a method that takes inspiration from descriptive geometry, a tool used

by architects for centuries, and turns it into a real-time design tool for PQ-meshes.

1.1 Prior Works

Geometrically-Constrained Approach

Planar quadrilaterals have been identiﬁed by practicians as an important optimi-

sation target for the construction of double-curved surfaces, as they avoid using

curved panels (Glymph et al. 2004). Previous research identiﬁed the need for integra-

tion of geometrical constraints within the design tools themselves and proposed

methods for shape generation of PQ-meshes (Schmiedhofer 2010). Several techniques

for generating exact planar quadrilateral meshes were proposed, mostly rely-

ing on afﬁne transformations, which preserve planarity, a notion illustrated in

Pottmann et al. (2007). For example, scale-trans surfaces, introduced in Glymph et

al. (2004) use composition of two afﬁne transformations: translation and homo-

thetic transformations. The designer control the shape with two curves, making

the process highly intuitive. Despite formal limitations, these shapes have been

used in many projects.

Constrained geometric approaches use shapes that are well known and

can be rationalised efﬁciently, for example, towards a high repetition of nodes

or panels (Mesnil et al. 2015). They suffer however from a lack of ﬂexibility and form a

restricted design space. This led to the introduction of post-rationalisation strat-

egies in order to cover arbitrary shapes with planar quadrilaterals (Liu et al. 2006).

Optimisation-Based Shape Exploration

Most recent methods propose hence to explore design space of feasible solu-

tions for a given mesh topology with the help of optimisation techniques (Deng et

al. 2015; Yang et al. 2011). The mesh is interactively deformed by the user with the help

of control handles. The overall smoothness is checked with discrete functions

of the vertices. To go further, an efﬁcient solver handling quadratic constraints

was presented in Tang et al. (2014) and used in Jiang et al. (2014). Projections and

subspace exploration are efﬁciently used for constrained-based optimisation in

Bouaziz et al. (2012), Deng et al. (2013, 2015). These methods provide great design

freedom, but illustrations shown in the cited references are local deformations

© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_7, ISBN 978-3-7281-3778-4

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66

of meshes. Design space exploration with exact PQ-meshes was also proposed

by composition of compatible afﬁne maps assigned to each mesh face and al-

lowed for handle-driven shape exploration (Vaxman 2012). This strategy was extend-

ed to other maps that preserve facet planarity by construction in Vaxman (2014).

The idea of this paper is to use the notion of projection, which is commonly

used in architecture, especially with plane view and elevations, and to link sub-

space exploration techniques with representation techniques based on projec-

tions in architecture.

Descriptive Geometry

Descriptive geometry is a technique of shape representation invented by the

French mathematician Gaspard Monge (Monge 1798; Javary, 1881). It is based on planar

orthogonal projections 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 de-

scriptive geometry: It describes an ellipsoid with a plane view, displayed with

some elevations. The curve network corresponds to the horizontal projection of

lines of curvature (Leroy 1857).

Because architectural objects have to deal mainly with gravity and vertical

forces, it makes naturally sense to separate projections in vertical and horizontal

planes. The idea to use these projections to guide structural design was used re-

cently in the framework of the thrust network analysis, where compression-only

structures are found from a planar network at equilibrium (Rippmann et al. 2012; Miki et

al. 2015). The objective of this paper is to show that descriptive geometry can be

turned into a general tool for the design of PQ meshes and their structural opti-

misation. The method, called the Marionette method, is presented in Section 2,

where the relationship between smooth and discrete geometry for PQ-meshes

is explained. Section 3 explores then some applications in architecture. Section4

shows ﬁnally the generality of the proposed method, which can be extended to

meshes other than the regular quadrilateral meshes and therefore constitute a

promising versatile tool to integrate intuitively fabrication constraints into archi-

tectural design.

© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_7, ISBN 978-3-7281-3778-4

http://vdf.ch/advances-in-architectural-geometry-2016.html

67

2. Marionette Meshes

2.1 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 (X Y ) plane in this section; the generalisation to

other projections is illustrated in Section 4.

Consider ﬁrst Figure 2: four points have a prescribed plane view A B C D in the

horizontal plane (P1). Three points A', B', and D' have prescribed altitudes zA , zB

,

and zD . In general, there is only one point C' with the imposed projection C so

that A', B', C', D' is planar.

The planarity constraint reads:

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 4.

Consider ﬁrst Figure 2: four points have a prescribed plane view ABCD in the horizontal

plane ( ). Three points A,Band Dhave prescribed altitudes zA,zBand zD. In general,

there is only one point Cwith the imposed projection Cso that ABCDis planar.

A

B

D

C

AzD

Aʹ

Bʹ

Cʹ

Dʹ

P1

P2

P3

P1

A

B

D

C

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

The planarity constraint reads:

det (

AB,AC,AD

) = 0 (1)

Expressing coordinates in a cartesian frame of (P1), and writing dBC =det (AB,AC),

) and dDC =det (AD,AC), if the points A,Band Dare not aligned,

then, one gets:

(−zA)=dBC

dBD ·(zD−zA)+dDC

dBD ·(zB−zA) (2)

Figure shows vertical lines used for construction, recalling the strings of a marionette,

which gives the name marionette quad. Note that the system is under-constrained if the

points are aligned, which 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 by using other projections, (see Section 4).

2.2 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 the altitude of two intersecting curves is prescribed.

4

(1)

Expressing coordinates in a cartesian frame of (

P

1

), and writing

dBC

=

det

(AB,

AC), dBD = det (AB, AD) and dDC = det (AD, AC), if the points A, B, and D are not

aligned, then, one gets:

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 4.

Consider ﬁrst Figure 2: four points have a prescribed plane view ABCD in the horizontal

plane ( ). Three points A,Band Dhave prescribed altitudes zA,zBand zD. In general,

there is only one point Cwith the imposed projection Cso that ABCDis planar.

A

B

D

C

zA

zAzD

zBAʹ

Bʹ

Cʹ

Dʹ

P1

P2

P3

P1

A

B

D

C

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

The planarity constraint reads:

det (AB,AC,AD) = 0 (1)

Expressing coordinates in a cartesian frame of (P1), and writing dBC =det (AB,AC),

det (AB,AD) and dDC =det (AD,AC), if the points A,Band Dare not aligned,

then, one gets:

(

zC−zA)=dBC

dBD ·(zD−zA)+dDC

dBD ·(zB−zA

) (2)

Figure 2shows vertical lines used for construction, recalling the strings of a marionette,

which gives the name marionette quad. Note that the system is under-constrained if the

points Band Dare aligned, which 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 by using other projections, (see Section 4).

2.2 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 the altitude of two intersecting curves is prescribed.

4

(2)

Figure 2 shows vertical lines used for construction, recalling the strings of a mar-

ionette, which gives the name marionette quad. Note that the system is under-

constrained if the points A, B, and D are aligned, which 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 by using other projections (see Section 4).

2.2 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 the altitude of two intersecting

curves is prescribed. Then, provided that the planar view admits no ’ﬂat’ quad

(i.e. quad where three points are aligned), equation(2) can be propagated through

a strip, and by there through the whole mesh. Indeed, on the highlighted strip

© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_7, ISBN 978-3-7281-3778-4

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68

A

B

D

C

zA

zAzD

zBAʹ

Bʹ

Cʹ

Dʹ

P1

P2

P3

P1

A

B

D

C

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

P1

P3

P2

Figure 3. Two elevations and a planar view deﬁne a unique Marionette Mesh.

© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_7, ISBN 978-3-7281-3778-4

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69

of Figure 3, the ﬁrst quad (top left) has three prescribed altitudes, and equation(2)

can be used. The same applies for all the quads of the strip.

For a N × M mesh, the propagation requires N M applications of equation(2),

the memory is 3N M. 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 3.1.

2.3 Link with Smooth Geometry

The proposed method has some interesting relations with smooth geometry.

The problem of covering curved shapes with planar panels is linked with the in-

tegration of conjugate curves networks (Liu et al. 2006; Bobenko & Suris 2008). Conjugate net-

works correspond to parameterisations (u, v) satisfying the following equation

(Bobenko & Suris 2008):

P1

P2

Figure 3: Two elevations and a planar view deﬁne a unique Marionette Mesh.

For a mesh, the propagation requires NM applications of equation (2), the memory

is 3 . The marionette technique guarantees hence that the number 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 3.1.

2.3 Link with smooth geometry

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 (Liu et al.,2006;Bobenko and Suris,2008). Conjugate networks correspond to

parameterisations ( ) satisfying the following equation (Bobenko and Suris,2008):

det

∂uf,∂

vf,∂2

uvf

= 0 (3)

Consider now that the components in xand yare ﬁxed, like in the problem solved by the

Marionette technique. We are looking for the height functions fzsatisfying equation 3.

Adopting the notation futo denote diﬀerentiation of fwith respect to u, equation (3) is

reformulated into:

det

fx

ufx

vfx

uv

fy

ufy

vfy

uv

fz

ufz

vfz

uv

= 0 (4)

Equation ( ) is deﬁned if the parameterisation in the plane (XY ) is regular, which means

if the study is restricted to height ﬁelds. An expansion of the determinant shows that the

5

(3)

Consider now that the components in x and y are ﬁxed, as in the problem solved

by the Marionette technique. We are looking for the height functions f z satisfying

equation 3. Adopting the notation fu to denote differentiation of f with respect to

u, equation(3) is reformulated into:

P1

P2

Figure 3: Two elevations and a planar view deﬁne a unique Marionette Mesh.

For a mesh, the propagation requires NM applications of equation (2), the memory

is 3 . The marionette technique guarantees hence that the number 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 3.1.

2.3 Link with smooth geometry

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 (Liu et al.,2006;Bobenko and Suris,2008). Conjugate networks correspond to

parameterisations ( ) satisfying the following equation (Bobenko and Suris,2008):

det ∂uf,∂

vf,∂2

uvf= 0 (3)

Consider now that the components in xand yare ﬁxed, like in the problem solved by the

Marionette technique. We are looking for the height functions fzsatisfying equation 3.

Adopting the notation futo denote diﬀerentiation of fwith respect to u, equation (3) is

reformulated into:

det

fx

ufx

vfx

uv

fy

ufy

vfy

uv

fz

ufz

vfz

uv

= 0 (4)

Equation ( ) is deﬁned if the parameterisation in the plane (XY ) is regular, which means

if the study is restricted to height ﬁelds. An expansion of the determinant shows that the

5

(4)

Equation(4) is deﬁned if the parameterisation in the plane (X Y ) is regular, which

means if the study is restricted to height ﬁelds. An expansion of the determinant

shows that the equation is a second-order linear equation in f

z

(u, v). The only

term of second order is f z

u v : 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 mari-

onette method corresponds to a propagation algorithm. Loosely speaking, it

can be shown that solutions of hyperbolic equations 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).

© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_7, ISBN 978-3-7281-3778-4

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70

has

one singularity: the central 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 4a, 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.

P

1

P

2

(a) Decomposition of a complex mesh into

simple patches.

(b) The corresponding lifted mesh

Figure 4: A Marionette Mesh with a singularity.

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 simple quad

domains without any singularity, for example by using the methods described in Tarini et al.

(2011) or Takayama et al. (2013). For example, Figure 4a has six domains, the mesh in Figure

5a has nine domains. These domains are four sided, and it is possible to extract independent

families of strip-domains, like displayed in Figure 5. Depending on the n-colorability of the

mesh, the number of families varies. The example showed is two-colorable. As a result, two

6

Figure 4. A Marionette Mesh with a singularity.

families of strips can be found and are shown in Figure 5b and 5c. 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.

(a) Initial mesh (b) Family of four strip-domains (c) Family of ﬁve strip-domains

Figure 5: Decomposition of a mesh into 2 families of strip-domains. Marionette Meshes can

be generated by choosing one guide curve across each strip-domain.

2.5 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 over-constrained.

Indeed, consider Figure 6: the plane view of a closed strip and the altitude of the points (Pi)

of one polyline are prescribed. If the altitude of the ﬁrst point used for the propagation P∗

0is

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 has no control on them. The last

point P∗

Nis therefore generally diﬀerent from the initial point P∗

0, leading to a geometrical

incompatibility of PQ-meshes.

P0

*

PN

*

P0=PN

Figure 5. Decomposition of a mesh into 2 families of strip-domains. Marionette Meshes can be generated by choosing

one guide curve across each strip-domain.

P0

*

PN

*

P0=PN

Figure 6. Closed Marionette Strip with incompatible closing condition induced by the prescription of the plane view of

the whole strip (orange) and the altitudes of the inner curve (blue).

© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_7, ISBN 978-3-7281-3778-4

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71

2.4 Marionette Meshes with Singularities

The modelling of complex shapes requires the introduction of vertices with a dif-

ferent valence, called singularities in the following. For example, the mesh dis-

played in Figure 4a has one singularity: the central 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 4a

, 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 de-

composed into simple quad domains without any singularity, for example, by

using the methods described in Tarini et al. (2011) or Takayama et al. (2013). For ex-

ample, Figure 4a has six domains and the mesh in Figure 5a has nine domains. These

domains are four sided, and it is possible to extract independent families 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 re-

sult, two families of strips can be found and are shown in Figure 5b and 5c. Exactly

one curve can be chosen across each strip-domain. Since strips are indepen-

dent, the height of these nine curves can be chosen independently and will not

over-constrain the problem.

2.5 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 be-

comes over-constrained. Indeed, consider Figure 6: The plane view of a closed strip

and the altitude of the points (Pi ) of one polyline are prescribed. If the altitude

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 im-

posed by the method, and the designer has no control on them. The last point

P

∗

0 is therefore generally different from the initial point P

∗

0, leading to a geomet-

rical incompatibility of PQ-meshes.

In the following, we develop a strategy to deal with the geometrical com-

patibility 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 propa-

gation of equation(2), we easily see that the altitude of the last point z*

N depends

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72

Figure 7. Architectural design with a closed Marionette Mesh, the altitude of the inner curve is prescribed, the designer

does not have control on the outer curve.

Figure 8. Some shapes with planar faces and a closed mesh generated with the method proposed in this paper.

© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_7, ISBN 978-3-7281-3778-4

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73

linearly on the altitude of the ﬁrst point z

∗

0 and on the altitudes 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:

In the following, we develop a strategy 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

altitude of the last point ∗

Ndepends linearly on the altitude of the ﬁrst point z∗

0and on the

altitudes of the points on the inner curve Z. It also depends on the in-plane projection of

the strip. Formally, there exists a vector Vand a scalar a, both functions of the plane view

so that:

V·

Z+a

·

z∗

0=z∗

N

(5)

We are interested in the case where z∗

0=z∗

N. There are two possibilities:

1. = 1, in this case, the condition restricts to V·Z= 0 and does not depend on z∗

0.

The vector is in the hyperplane of V, which leaves N−1 degrees of freedom.

2. = 1: there is only one solution for z∗

0. This is the most constrained case: the designer

can only control the inner curve of the strip.

The meshes with one solution are less ﬂexible, but they can still generate interesting shapes,

like the one displayed on Figure 7, which recalls the examples of Figure 6. The designer has

a total control on the altitude of the inner curve and the plane view, but cannot manipulate

freely the outer curve. Note that the strings of the marionette are here materialised as

columns in the rendering, illustrating the geometrical interpretation of the method.

Figure 7: Architectural design with a closed Marionette Mesh, the altitude of the inner curve

is prescribed, the designer does not have control on the outer curve.

The most interesting case occurs when the designer has potentially the control of two

curves. It relies on a condition on the planar view explained above. A simple case where this

8

(5)

We are interested in the case where z

∗

0 = z*

N . There are two possibilities:

1. a = 1, in this case, the condition restricts to V · Z = 0 and does not depend

on z

∗

0. The vector z is in the hyperplane of V, which leaves N − 1 degrees of

freedom.

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

∗

0. This is the most constrained case: the

designer can only control the inner curve of the strip.

Closed Meshes

The meshes with one solution are less ﬂexible, but they can still generate inter-

esting shapes, like the one displayed on Figure 7, which recalls the examples of

Figure 6. The designer has a total control on the altitude of the inner curve and the

plane view, but cannot manipulate freely the outer curve. Note that the strings

of the marionette are here materialised as columns in the rendering, illustrating

the geometrical interpretation of the method.

The most interesting case occurs when the designer has potentially the con-

trol of two curves. This relies on a condition on the planar view explained above.

A simple case where this condition is fulﬁlled is when it has a symmetry. In this

case, there is a N −1 parameters family of solutions for the altitude 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 notations of equation(5). This

operation is straightforward and allows one to control the elevation of a second

curve, like for open meshes. An example of this strategy is displayed in Figure8,

where all the meshes have the same planar view.

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

Figure 10. A plane view (thin lines) with a prescribed boundary (thick blue lines).

Figure 11. A result of an optimisation procedure: the shell structure is a Marionette Mesh (top view and prescribed

curves on the middle) minimising total elastic energy. On the right: red areas indicate compression.

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3. Architectural Design with

Marionette Meshes

3.1 Computational Set-up

The algorithms described in this paper have been implemented in the visual-scripting

plug-in GrasshopperTM for the modelling software RhinoTM. This allows inter-

action with other numerical tools necessary for architectural design, like ﬁnite-

element analysis software. An example of interaction between fabrication-aware

shape generation and structural analysis is shown in Section 3.3.

Marionette Meshes only require the solution of a linear system. The com-

putation 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.

In our framework, the planar views are generated with NURBS patches, and

the elevation curves are drawn as Bézier curves. The smoothness of the ﬁnal

mesh depends thus on the smoothness of the in-plane parameterisation. A C 0

projection yields a C 0 solution to the hyperbolic equation(4), so that shape func-

tions with creases can easily be propagated through the mesh. Figure 9 shows a

corrugated shape generated from a

C

0

planar view 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. (2015).

3.2 Shape Exploration with Marionette Meshes

The framework introduced here intrinsically accounts for the planarity of panels.

Its mathematical formulation is, however, suited for many architectural constraints.

Hard constraints must be fulﬁlled exactly, whereas soft constraints are included

into the function to minimize (Nocedal & Wright, 2006). 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.

Hard constraints deﬁned by linear equations are treated effectively within the

proposed framework. Examples of linear constraints are prescribed volume or a

maximal allowable altitude. The marionette method imposes N M − (N + M − 1)

out of N M parameters, this means that another N + M − 1 linear constraints can

be applied without over-constraining the optimisation problem.

Perhaps the most common application of hard constraint in architecture is the

prescription of a boundary, as depicted in Figure 10. In this ﬁgure, the planar view is

imposed and the user prescribes the altitude of some points of the mesh along a

curve (white circles). In this case, the number of prescribed points is superior to

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76

the number of degrees of freedom, and the problem might be over constrained.

It might hence be preferable to turn this problem into a soft constrained prob-

lem with a quadratic function to minimize. In the same way, for really complex

shapes with many singularities or highly constrained boundary, other methods

will probably be more efﬁcient, more relevant, and maybe more intuitive, like for

example Jiang et al (2014).

3.3 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 parameterised

design space and the coupling with advanced analysis software seems particu-

larly promising (Preisinger & Heimrath, 2014). Indeed, non-linear criteria, like the buckling

capacity, are of high importance for practical design of thin shell or grid shells

(Firl & Bletzinger, 2012).

An illustration of the potential of Marionette Meshes for a structurally in-

formed architectural design is proposed in Figure 11: The shell is a Marionette mesh

spanning over an ellipse. The plane view is inspired by Figure 1. The mesh is consti-

tuted of six NURBS patches and has two singularities (white dots in the image);

guide curves are found with the method proposed in this paper. The boundary

curve is constrained in the horizontal plane (blue curve on Figure 11). One curve

in the other direction (orange curve in

Figure 11

) deﬁnes the whole elevation of

the dome. The shell is submitted to gravity load. All 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 gen-

eration of a 1000 faces mesh requires less than 1 ms with the Marionette tech-

nique, far less than the assembly and computation of a shell model with FEM.

The structure is optimised towards a minimum of the total elastic energy

by means of genetic algorithms. The design parameters are the four altitudes

of the control points controlling the shape of the guide curve. It is noticed that

tension areas, depicted in blue in Figure 11, are almost non-existent on the inner

and upper face of the shell. Hence, if deﬁned properly with an accurate number

of singularities, the design space offered by Marionette Meshes is wide enough

to ﬁnd compression-dominant shapes by the means of structural optimisation.

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4. Generalisation of the Method

4.1 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 pro-

jections, like in Figure 12.

Some projections are of practical interest for archetypal projects. Towers and

facades can be modelled with cylindrical projections. Stadia can be designed

using projections on torus or on moulding surfaces, the offset directions cor-

responding to the normals of the smooth surface. Moulding surfaces ﬁt natu-

rally the geometry of stadia (see Figure 13a) and have some interesting features,

discussed in Mesnil et al. (2015) :

• Their natural mesh contains planar curves, which are geodesics of the

surface: The planarity is preserved by the marionette transformation.

• They are naturally meshed by their lines of curvatures, which gives a

torsion-free beam layout on the initial surface, and on the final shape.

4.2 Extension to Other Patterns

The method proposed in this paper can be extended to other polyhedral patterns.

As noticed by Deng et al. (2013), tri-hex meshes (also known as Kagom lattices)

have the same number of degrees of freedom as quad meshes. There is there-

fore a straight forward way to lift Kagome lattices with the marionette technique.

Figure 14a shows the guide curves for the Kagome pattern. Other isolated points

are required to lift the mesh. The altitude of these points can be chosen in or-

der to minimise the fairness energy introduced in Jiang et al. (2014), which is not

difﬁcult under linear constraints. Figure 14c shows a pattern introduced in Jiang et

Aʹ

Bʹ

Cʹ

Dʹ

Figure 12. A Marionette quad with non-parallel guide lines.

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78

(a) Reference moulding surfaces (b) Non-symmetrical design (c) Symmetrical design

Figure 13: Design of stadia obtained from a projection on a moulding surface: the prescribed

curves are the inner ring and a section curve.

4.2 Extension to other patterns

The method proposed in this paper can be extended to other polyhedral patterns. As noticed

by (Deng et al.,2013), tri-hex meshes (also known as Kagom lattices) have the same number

of degrees of freedom as quad meshes. There is therefore a straight forward way to lift

Kagome lattices with the marionette technique. Figure 14a shows the guide curves for the

Kagome pattern. Other isolated points are required to lift the mesh. The altitude of these

points can be chosen in order to minimise the fairness energy introduced in (Jiang et al.,

2014), which is not diﬃcult under linear constraints.. Figure 14c shows a pattern introduced

in (Jiang et al.,2014): the mesh is derived from an hexagonal pattern and three guide curves

can be used to lift the mesh.

(a) Kagome lattice (b) Dual Kagome lattice (c) Hex pattern

Figure 14: Marionette method applied to several patterns, white dots correspond to pre-

scribed altitudes.

For example, Figure 15 shows a Kagome lattice covered with planar facets generated with

the marionette method. The design started from a planar view generated with a NURBS

13

Figure 13. 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 surfaces (b) Non-symmetrical design (c) Symmetrical design

Figure 13: Design of stadia obtained from a projection on a moulding surface: the prescribed

curves are the inner ring and a section curve.

4.2 Extension to other patterns

The method proposed in this paper can be extended to other polyhedral patterns. As noticed

by (Deng et al.,2013), tri-hex meshes (also known as Kagom lattices) have the same number

of degrees of freedom as quad meshes. There is therefore a straight forward way to lift

Kagome lattices with the marionette technique. Figure 14a shows the guide curves for the

Kagome pattern. Other isolated points are required to lift the mesh. The altitude of these

points can be chosen in order to minimise the fairness energy introduced in (Jiang et al.,

2014), which is not diﬃcult under linear constraints.. Figure 14c shows a pattern introduced

in (Jiang et al.,2014): the mesh is derived from an hexagonal pattern and three guide curves

can be used to lift the mesh.

(a) Kagome lattice (b) Dual Kagome lattice (c) Hex pattern

Figure 14: Marionette method applied to several patterns, white dots correspond to pre-

scribed altitudes.

For example, Figure 15 shows a Kagome lattice covered with planar facets generated with

the marionette method. The design started from a planar view generated with a NURBS

13

Figure 14. Marionette method applied to several patterns, white dots correspond to prescribed altitudes.

Figure 15. Free-form design covered by planar Kagome lattice.

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79

al. (2014): The mesh is derived from an hexagonal pattern and three guide curves

can be used to lift the mesh.

For example, Figure 15 shows a Kagome lattice covered with planar facets

generated with the marionette method. The design started from a planar view

generated with a NURBS patch, a Kagome was then generated following the

isoparametric lines and lifted with the marionette technique. One of the guide

curve is the parabolic arch of the entrance, the other is an undulating curve fol-

lowing the tunnel. Like for PQ-meshes, the computation is done in real time.

5. Conclusion

We have introduced an intuitive technique for interactive shape modelling with

planar facets. It is based on descriptive geometry, which is used by architects and

engineers. The concept has many applications, in particular the modelling of PQ

meshes with or without singularity. Some examples show the formal potential

of 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 Mario-

nette 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. The choice of appropriate projections, while

obvious for many shapes of relatively low complexity, is a limitation to the gen-

erality of the method compared to previous methods developed in the ﬁeld of

computer graphics. The Marionette technique should be seen as an intuitive way

to model shapes, and is complementary with other less-intuitive methods that

perform well on surface-ﬁtting or local exploration problems.

We made a comment on the 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.

The underlying smooth parameterisation of marionette meshes could hence open

new possibilities for efﬁcient parameterisation of fabrication-aware design space

in structural optimisation problems.

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80

Acknowledgements

This work was done during Mr. Mesnil doctorate within the framework of an industrial agreement for training through re-

search (CIFRE number 2013/1266) jointly ﬁnanced by the company Bouygues Construction SA and the National Association

for Research and Technology (ANRT) of France.

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