__________________________________________________________________________________

DIPLOMARBEIT

Freeform Geometries in Wood Construction

ausgeführt zum Zwecke der Erlangung des akademischen Grades eines Diplom-Ingenieurs

unter der Leitung von

o.Univ.Prof. Dr. Helmut Pottmann

E104Institute of Discrete Mathematics and Geometry

Geometric Modeling and Industrial Geometry Research Unit

eingereicht an der Technischen Universität Wien

Fakultät für Architektur und Raumplanung

von

1029061

Wien, am 01.10.2013eigenhändige Unterschrift

For my parents.

Abstract

Despite being one of the oldest construction materials on earth and

having numerous advantages over modern high tech construction ma-

terials, wood has been greatly marginalized in the construction of to-

day’s increasingly popular freeform architectural shapes. Architectural

freeform shapes must be rationalized prior to their building due to their

large scale. This work explores planar quadrilateral mesh panelizations,

rationalizations with developable mesh strips and rationalizations with

geodesic patterns in combination with new computational tools and

scripting as ways to use wood for the construction of voluptuous free-

form architectural structures. For each of the above rationalization tech-

niques a project is designed, rationalized for and detailed for the manu-

facturing in wood.

Kurzfassung

Obwohl Holz einer der ältesten Baustoffe der Welt ist und gegenüber

anderen modernen High-Tech Baustoffen über viele Vorteile verfügt,

wurde dieses im Bau der heute immer populäreren Freiformstrukturen

stark marginalisiert. Architektonische Freiformflächen müssen auf-

grund ihrer Größe vor dem Bauen rationalisiert werden. Diese Arbeit

untersucht Nutzungsmöglichkeiten des Baustoffs Holz beim Bau von

Freiformstrukturen. Dies erfolgt durch die Kombination von Rational-

isierungen von Freiformflächen mit planaren Vierecksnetzen, abwick-

elbaren Streifen und geodätischen Mustern mit neuen rechnerischen

Werkzeugen und Scripting. Für jede der genannten Rationalisierungs-

methoden wurde ein Projekt entworfen, rationalisiert und für die Ferti-

gung in Holz detailliert.

Aknowledgements

There are so many people who stood by me during my studies, especially

during the last nine months while I was working on this thesis. I’d like to

start by thanking Michael Eigensatz for introducing me to the exciting

world of freeform geometry through his class at TU Wien, and for being

a valuable teacher to this day. My thanks goes to the entire Evolute

team with whom I was privileged to work with for over a year. This thesis

would not be possible without all the knowledge that I obtained while

working at Evolute. A special thanks goes to Alexander Schiftner who

continues to be a great support and teacher for over a year now. Many

thanks goes to my supervisor prof. Helmut Pottmann. It was a pleasure

to work under his supervision. I’d also like to thank Florian Rist who

supported me selflessly while I was building the models that are shown

in this work. Further thanks go to my colleagues Moritz Rosenberg and

Benjamin Straßl, who read my work in advance and gave me valuable

feedback.

My most heartily gratitude goes to Ivana who is the most reliable person

in the world and who is always there for me when I need her.

Last but not least I wish to thank my parents and my sister Marija, not

only for always being there for me and giving me everything that I ever

needed for completing my studies, but in first line for being a loving

family who always encouraged me to learn.

Table of Contents

1. Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Aim of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Method / Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Disposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2. Wood as a Construction Material 7

2.1 The Structure of Timber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Types of Timber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Softwoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Hardwoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Engineered Wood Products (EWPs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 Plywood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.2 Glued Laminated Timber (Glulam) . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.3 Laminated Veneer Lumber (LVL) . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.4 Fibreboards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3. The Geometry of Freeform Architecture 15

3.1 Traditional Surface Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Rotational Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.2 Translational Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.3 Ruled Surfaces and Developable Surfaces . . . . . . . . . . . . . . . . . . . . . 16

3.1.4 Pipe Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.5 Offset Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Freeform Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.1 Freeform Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.2 Bézier Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.3 B-Spline and NURBS Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.1 Subdivision Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Freeform Surface Rationalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.1 Non-Rationalized Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.2 Pre-Rationalized Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.3 Post-Rationalized Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4. PQ Meshes 30

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Application in Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Geometric Properties of PQ Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.1 Planarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.2 Conjugate Network of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3.4 Mesh Offsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3.5 The Discrete Gaussian Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4 Project: Fair Stand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4.1 PQ Mesh Creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.2 The Scripting Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4.3 The Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5. Developable Surfaces and DStrips 73

5.1 Developability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2 DStrip Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2.1 Principal Strip Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2.2 DStrips Between Two Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.3 DStrip Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3.1 Input Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3.2 Evaluation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3.3 Test With Developable Reference Surface . . . . . . . . . . . . . . . . . . . . . 81

5.3.4 Test With Arbitrary Reference Curves . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4 Project: Bouldering Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4.1 The Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6. Geodesics Curves on Freeform Surfaces 105

6.1 Geodesic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105

6.2 Application in Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105

6.3 Algorithmic Panelization of Surfaces with Geodesic 1-Patterns . . . . . . . . . . .108

6.3.1 Designing 1-Patterns of Geodesic Curves . . . . . . . . . . . . . . . . . . . . . .110

6.3.2 Creating Panels from Geodesic 1-Patterns . . . . . . . . . . . . . . . . . . . . .113

6.4 Project: 21er Raum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116

6.4.1 The Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .122

6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124

7. References 126

8. Figure Credits 129

1. Introduction

1.1 BackgroundSmooth architectural surfaces

The modernist movement began its over five decades long hegemony in the field of architec-

tural theory at the CIAM[1] Congress in 1928, when LeCorbusier and Walter Gropius presented

a programme for a future architectural revolution. Modernism became a rigid orthodoxy that

failed to create a humanely built environment. In the 1980’s a new style, Deconstructivism

emerged as a reply to the hold that the old style established on architecture [Powel 2004].

The new style breaks the bonds established by the previous modernist movement. Instead

of searching for a universal architectural formal language, deconstructivist practices invest in

embodying the differences within and between diverse physical, cultural and social contexts

in formal conflicts. The most paradigmatic architecture of that period, such as the Sainsburry

Wing of the National Gallery in London by Robert Venturi or Peter Eisenman’s Wexner Centre

in Ohio, attempt to create a formal architectural representation of contradiction [Lynn 2004].

In the early 1990’s architecture was divided between two camps of philosophical thought. The

one camp, deconstructivism, would have architecture break under the stress of difference while

the other, modernism, would have it stand firm. In his seminal publication Folding in Architec-

ture in 1993 Greg Lynn proposes new pliant and smooth[2] forms that would provide an escape

from the differences between modernist and deconstructivist formal languages. Greg Lynn’s

smooth, pliant and voluptuous forms are, from a geometric point of view, called freeforms in

this work. With the advent of computer technology and CAD applications since Lynn’s Folding

in Architecture it was possible to push the boundaries of freeform architecture ever further.

Patrick Schumacher argues that the recent emergence of parametric modelling and scripting

is comparable to the discovery of perspective drawing in the renaissance, meaning that it rep-

resents a paradigm shift in architectural theory. Today, freeform architecture is slowly passing

from avant-garde architectural practices into the mainstream and more and more architectural

practices design smooth complex shapes [Schumacher 2012]. The emergence of smooth sur-

faces in architectural design posed new questions to the engineers whose task it was to build

those structures. A new discipline, the architectural geometry emerged in recent years trying to

1 Congrès International d’Architecture Moderne2 smoothness - “the continuous variation” and the “continuous development of form” [Deleuze 1987 p.478]

1

answer those questions. Smooth surfaces that are part of a designed industrial object like a car

or a household appliance or a kid’s toy, can usually be produced in one piece without breaking

the surface up. Further, objects that are part of a discourse of industrial design are in most cases

produced in large quantities thus it is rentable to produce complicated moulds for those parts

if they are going to be used numerous times. Freeform surfaces of an architectural scale have

to be divided into smaller parts to become feasible for production and construction. The divi-

sion of a large surface into smaller parts, panels, is called panelization. If the surface is divided

with arbitrary lines or cutting planes the resulting panels will most probable all be unique and

doubly curved. For the production of a doubly curved panel it is necessary to produce a custom

mould that can be used only for that panel alone. Architectural geometry provides different

possibilities for surface rationalization and discretization, in order to simplify the structures and

make them feasible, rentable and buildable.

Wood in freeform architecture

Wood is one of the oldest building materials on earth. People have used wood for thousands

of years to build their homes because it is a reliable natural material, that is relatively easily pro-

cessed. It was the dominant building material until the discovery of structural steel in the 18th

century and reinforced concrete in the 20th century. In the last decades wood was rediscovered

as building material due to new engineered wood products and to a shift towards sustainable

building. Wood offers numerous advantages over other high tech building materials. Besides

being a completely recyclable natural product that eliminates CO2 from the atmosphere, pro-

duces oxygen, and stabilises the ground, wood is a lightweight material that has an up to six-

teen times better load to weight ratio than steel and an up to five times better load to weight

ratio than concrete [McLeod 2010].

Wood can be used to produce a vast spectrum of different products from load bearing struc-

tural members to thin panels for cladding purposes. The unique properties of wood and its

products allow it to be relatively easily processed with different CNC[3] tools. Further, its molecu-

lar composition allows wood to adopt a curved shape through pure bending. All the properties

above make wood a good material for constructing freeform structures.

3 CNC - Computerized Numerical Control

2

1.2 Aim of the Thesis

Despite many advantages of wood over other materials, it is mostly ignored in the domain

of freeform architecture. The majority of freeform structures that are built to date are glass-

steel structures. The aim of this thesis is to investigate the possibilities of constructing freeform

structures with wood using knowledge about PQ meshes[4], DStrips[5] and geodesic patterns

from the field of architectural geometry in combination with computational tools and para-

metric scripting.

1.3 Method / Approach

Three concepts from architectural geometry for the discretization of freeform surfaces, that are

promising in combination with wood constructions are discussed. For each of the concepts,

PQ meshes, DStrips and geodesic patterns, one project is designed and rationalized with the

respective methods. Further, a physical model is presented as proof of concept for each of the

three approaches.

1.4 Limitations

Due to limitations in space, time and budget the physical models are scaled models of the

designs and no real prototypes. They serve well to illustrate the presented idea and to show

their feasibility, but only full scaled models would be conclusive proofs, especially in the case of

DStrips (chapter “5. Developable Surfaces and DStrips”).

There is no commercially available software on the market yet that performs according to what

is described in chapter “6.3 Algorithmic Panelization of Surfaces with Geodesic 1-Patterns” so

the described new methods for panelization with geodesic curve patterns could not be tested.

4 Planar Quadrilateral (quad) meshes5 DStrip - developable strip

3

Aim of the Thesis

1.5 Disposition

The thesis is structured as follows:

chapter 2 gives an introduction to wood as a construction material and to theengineered wood products that are used later in the projects of chapters 4, 5 and 6.

chapter 3 gives a profound theoretical background on the geometry that isnecessary to follow the work in the later chapters.

chapter 4 explores PQ meshes and the possibilities of using them in woodenfreeform structures.

chapter 5 explores DStrips and the possibilities of using them to cover freeformsurfaces with panels of plywood by means of bending the panels without tearing or breaking them.

chapter 6 explores geodesic patterns and the possibilities of using them to coverfreeform surfaces with wooden planks.

4

2. Wood as a Construction Material

Wood is one of the oldest construction materials and for centuries it has been the dominant

construction material worldwide. The introduction of steel to the building industry in the 18th

century was however a paradigm shift in architectural history. Steel could be manufactured

at large scale without thinking of sustainability or the fossil fuel which is spent in the process.

Steel, which allowed larger scale structures and larger spans, replaced wood in all major build-

ing projects leaving it only the niche of housing construction where wood with its natural feel,

low cost and ease of manipulation and processing enjoys continuous respect. The 20th cen-

tury brought a new invention which was going to change architecture once more — concrete

and reinforced concrete. Concrete is dominating the building industry until this day, but wood

has made its comeback as well. Since a few decades ago, wood started returning as construc-

tion material in larger structures. In the beginning wood as a construction material succeeded

because of its easy processing and overall availability, however there are other factors which

speak for the usage of wood in construction nowadays. Some of those factors are listed below:

development under environment friendly conditions

production, manufacturing and processing without mentionable use of fossil fuels

good dead weight to load bearing capacity ratio

different wood species with different visual qualities

good isolator and heat accumulator

availability of high quality connection techniques

possibility of prefabrication

Wood has proven to be one of the most sustainable building materials. It is a self growing mate-

rial which is completely recyclable The tree regulates our climate, stabilizes the ground and pu-

rifies the air by producing oxygen from carbon dioxide [McLeod 2010]. The development of the

Engineered Wood Products (EWPs) in the last decades contributed well to the return of wood

into the building industry. EWPs allow bigger sections and longer members than it is possible

to achieve with traditional sawn timber members [Porteous and Kermani 2007].

7

Wood as a Construction Material

2.1 The Structure of Timber

The tree has essentially three major parts in its structure which are easy distinguishable. The

roots of the tree are growing into the soil from where they absorb minerals and transfer them to

the other parts of the tree. The roots also act the trees foundation. The trunk is the middle part

of the tree which transports water and minerals from the roots to the crown and resists gravity

and wind loads. The crown is composed of branches and twigs which carry the leaves. Chemi-

cal reactions which produce oxygen, sugar and cellulose take place here. The produced sugar

and cellulose cause the growth of the tree. Our point of interest is mainly the trunk as this part

of the tree provides valuable wood for producing structural elements. The main features of the

tree trunk are visible in its section (Figure 2.1). It is comprised of several layers of material which

are layered circular from the pith in the centre of the cross section. The section tells a lot about

the life of a tree. The clearly visible concentric rings in the section are the annual rings, also

called the growth rings. Underneath the dry outer layer of the tree, the bark, there is a thin layer

which is responsible for the tree’s growth, the cambium [Bablick 2009]. Underneath the cam-

bium, new wood cells are formed over the old wood. On the other side, between the cambium

and the bark new bark cells are formed. In regions with temperate climate, the tree produces a

new layer of wood under the cambium each year, forming one annual ring. The growth process

starts in the spring and comes to an end in the winter. In such regions, where a definite growing

season exists, the annual rings are visibly divided into two layers: the springwood or earlywood

Pith

Juvenile wood

Annual rings

Heartwood

Sapwood

Rays

Outer bark

Inner bark

Cambium

Figure 2.1: Illustration of a cross section of a tree trunk

8

and the summerwood or latewood. The springwood forms during a relatively fast growth pe-

riod in the spring and consists of relatively large hollow cells whereas summerwood consists of

cells with thick walls and small hollow areas. The central core of the wood is called hardwood.

The hardwood is mainly made up of dead cells which have no function in the transport of

water or minerals, but it has an important function of giving mechanical stability to the trunk.

The lighter coloured layer outside the hardwood and underneath the bark is the sapwood. It is,

depending on the species, 25 - 170 mm wide. The sapwood is made up of dead and living cells

which have the function to transport sap from the roots to the crown of the tree. Over time,

as new layers of wood grow underneath the cambium, sapwood changes to hardwood, but

the size, shape and number of cells remains unchanged. Sapwood and hardwood have nearly

equal strengths and weights. Hardwood is a better choice for construction because it has a

higher natural resistance towards attacks by fungi and insects. Most of the wood cells, which

are usually long tubular cells, are oriented in the direction of the trunk. The only exceptions are

the cells called rays, which run radially across the trunk. The rays’ purpose is to transport miner-

als between the pith and the bark [Porteous and Kermani 2007].

2.2 Types of Timber

According to their botanical origin, trees and commercial timbers are divided into two groups:

hardwoods and softwoods. This classification has not any bearing on the actual hardness of the

wood. It is therefore possible to have some physically softer hardwoods, like wawa from Africa,

than other physically harder softwoods like the pitchpines [Porteous and Kermani 2007].

2.2.1 Softwoods

Softwoods are typically trees with a quick growth rate, generally evergreen trees. They can be

felled after 30 years of growth. The fast growth and early felling results in low-density timber

with a relatively low strength. Unless they are treated with preservatives, softwoods generally

exhibit poor durability quality. The biggest advantage of softwoods is that they are easily avail-

able and comparatively cheaper because of their quick growth and speed of felling [Porteous

and Kermani 2007].

9

Types of Timber

2.2.2 Hardwoods

Compared to softwoods, hardwoods grow at

a much slower rate, sometimes over 100 years.

This results in more dense timber, giving the

timber more strength. Hardwoods posses

a higher durability than softwoods, and are

therefore less dependent on preservative sub-

stances. Due to the long growth period hard-

woods are often more expensive compared to

softwoods [Porteous and Kermani 2007].

2.3 Engineered Wood Products

(EWPs)

The size of the tree from which a wood prod-

uct is sawn limits the quality and size of the fi-

nal product. If one want’s to build a large scale

structure in wood, the readily available sawn

sections of wood will not meet the demands of

this structure. Engineered wood products are

developed to overcome those size limitations,

and to make a huge variety of forms in wood

possible. EWPs have many comparative advan-

tages over solid sawn timber. Large lengths

and sections can be produced from small logs

which offers economical advantages because

trees from which a large section could be cut

are rare and expensive [Porteous and Kermani

2007].

2.3.1 Plywood

Plywood was the first EWP to be invented. It

is a flat panel made by bounding together at

Figure 2.2: Bending plywood, thickness 7mm, bend-ing radius 25 cm

10

least three layers of veneer (also called plies or

laminates) laid out with their grain directions

perpendicular to each other. The 2 - 4 mm thick

veneers are always combined in an odd num-

ber and then bounded under high pressure.

The outer laminates (face ply and back ply) are

always made of veneer, whilst the inner lami-

nates can be made of veneer as well as of sliced

or sawn wood. The inner laminates are the core

of the plywood. It is possible to make plywood

resistent to water by using special waterproof

adhesives. Those plywoods can be used in the

exterior and as structural plywood. Plywood is

available in fairly large sheets (1200 mm x 2400

mm) and it is relatively easily bendable per

hand in larger radii. A special type of Plywood

is bending Plywood (Figure 2.2) which thanks

to its structure can be bent, depending on the

plywood’s thickness, to a radius of 25 cm in per-

pendicular direction to the slope of grain [Por-

teous and Kermani 2007].

2.3.2 Glued Laminated Timber (Glulam)

Glued laminated timber, Glulam (Figure 2.3) is

manufactured by means of binding together

at least three small sections of timber boards

(laminates) with adhesives. The timber boards

are laid up so that their grain direction is essen-

tially parallel to the longitudinal axis. This tech-

nology enables the production of straight and

curved members. Timber boards with thickness

Figure 2.3: Glulam member

Figure 2.4: Laminated veneer lumber

Figure 2.5: Medium-Density Fibreboard (MDF) panels

11

Engineered Wood Products (EWPs)

of 33 - 50 mm are used as laminates in straight or slightly curved members, whereas much thin-

ner laminates (12 mm to 33 mm) are used for the production of curved glulam members. The

boards are in both cases 1.5 - 5 m long. They are first finger joined and then placed randomly

in the glulam member. The laminates are then glued with a carefully controlled adhesive mix

and placed in mechanical or hydraulic jigs of appropriate shape and size. The laminates rest in

the jigs until the adhesive is cured and the glulam member takes its final shape. The final step is

cutting, shaping and finishing of the glulam member [Porteous and Kermani 2007].

2.3.3 Laminated Veneer Lumber (LVL)

Laminated veneer lumber was first produced about 40 years ago (Figure 2.4). It is produced

from thin veneers similar to those encountered in the production of plywood. Unlike in ply-

wood, the successive veneers of LVL are oriented in the same grain direction, except for a few

sheets of veneer which are laid up perpendicularly to the longitudinal direction to enhance the

overall strength of the member [Porteous and Kermani 2007].

2.3.4 Fibreboards

Fibreboards such as high-density fibreboard (HDF), medium-density fibreboard (MDF) , tem-

pered hardboard, cement-bonded particleboard etc. are used extensively in housing construc-

tion and furniture production (Figure 2.5). For the production of fibreboards wood fibres are

mixed with adhesives to form a mat of wood. They are pressed until the adhesive is cured and

afterwards cut to the required sizes. The quality of fibreboards ranges from general purpose

boards which are designed only for the use in interior dry spaces to heavy-duty load-bearing

boards which can be used in construction, even in humid conditions. A special type of fibre-

boards is kerfed MDF which is possible to be bent, depending on it’s thickness, to a radius of

just 25 cm [Porteous and Kermani 2007].

12

3. The Geometry of Freeform Architecture

Freeform architecture is a term which is nowadays established in denoting architectural forms

which are composed of one or more freeform surfaces, also known as complex geometry sur-

faces. It is necessary to take a step back and see what types of traditional surface classes exist

in order to define freeform surfaces as a family of surfaces which cannot be classified as any of

those types of surfaces.

3.1 Traditional Surface Classes

The surfaces which are classified as traditional surfaces are generated by sweeping a profile

curve undergoing a smooth motion. Rotational, translational, ruled, helical and pipe surfaces

belong to the class of traditional surfaces. This subchapter gives a short overview of traditional

surface classes. For detailed information about the types of surfaces presented below see [Pott-

mann et al. 2007].

3.1.1 Rotational Surfaces

Rotational surfaces (or surfaces of revolution) are surfaces in Euclidean space created by rotat-

ing a planar or spatial curve c, the generatrix, around an axis A. Every point p of the generating

curve c describes a circle cp whose supporting plane Sc lies orthogonally to the axis A, therefore

surfaces of revolution carry a set of circles in parallel planes, parallel circles.

Meridian curves are congruent planar curves which are generated by intersecting a rotational

AA A

cp

m

M

Sccp

Figure 3.1: A curve c is rotated around the axis A to generate a rotational surface

15

The Geometry of Freeform Architecture

p

c

g

Figure 3.3: A ruled surface

Figure 3.2: (a) extrusion surface and (b) translational surface

surface with planes M which contain the axis

A. The meridian curves and the parallel circles

of a rotational surface intersect at right angles,

thus forming a net of orthogonal curves on the

surface, due to the fact that the supporting

planes Sc of the parallel circles cp and the me-

ridian planes M of the meridian curves m are

orthogonal.

3.1.2 Translational Surfaces

If we take two curves k and d which intersect in

one point o, the origin and translate the profile

curve k along the path curve d we will generate

a translational surface (Figure 3.2). If the path

curve d is a straight line, then an extrusion sur-

face is generated [Pottmann et al. 2007].

3.1.3 Ruled Surfaces and Developable

Surfaces

Ruled surfaces are a special type of traditional

surfaces that contain a continuous family of

straight lines called generators or rulings. Ruled

surfaces have the advantage over other tradi-

tional surfaces in an architectural context that,

due to the existence of the rulings, they can be

easier built [Flöry et al. 2012].

Cylinders, cones, one-sheet hyperboloids and

hyperbolic paraboloids are some of the sim-

plest ruled surfaces. Generally, ruled surfaces

are created by moving a point p of a straight

line segment g along a curve c and changing

the line’s direction continuously. [Pottmann

k

d

k

do

o

16

d

Sd

S

Figure 3.5: The offset surface Sd lies at constant dis-tance d to the surface S.

et al. 2007] Another interesting type of ruled

surfaces are developable surfaces which bear

special potential in architectural applications.

[Pottmann et al. 2008] Developable surfaces

will be discussed in detail in chapter 5.

3.1.4 Pipe Surfaces

A pipe surface is the envelope of spheres of

equal radius r whose centres lie on a curve c,

called the spine curve or central curve c. The

pipe surface can also be seen as a family of cir-

cles with the radius r lying in the normal planes

of a spatial curve. [Pottmann et al. 2007]

3.1.5 Offset Surfaces

An offset surface Sd of the surface S is the sur-

face with a constant normal distance d to the

original surface S. The offset surface Sd and the

surface S share their normals. Further, the tan-

gent planes of S and Sd in corresponding points

are parallel, therefore offset surfaces are also

called parallel surfaces. [Pottmann et al. 2007]

3.2 Freeform Surfaces

Rotational-, translational-, ruled-, helix-, pipe

surfaces and surface offsets are not closely suf-

ficient to meet the high demands of today’s

state-of-the-art architectural designs. More

complex shapes are nowadays designed using

freeform surfaces because those surfaces of-

fer more flexibility compared to the traditional

surfaces. We will discuss three types of surfaces

c

cr

Figure 3.4: A pipe surface as (a) an envelope of spheres, and (b) a collection of circles lying normal to

the curve c.

17

Freeform Surfaces

in this chapter: Bézier surfaces, B-Splines and

NURBS surfaces and subdivision surfaces. Bézi-

er and B-Spline surfaces are as a matter of fact

special types of subdivision surfaces [Pottmann

et al. 2007]. A base for understanding freeform

surfaces is the knowledge of freeform curves.

3.2.1 Freeform Curves

Before the invention of computers, the design-

er had to draw the freeform curve by hand us-

ing some mechanical aids. The pencil was led in

a smooth way across the paper. The quality of

the curve depended on the skill of the design-

er. The drawing of wide stretching curves was

in particular difficult because the entire arm

had to be moved smoothly. To draw very long

smooth curves designers used a mechanical

aid to guide their hands. Such tools were called

splines. Splines were usually bendable wooden

or metal rods whose shape was controlled by a

few points where the rod was fixed with special

weights —ducks.

3.2.1.1 Bézier Curves

Bézier curves are among the most widespread

freeform curves, mainly because of their sim-

plicity and their ease of use. Bézier curves are

constructed via the Casteljau algorithm, which

is based on repeated linear interpolation. They

are completely defined by their control poly-

gon. [Pottmann et al. 2007]

d0

d1

d2

d3

d4

d5

d0

d1

d2

d3

d4

d5

(a)

(b)

Figure 3.6: Image showing a degree 3 B-Spline curve with six control points (a) consisting of three cubic

Bézier curves (b).

18

3.2.1.2 B-Spline Curves

B-Spline are more powerful than Bézier curves

because they offer local shape control. B-Spline

curves consist of Bézier curve segements of the

same degree which are connected at their end-

points with the highest possible smoothness.

B-Splines can be generated by curve subdivi-

sion. This is a process in which a given coarse

polygon is subdivided iteratively with Chaikin’s

algorithm or Lane-Riesenfeld’s algorithm. [Pott-

mann et al. 2007]

3.2.1.3 NURBS Curves

NURBS1 curves are the third and most sophis-

ticated type of freeform curves. They offer fur-

ther fine-tuning capabilities via weights asso-

ciated with the control points. The weight of a

control point represents the power with which

it pulls the curve towards itself. Essentially, B-

Spline curves are special cases of NURBS curves

wherein all weights in the control points are

equal. [Pottmann et al. 2007]

3.2.2 Bézier Surfaces

I will first discuss a simple special case of Bézier

surfaces, the translational Bézier surfaces, and

then move on to the general Bézier surfaces.

Translational Bézier surfaces

Translational Bézier surfaces can be created

from Bézier curves. Two curves are needed for

1 NURBS — NonUniform Rational B-Spline

d0

d1

d2

d3

d4

d5

d0

d1

d2

d3

d4

d5

d0

d1

d2

d3

d4

d5

(b)

(c)

(a)

Figure 3.7: A B-Spline is a special type of NURBS curves with equal weights in all control points (a). By changing the weight in a control point d3 of a NURBS curve, the curve can be either detracted (b) or attract-

ed by that point (c).

19

Freeform Surfaces

b20

b30

b00

b01

b11

b12

b13 b02

b12

b22

b32

b10

Figure 3.9: A Bézier surface of degree (3,2) with a boundary polygon and a boundary curve shown in

red

the creation of a translational Bézier surface,

one Bézier curve of degree m (bm) and one Bézi-

er curve of degree n (bn). In order for the curves

to be suited for generating a translational sur-

face, they need to share a common endpoint

b00. The control points of the Bézier curves are

denoted with the double index notation. The

control points of the quadratic curve b2 in the

example shown in Figure 3.8 will be called b00,

b10 and b20. The control points of the cubic curve

b3 are b00, b01, b02 and b03.

The translational surface of those two Bézier

curves will carry one family of quadratic Bézier

curves b2 and one family of cubic Bézier curves

b3. In order to distinguish between the param-

eters of those two Bézier curves, the parameter

b2 is denoted as the u parameter and the pa-

rameter along b3 as the v parameter of the sur-

face. [Pottmann et al. 2007]

General Bézier surfaces

General Bézier surfaces are a straightforward

extension of translational Bézier surfaces. The

Bézier surface is defined by its control mesh.

The mesh consists of an array of points in

space, connected to a quadrilateral mesh with

row and column polygons. The control points

are described with two indices. The first index

0,1,...,m denotes the row in which the vertex

is located, while the second index 0,1,...,n de-

scribes the vertex column. The number of con-

T

b00

b01

b3b2

b02

b03

b13

b23

b21

b22

b12

b11

b10

b20

Figure 3.8: A Bézier curve of degree 3 is translated along a Bézier curve of degree 2 in order to create a

translational Bézier surface .

20

trol points is therefore (m+1)(n+1). The surface

still contains two families of Bézier curves, the

first family in u direction of degree m and the

second family in v direction of degree n. Due to

this fact, a Bézier surface has a degree of (m,n).

[Pottmann et al. 2007]

3.2.3 B-Spline and NURBS Surfaces

Since Bézier surfaces are constructed from Bézi-

er curves, they share the same drawbacks. When

the degree of the Bézier surface becomes too

high it stops representing its control mesh ac-

curately. The other drawback is that Bézier sur-

faces do not feature local control, hence chang-

ing the position of one control point will have

effect on the whole surface. This makes the de-

sign process unnecessary harder than it should

be. B-Spline surfaces overcome the problems

that Bézier surfaces have. They are also defined

by a quadrilateral control mesh, but they allow

the user to chose the degrees for the u- and v-

curves. Similar to the case with freeform curves

where the NURBS curve is the most sophisti-

cated, NURBS surfaces are among the most

powerful freeform surfaces. They have all the

same possibilities as B-Splines, offering in addi-

tion the possibility to adjust the weight of each

control point individually, which will make the

point pull the surface towards itself with more

or less power. Similar as with freeform curves, a

B-Spline surface is a special type of NURBS sur-

p

p

p

Figure 3.10: A NURBS surface with different weights in the control point p. (a) NURBS surface with decreased

weight in control point p, (b) B-Spline surface with equal weights in all control points, and (c) NURBS sur-

face with increased weight in control point p.

21

Freeform Surfaces

face wherein all weights in the control points are equal.

[Pottmann et al. 2007]

3.3 Meshes

Meshes can be seen as discretizations of smooth surfaces.

They are basically collections of points in space (vertices)

connected by edges to form polygons —faces. Usually one

type of faces dominates the mesh e.g. triangular-, quadri-

lateral- or even hexagonal faces. In this case one speaks of

triangle dominant meshes, quad-dominant meshes etc.

If the mesh consists of only faces of one type then one

speaks of for instance quad meshes. A mesh is traditional-

ly stored in the computer with the help of two arrays. The

first array — the array of vertices stores each vertex with

a unique index. For each index the x, y and z coordinates

of that vertex are stored. The second array — the array of

faces stores all mesh faces in a second list [Pottmann et al.

2007]. It is sufficient to store only the indices of the face’s

adjacent vertices from the first array. This way the mesh is

stored efficiently by essentially storing the 3D positions

of all vertices in the first array and their connectivity in the

second array. The arrays in which the mesh from Figure

3.11 is stored are shown in Table 3.1.

5 6

7

9 1011

13 14

15

1 2

12

8

4

0 3

I

E

HG

D

A

F

B C

Figure 3.11: A pure quadrilateral mesh with nine faces (A-I) and sixteen vertices (0-15)

Array of vertices 0 = (0.0, 0.0, 3.0) 1 = (5.0, 0.0, 1.5) 2 = (10.0, 0.0, 1.5) 3 = (15.0, 0.0, 3.0) 4 = (0.0, 3.0, 1.5) 5 = (5.0, 3.0, 0.0) 6 = (10.0, 3.0, 0.0) 7 = (15.0, 3.0, 1.5) 8 = (0.0, 6.0, 1.5) 9 = (5.0, 6.0, 0.0)10 = (10.0, 6.0, 0.0)11 = (15.0, 6.0, 1.5)12 = (0.0, 9.0, 3.0)13 = (5.0, 9.0, 1.5)14 = (10.0, 9.0, 1.5)15 = (15.0, 9.0, 3.0)

Array of facesA = (0, 1, 5, 4)B = (1, 2, 6, 5)C = (2, 3, 7, 6)D = (4, 5, 9, 8)E = (5, 6, 10, 9)F = (6, 7, 11, 10)G = (8, 9, 13, 12)H = (9, 10, 14, 13)I = (11, 10, 14, 15)

Table 3.1: Two lists in which the mesh from Figure 3.11 is stored

22

Meshes can represent a coarse discretization of a smooth sur-

face, like on some buildings or very smooth discretizations like

those which are used in today’s animated videos. The latter

surfaces look perfectly smooth when rendered even though

they are constructed of very tiny mesh faces.

3.3.1 Subdivision Surfaces

Regular quadrilateral meshes, which have only vertices of va-

lence four, have certain topological restrictions for the mod-

elling of more complex shapes. The same applies to B-Spline

surfaces as they can be seen as refinements of their control

polygons which are basically quad meshes. In order to model a

complex shape, we can use irregular meshes with singularities.

Singularities are types of mesh vertices of a different valence

from the dominant type of vertices. The valence of a vertex is

the number of its adjacent edges, for example in a quad domi-

nant mesh most of the vertices would be of valence four. A ver-

tex with three or five adjacent edges would be of valence three

(red circle in Figure 3.12) or valence five (blue circle in Figure

3.12) and thus those vertices would be denoted as a singulari-

ties. Figure 3.12 shows a coarse mesh (a) subdivided two times

(b) and (c) with the Catmull-Clark algorithm. Note how the sub-

(a) (b) (c)

Figure 3.12: Subdivision surface. (a) coarse mesh (b) one step of subdivision refinement (c) two steps of subdivi-sion refinement

23

Meshes

divided mesh inherits the singularities from it’s parent [Pottmann et al. 2007].

3.4 Freeform Surface Rationalization

Freeform surfaces have been used in industrial design many years before they could be used in

architecture. One reason for that is the scale of the individual projects. While small scale designs

e.g. a rubber duck, can be manufactured in one piece, this approach would be impossible in

architecture. A freeform surface which has the scale of a building has to be divided into smaller

parts in order to be buildable. This segmentation of a surface is called panelization. Surfaces

can be panelized with different methods according to the desired outcome. [Schiftner et al.

2012] present a rough classification of design approaches for realizing a freeform surface. Ac-

cording to them, a freeform project might be realized as non-rationalized, pre-rationalized or

post-rationalized.

3.4.1 Non-Rationalized Structures

A freeform surface might be realizable without prior rationalization. In this case the pattern of

structure and panels can be chosen freely. Mostly, the best choice in this situations is the inter-

section of the freeform surface with a regular grid of planes. An example of this method is the

Fashion and Lace Museum in Calais, designed by the Paris based architects Alain Moatti and

Henry Rivière (Figure 3.13).

3.4.2 Pre-Rationalized Structures

The approach of pre-rationalization is to generate a design by using only certain classes of

surfaces e.g. translational and rotational surfaces or developable surfaces, which Frank Gehry

Figure 3.13: The facade of the Fashion and Lace museum, Calais is constructed with doubly curved glass panels

24

made famous in his designs (Figure 3.14). Many of Gehry’s designs are modelled in paper or

thin metal or plastic sheets that are easily bendable, to ensure developability of the final sur-

faces [Shelden 2002].

3.4.3 Post-Rationalized Structures

In the case of post-rationalization an ideal surface is selected as reference and rationalized so

that the result closely resembles it and meets certain rationalization criteria, such as cost and

quality of the surface and the substructure, panel size, planarity etc. Triangulation might be the

most prominent rationalisation method because it is the most straightforward method (Figure

3.15 - a). Curved panels are not possible in triangular meshes due to the fact that a surface

through three points is planar by definition. Another drawback is the lack of a clean offsetability

of triangulated freeform surfaces, which results in more complicated nodes and substructures.

Planar quad panelizations

Planar quad panelizations (Figure 3.15 - b) are often the better choice over triangulation. In or-

der to build large-scale freeform shapes there is a basic need to subdivide it into smaller panels

which can be manufactured and transported at a reasonable cost. Generally, for the production

of curved panels one has to build a custom mold for each panel. Those moulds are expensive

and they are tried to be avoided when possible. Recent developments in architectural geom-

etry enable us, among others, to optimize the panels for planarity. This way a lot of curved

panels can be flattened (depending on the shape of the reference surface), while preserving

the desired aesthetic quality.

Planar quad meshes offer possibilities for offsetting, thus a simpler substructure can be de-

Figure 3.14: Jay Pritzker Pavilion, Chicago by Frank O. Gehry (left) architectural model (right) photograph of the completed building.

25

Freeform Surface Rationalization

signed. Instead of having six beams meeting in a

node, like it is the case with triangular meshes, in a

quadrilateral mesh there are only four beams meet-

ing in each node. PQ (planar quadrilateral) meshes

have, in comparison to triangular meshes, fewer

panels, less cutting waste and less total joint length.

Developable strip panelizations

Developable strip panelizations (Figure 3.15 - c) are

one-directional refinements of a planar quad pan-

elization. A fairly coarse PQ mesh is subdivided with

strip subdivision in only one direction. This process

results in densely subdivided mesh strips which are

developable if all of its subdivided quad faces are

planar.

There are numerous benefits from the design ap-

proach with DStrips. Developable strips can be man-

ufactured from planar sheets without the necessity

for any moulding or advanced bending machines.

This leads to a significant reduction of cost. The fact

that they are produced by bending material in one

direction makes them a semi-discrete representa-

tion of a freeform surface, thus they are perfectly

smooth in one direction as opposed to paneliza-

tions with PQ panels. Another benefit of develop-

able surfaces is that offsets of a developable surface

are developable again, which means that multilay-

ered structures with developable strips are easy to

achieve. The entire substructure of a DStrip model

can be made of developable materials [Schiftner et

al. 2012].

(a)

(b)

(c)

Figure 3.15: Freeform surface rationalization. (a) A freeform surface rationalized with triangular pan-els. (b) A freeform surface rationalized with planar

quadrilateral panels. (c) A freeform surface rational-ized with developable strips

26

4. PQ Meshes

4. PQ Meshes

4.1 Introduction

In section "3.3 Meshes" a basic definition of a mesh was given. We have seen that a mesh can

be seen as a discrete representation of a smooth surface in "3.4.3 Post-Rationalized Structures".

This chapter aims on expanding the topic of PQ meshes and their usability in construction,

especially in wood structures. In the next section we will se how planar quadrilateral paneliza-

tions have been used in architecture so far. Afterwards, in section 4.3, the geometric concepts

and properties of PQ meshes, as well as the different types of meshes are explained. The section

4.4 shows a small demonstration project in which a PQ mesh is used in combination with script-

ing to create a multi-layered freeform wood structure.

4.2 Application in ArchitectureUpon analysing some of the most prominent built structures one can conclude that the standard

method for covering a curved surface with panels is the triangulation of the surface. The

majority of curved surfaces in architecture which are panelized, are panelized with triangles. The

advantages of a quadrilateral panelization over triangular panelizations have been mentioned

before in section "3.4.3 Post-Rationalized Structures", offsetability of the meshes, simpler joints

and less running meters of total joint length being the most obvious from a practical point

of view. Some less obvious advantages of quad meshes over triangular meshes include the

following: quad panelizations are less obtrusive then triangular ones. Transparent structures

(Figure 4.2) let more daylight trough the facade because they are less dense then triangular

panelizations and the structure creates less obtrusive shadows [Schmiedhofer et al. 2008]. In

recent years there are more and more projects with curved surfaces that were successfully

covered by a quadrilateral panelization. The Sage Gateshead by Foster + Partners (Figure 4.2

Figure 4.1: The Dongdaemun Design Plaza by Zaha Hadid (left), a graphic showing the planar quad panels in blue and cylindrical panels in red. The gray panels are doubly curved (right)

30

a) which opened in 2004 features a steel

glass facade made out of planar quadrilateral

panels. The building was designed using

translational surfaces, so the panelization

was a straightforward process. This method is

very limiting and will not work with arbitrary

freeform shapes [Schmiedhofer et al. 2008].

The Nationale-Nederlanden building, popularly

called the Dancing House, by Frank Gehry in

Prague (Figure 4.2 b) features a glass facade

made out of planar quadrilateral panels.

This building was opened in the year 1996,

long before optimization algorithms for PQ

meshes existed. The panels are laid out so that

three points of each quad lie on the designed

surface and the fourth point is projected to

the plane which is defined by the other three

points. This results in a structure made out of

completely flat glass panels which have quite

large gaps between them. A similar approach

has been used for the facade of the Yas Marina

Hotel by Asymptote Architecture and Evolute

(Figure 4.2 c). In this case the panels have been

rotated away from the surface intensionally to

create the distinctive look of the building while

taking advantage of the flat panels which can

be achieved this way. The above methods are

especially useful in applications similar to the

two mentioned projects where the facade is

used for pure visual reasons and there are no

requirements of thermal stability or protection

from acoustics or the elements imposed on

the structure. Analogously, such structures

(a)

(b)

(c)

Figure 4.2: (a) The Sage by Foster + Partners, (b) The Dancing House by Frank Gehry, (c) The Yas Marina

Hotel by Asymptote

31

Application in Architecture

can be used in interior applications, where

applicable, without any problems. They

reduce the cost of the building drastically

because it can be completely covered with

flat material.

A much more interesting project from the

point of view of PQ meshes is the Dongdae-

mun Design Plaza building by Zaha Hadid ar-

chitects in Seoul (Figure 4.1). Evolute created

a layout of quadrilateral panels to cover the

entire exterior surface of the building. This

panel layout comes close to a PQ paneliza-

tion. In some cases it is not possible to achieve

a PQ panelization in which all panels are pla-

nar, but if the majority of the panels are pla-

nar the panelization can be considered a PQ

panelization. The panelization of the Dongda-

emun Design Plaza is not a true PQ paneliza-

tion because the share of planar panels in the

total number of panels is not sufficient. At the

time of this work, there is no built architec-

tural freeform project with a true PQ paneli-

zation to show as reference. A very interesting

structure however, is the steel glass roof for

the Department of Islamic Arts at Louvre Mu-

seum (Figure 4.3). It was designed by Mario

Bellini Architects and Rudy Ricciotti in 2008

and built in 2012.

The glass structure is covered by a triangular

surface and is not visible from the outside (a).

Underneath the outer layer of the roof there

is a hybrid PQ-triangular structure (b). The hy-

brid structure consists of totally planar quad-

(b)

(c)

(d)

(a)

Figure 4.3: Department of Islamic Arts at Louvre Mu-seum.

32

rilateral panels and triangles. A spacial optimi-

zation technique, that allows the penalization

of a freeform surface this way, has been used.

Figure 4.4 shows the same reference surface

panelized similar to the panelization that has

actually been built.

Starting from the reference surface (Figure

4.4 - a) a fairly coarse quad mesh is construct-

ed first. The size and proportions of the quads

in this mesh will have direct influence on the

size and proportions of the final panels. The

quad mesh is optimized for a smooth layout

of the edges on the surface and for optimal

surface closeness. It doesn't need to be op-

timized for planarity because the planarity

of the faces is achieved in a different way in

this situation. The next step was to subdivide

the mesh in order to return its "diagonalized"

mesh . This subdivision algorithm adds a ver-

tex in the centroid of each face and connects

it to the existing face vertices of the original

mesh. When applied to a quad mesh, the al-

gorithm results in a mesh whose edges form

polylines that are diagonal to the polylines

formed by the edges of the input mesh. The

diagonalized mesh had to be optimized to

the reference surface in such a way that it is

possible for the mesh's boundary vertices to

"overflow" the boarder of the reference sur-

face and allow the first next group of vertices

to snap to its boundary (c). This step was nec-

essary because of what happens in the next

step, which is a "dual" subdivision. The dual

(a)

(b)

(c)

(d)

(e)

Figure 4.4: Geometry of the roof structure of the De-partment of Islamic Arts at Louvre.

33

Application in Architecture

subdivision algorithm adds a vertex in the centroid of each face, but instead of connecting it

to the face's existing vertices like the previously used algorithm, it connects the newly created

vertices to each other. The dual subdivision results in the mesh whose boundary is aligned to

the boundary of the reference surface (Figure 4.4 - d). The colouring of the faces represents

their planarity value, where blue means that the face is completely planar, whilst the faces that

are curved the most are coloured red. After the dual subdivision the mesh consists of alternat-

ing diagonal rows of completely planar quads and rows of curved quads. The final step is to

insert diagonal edges along the rows of curved faces, dividing each curved quad face into two

triangles, and ultimately making all faces in the mesh completely planar.

34

4.3 Geometric Properties of PQ Meshes

4.3.1 Planarity

Quadrilateral meshes, as opposed to triangular meshes, typically do not consist of planar faces.

The planarity of a mesh's faces is important when it comes to producing the shapes. In most

cases, the mesh's faces directly translate into physical panels which are usually made of glass

or metal. The curved nature of the faces inflicts many problems in the production process of

those panels. Curved panels are more expensive to produce than flat panels, especially dou-

ble curved panels for which a custom mold needs to be manufactured. In recent years it has

become possible to optimize quadrilateral meshes in such a way that their faces become flat,

while maintaining pleasing aesthetical mesh qualities.

Measuring planarity

The planarity of a quadrilateral face is measured by the closest distance between its diagonals,

as shown in Figure 4.5. The diagonals of a completely flat face lie on the face itself and thus, in

that case the distance between the diagonals is zero. The optimization of quadrilateral meshes

aims on minimizing this distance in order to achieve more planar faces.

It should be noted that depending on different restrictions, such as the desired shape the mesh

should take, or maintaining “nice“ or fair polylines in the mesh, it will not be possible to create

completely planar faces while respecting all restrictions. The optimization process is a trade off

between planarity, aesthetics, closeness to the design intent and other possible constraints.

There are some tolerances in materials, especially in wood, which allow the panels to be cold

bend into position, so the faces do not have to be completely flat, but their planarity needs to

be within those tolerances.

d

Figure 4.5: The planarity of a quadrilateral face is expressed as the closest distance between its

diagonals.

35

Geometric Properties of PQ Meshes

Scale invariant planarity

The scale invariant planarity is the planarity value of a quad face divided by the mean length

of its diagonals. The scale invariant planarity is a more general value compared to the absolute

planarity of a mesh face. The scale invariant planarity does not depend on the face's size and

thus it is easier compared to values from different meshes. Glass manufacturers usually have

the bending tolerances for their products expressed in form of scale invariant planarity.

4.3.2 Conjugate Network of Curves

Let's assume two families of curves, A and B, on a smooth surface (Figure 4.6). If we pick a curve

c in the network and compute a tangent to the curve from the other family in each of its points,

we will end up with a ruled surface which touches S along the curve c, the tangents being the

rulings of the surface. The two families of curves, A and B, are conjugate if the ruled surface

is developable, or simply a tangent developable surface along c. In simpler surfaces, such as

rotational and translational ones, a conjugate network of curves is even simpler to find. The

network of meridian curves and parallel circles in a rotational surface is a conjugate network of

curves. In a curve network which results by translating one curve along the other i.e. a trans-

lational surface, there is a cylinder on each curve, which consists of the tangents to the other

family of curves. Since the cylinder is developable it is safe to say that this network of curves is

conjugate.

family A

family B

c

S

Figure 4.6: A conjugate curve network on a freeform surface

36

4.3.2.1 Principal Curvature Lines

There is an endless number of conjugate curve networks in any given surface S. If one family of

curves is given it is possible to compute its conjugate family of curves by determining the tan-

gent developable surface along each curve c of family A. When in each point on the surface S a

ruling is computed, the rulings can be used to compute a family B of curves whose tangents are

exactly those previously calculated lines, the rulings of the tangent developable surfaces along

the curves of family A. In some cases it can be helpful to be able to compute an arbitrary con-

jugate network of curves based on one family of input curves, e.g. if we wish to create a mesh

layout following a specific path on the surface. However, in most cases, we will be interested in

the special case of the conjugate network of principal curvature lines. The network of principal

curvature lines is a unique network (except for simple shapes such as the sphere or the plane)

which besides being conjugate is also orthogonal. The usage of principal curvature lines as aid

in the layout of PQ meshes is very advisable because , quadrilateral meshes which follow the

principal curvature lines have the most potential to be optimized towards planarity.

PQ meshes which follow the principal curvature lines are also promising to be aesthetically

pleasing because such meshes will have nearly rectangular shaped faces.

Figure 4.7: A principal curve network on the same freeform surface as used in Figure 4.6

37

Geometric Properties of PQ Meshes

4.3.3 Optimization

The work of Liu et al. (2006) has led to a planarization algorithm, which take as an input a quad

mesh whose faces are not planar and returns a planar quad mesh that approximates the same

smooth surface. This process of moving the vertices in space to make the mesh faces as planar

as possible is known as optimization. It should be noted that the term optimization is much

broader and does not apply solely to the planarization of mesh faces. The optimization will, in

most cases, not be able to produce completely planar faces while maintaining a visual pleas-

ing mesh layout and remaining inside diverse other constrains, but we can ask for faces whose

planarity is within some tolerances that allow the usage of planar materials.

Limitations

The outcome of the optimization process depends on the input surface and especially on the

input mesh. We cannot provide the algorithm with an arbitrary mesh and expect a perfect re-

sult. The best optimization results will generally be obtained if the network of principal curva-

ture lines from the underlying smooth surface can be used as basis for the mesh. However, this

approach can result in undesired singularities and large variations in face sizes, depending on

the flow of principal curvature lines. This problems can be largely reduced if the designer keeps

these limitations in mind during the design stage and aims at a design which is better suited for

the discretization with PQ meshes.

Subdivision and optimization

A good possibility for the workflow when discretizing a freeform surface with a PQ mesh is

shown in Figure 4.8. Starting from the input surface (a) one first creates a mesh that very rough-

ly approximates the input surface. The mesh is then edited towards the smooth surface by

alternating between a quad-based subdivision algorithm and the planarization algorithm. The

input surface is used as reference for the optimization. This way the model is optimized at dif-

ferent levels of detail and it comes closer to the input shape after each subdivision and optimi-

zation step ( c and d). For best results, the optimization should occur after each subdivision step

because the subdivision ruins the planarity of faces [Pottmann et al. 2007]. Figure 4.8 (d) shows

how the optimization algorithm can twist the mesh in a certain direction. This occurs due to the

fairness parameters in the optimization algorithm which tend to make all polylines of edges as

smooth as possible. This is not a major problem since there is a number of ways to control the

optimization algorithm. In this case a coplanarity plane p has been used. A collection of mesh

vertices that belong to the same polyline of edges has been set to be coplanar to the plane.

38

(a)

(b)

(c)

(d)

(e) p

(f )

Figure 4.8: The process of subdivision and optimiza-tion. From input surface to optimized PQ mesh.

This makes the optimization algorithm move

those vertices as close as possible to the speci-

fied plane, which results in a polyline of edges

that lies coplanar on the plane (e). Figure 4.8 (f )

shows the final PQ mesh after another step of

subdivision with the Catmull-Clark algorithm

and another round of optimization. The col-

our coding shows the planarity of the faces.

The dark blue faces have the lowest planarity

value, hence they are close to completely pla-

nar, whereas the yellow panels have a slightly

higher curvature and planarity value.

39

Geometric Properties of PQ Meshes

Figure 4.10: A geometric support structure, formed by connecting corresponding vertices of two parallel

meshes M and M*.

M

M*

4.3.4 Mesh Offsets

Mesh offsets play an important role in multi-

layered architectural applications [Eigensatz et

al. 2010]. Multi-layered architectural structures

with beams and panels yield several meshes on

which each layer of the construction is based

[Pottmann et al. 2007]. The panels of such struc-

tures correspond to the mash faces. The nodes

are formed in the mesh vertices and the beams

correspond to the edges, through which the

central plane of the beam passes. Even in very

simple structures which consist of one layer of

panelling and one layer of beams beneath it, an

offset mesh is needed to orient the beams and

the nodes.

It is obvious that, in order to achieve an effi-

cient structure, those meshes should be paral-

lel. Two meshes M, M* are parallel (Figure 4.9)

if they are combinatorially equivalent and the

corresponding edges are parallel. The meshes

are combinatorially equivalent when there is a

direct correspondence between their vertices,

edges and faces. This notion is not restricted

only to quadrilateral meshes, but the planarity

of faces must be given.

The ideal case for most structures is if the cen-

tral planes of incoming beams accommodate

the node axis A in the adjacent vertex. This case

results in torsion free nodes in which both sides

of incoming beams which are all of the same

height align perfectly. Suitable node axes of M

can be obtained via a parallel mesh M* by con-

necting the corresponding vertices. The axes

M

M*

Figure 4.9: The Parallel meshes M and M* are combi-natorially equivalent and their corresponding edges

are parallel.

40

should be approximately orthogonal to M, thus

they are obtained from a mesh M* lying at a ap-

proximately constant distance to M. In this case

M* is a offset of M. For simplification [Pottmann

et al. 2007] introduce the concept of geometric

support structure (Figure 4.10) which focuses

only on the central planes of the supporting

beam layout. They define it as a collection of

planar quads which connect corresponding

parallel edges of two parallel meshes M, M*

which are used for the definition of node axes.

According to [Pottmann et al. 2007] an offset

mesh Md of a PQ mesh M is parallel to M and

lies at a constant distance to M. The way of de-

fining the distance needs to be specified. There

are three possibilities for this: vertex offsets,

edge offsets and face offsets. The three meth-

ods all have different outcomes and utilize dif-

ferent approaches. In order to continue with

the explanation of these concepts a digression

to the discrete Gaussian image is in order.

4.3.5 The Discrete Gaussian Image

For a pair of meshes M, Md; Md being the offset

mesh of M, the discrete Gaussian image (Gauss-

ian image mesh) is defined as the scaled dif-

ference mesh S = (Md - M)/d of the two parallel

meshes M and Md. The resulting scaled differ-

ence mesh S of two parallel meshes M and Md

is parallel to M and Md. The distance properties

between M and Md are reflected in the distanc-

es between the Gaussian image mesh S and

the origin of the unit sphere S* (Figure 4.11). S

approximates the unit sphere S* because the

Md

M

S*

S

Figure 4.11: The scaled difference mesh S of the par-allel meshes M and Md lying on the unit sphere S*.

41

Geometric Properties of PQ Meshes

distance d between M and Md is constant. The

difference of the two meshes (M and Md) was di-

vided with d and therefore the Gaussian image

S must have the distance 1 to the origin, thus

S* is a unit sphere. Considering the above (PQ

mesh M with offset mesh Md at constant dis-

tance d and the Gaussian image mesh S = (M-

Md)/d ) all offset properties are encoded in the

Gaussian image mesh S. [Pottmann et al. 2007]

4.3.5.1 Vertex Offsets

Vertex offsets result in a special type of meshes,

known as circular meshes. Circular PQ meshes

share the property that their PQ faces have cir-

cum-circles (Figure 4.12). In vertex offset mesh-

es this is due to the fact that the vertices of the

Gaussian image mesh S lie on the unit sphere

S* by definition. The faces are planar and the

planes on which the faces lie intersect the unit

sphere in a circle which circumscribes the face

of that plane [Pottmann et al. 2007]. If a quad

face a, b, c, d has a circum-circle, then the sum

of two opposite angles in that face equals 180

degrees. A paralel quad a1, b1, c1, d1 would have

the same angles, hence it would have a circum-

circle as well.

4.3.5.2 Edge Offsets

Edge offsets have the property that their Gauss-

ian image mesh has inscribed circles. This is be-

cause all edges of the Gaussian image mesh of

a pair of meshes M and Md at constant edge-to-

edge distance d lie tangent to the unit sphere.

They are called Koebe meshes. The Gaussian

a

a1

b

b1

c

c1

d

d1

Figure 4.12: Two parallel quads with their circum-circles.

S*

S

Figure 4.13: The edges of a Koebe mesh S are tangent to the unit sphere S*.

42

image mesh S is characterized by the property

that its faces intersect S* in circles, which are

the inscribed circles of those faces (Figure 4.13).

This way a circle packing is obtained on S. Vari-

ous tools for the computation of a Koebe mesh

S are available. As mentioned above, when the

Koebe mesh is obtained it stores all necessary

information from which an endless number of

offset meshes can be obtained. A offset mesh

Md is calculated as Md = M + d.S. Meshes with

edge offsets also have the property that incom-

ing edges form the same angle with the node

axis in the adjacent vertex, which would result

in perfectly aligned beams of constant height

which meet at those nodes. Meshes with edge

offsets would be an ideal base for multi-layered

architectural structures. Unfortunately this kind

of meshes do not allow the approximation of

arbitrary shapes therefore, in most cases they

are not suitable for the task. [Pottmann et al.

2007]

4.3.5.3 Face Offsets

Face offsets are the most interesting type of off-

sets for architectural purposes. They are char-

acterized as conical meshes because all face

planes of such a mesh, which meet at any mesh

vertex, are tangent to a cone of revolution C

in that vertex along the rulings r1,..., r4 (Figure

4.14). The node axis n in the vertex is also the

rotational axis of the cone. The corresponding

planes on which the mesh faces lie in two off-

set meshes M and Md are parallel in this case.

Further, the Gaussian image mesh S has face

n

v

r1

r2

r3

r4

C

Figure 4.14: The faces of a conical mesh that meet in one vertex v are tangent to a cone of revolution C. The

vertex normal n is the cone's axis.

C

S*

S

Figure 4.15: The faces of the planar Gaussian image mesh of a mesh with face offsets (conical mesh) are

tangent to the Unit sphere.

43

Geometric Properties of PQ Meshes

planes, which are tangent to the unit sphere and thus lie at constant distance 1 to the origin of

the unit sphere S* (Figure 4.15). Because the adjacent PQ faces of a vertex s1 in the Gaussian im-

age mesh S are tangent to the unit sphere S* and their planes, which pass through that vertex,

envelope a cone of revolution C1* the faces meeting at that vertex are also tangent to the same

cone. The cone in this case has an axis which passes through the centre of the unit sphere S*. If

we now consider that corresponding face planes in parallel meshes are parallel then we will see

that the face planes meeting in any vertex m1 of any parallel mesh M are also tangent to a cone

of revolution C. All that needs to be done is to translate the cone C1* from the vertex s1 to its cor-

responding vertex m1 to get the cone C1 in vertex m1 of the parallel mesh M. The big advantage

in architectural applications is that the vertex md1 in the mesh Md lies in the same node axis A1

as the vertex m1 of the mesh M. This is the same axis A1 which passes through the origin of the

unit sphere S*. This means that the mesh M has an endless number of parallel offset meshes

Md which lie on any distance d, where all corresponding vertices of the meshes share a com-

mon axis. This makes it easy to create multi-layered architectural structures in which all layers

are parallel and the beams of constant height align perfectly in all nodes, creating torsion free

nodes and an overall aesthetically more pleasing structure. [Pottmann et al. 2007]

Application

There are two things that make parallel meshes with face offsets, conical meshes, accessible

for optimization. The first thing is the so called angle condition. A PQ mesh is conical if the

sum of opposite edge angles in a vertex is equal. This condition makes it easy to optimize a

mesh toward a conical mesh in special software applications. This angle condition says another

thing about conical meshes. It is a "discrete condition for orthogonality of the two mesh polygons

passing through a vertex" [Pottmann et al. 2007]. This brings me to the second thing which is

helpful in practical application. The notion above interprets conical meshes as orthogonal and

conjugate in a discrete sense. In praxis the mesh can be laid out on the surface that one wishes

to approximate, by following the network of principal curvature lines of that surface, ideally.

The mesh can also be laid out based on another network of conjugate curves on the surface,

but the principal curvature lines promise the best results. Such a mesh can then be optimized

toward a conical mesh, much easier than an arbitrary mesh, using the angle condition.

44

4.4 Project: Fair Stand

In this section I am going to present a project that was developed using the knowledge about

PQ meshes from this chapter. This project, the first of the three projects in this work, is also

the most complex one. The term "complexity" has nothing to do with any kind of complicated

concepts, it rather comes from the sheer number of unique flat pieces, all of which have been

generated through scripting, that are assembled into a freeform structure. A project like this

would almost certainly not be possible without the power of scripting. The time it would take

to create the geometry with traditional methods would most probably not be rentable. We will

return to the topic of the importance of scripting is such complex projects at a later point.

For this project, as is the case for the other two projects in this work, I was looking to design a

structure that will fulfil a simple program, so that I could focus my attention on the geometry of

the structures. I choose to design a trade fair stand for an Austrian wood manufacturer who has

experience in building freeform timber structures. The stand should be built by the company

themselves and showcase their know how in woodworking. The stand is planned to be exhib-

ited at the fair Bau 2015 in Munich. The fair Bau is "World's Leading Trade Fair for Architecture,

Materials, Systems" ,as their slogan tells us. The chosen wood manufacturer from Austria exhib-

ited already at Bau 2013. The stand that they used in 2013 was a 6 meters long and thirteen me-

ters wide end stand, where one of the thirteen meter long sides was blocked by a neighbour-

ing stand. I have assumed the same stand type for my project because there is no information

available yet about which stand the company will use at Bau 2015. The project is intended to

give an example of the use of PQ meshes in combination with scripting by creating a system

that is adoptable to different situations. Should the stand's venue, size or type change, it will

be possible to make adjustments to the stand's form while maintaining the same construction

system, by simply reusing the existing script on a different PQ mesh. The stand's design is domi-

nated by a large, curved structure which acts like a partition wall for the whole space. The wall

divides the space into a larger front part that is the main exhibiting area and a smaller back part

which consists of a small store room, for storing furniture and exhibits that are not used at the

moment, and a meeting space right behind the corner of the wall. The meeting space can be

transformed to a space for video projections by moving furniture to and from the neighbour-

ing storage room. The wall is the defining structure of the stand and the central attraction end

exhibition piece. Exhibition space and exhibit are merging into one single structure that flows

along the long axis of the stand, dividing it into public space and semi private space. Except for

the store room, the space never becomes truly private, the stand rather promotes transparency

45

Project: Fair Stand

and openness by dividing spaces with a semi opaque structure. The wall's task, beside creating

the space division inside the stand, is also to attract visitors. Its height ensures that it will be

seen from a relatively large distance, and the unusual design defined with curved lines should

attract the visitor's attention and invite them to come closer. The colour of the wall's outer

layer, the Planar quadrilateral panels made of acrylic glass, changes gradually from white in the

central area to transparent at the ends. This feature should invite visitors to take a closer look

at the structure behind the panels, move around the wall and examine the structural details of

the load bearing wood structure. For the visitors that arrived at the stand's edge or entered it

already, the wall will act as a backdrop to the happening at the main exhibition space in front of

it. The central part of the wall will be used as projection background for videos and information

about the company and its products. The freeform structure, becomes the company's greatest

exhibit that stands large and proud, organizing all that happens around it, while also standing

back in order to allow space for all other exhibits that there might be.

Figure 4.16: Explosion diagram showing the three layers of the

structure

46

Figure 4.17: Top and side views of the fair stand. Scale 1:75

4.4.1 PQ Mesh Creation

After the shape of the stand's wall was designed and before the wood structure could be cre-

ated, the shape needed to be discretized with a PQ mesh first. This particular mesh that I was

creating had, besides resembling the reference surface as closely as possible, to obey the fol-

lowing constraints:

Fairness

The term fairness is used in numerous publications to describe a visually pleasing mesh layout.

The optimization for the fairness property aims on minimizing the kink angle between two

consecutive mesh edges.

Planarity

The mesh has to have planar faces within given tolerances. The faces will be build of six milli-

metre thin acrylic glass which is relatively bendable, so the planarity of the faces doesn't need

to be as precise as it would need to be if the panels where to be built of glass.

Coplanarity

During the optimization process the algorithm moves vertices in space in order to create pla-

nar faces between them. In some cases we need the vertices to obey different criteria as well.

In this case the coplanarity of vertices in certain areas, along the polylines of the mesh which

match up with the two supporting walls behind the structure, is almost as important as the

planarity of the mesh's faces for a successful structure. The walls behind the freeform wall will

be used as anchors for the structure, so the vertices, which represent the nodes of the structure

that will be attached to the walls, had to be arranged in such a way that they match up with the

wall's vertical axis.

The conical property

As it was mentioned before, conical meshes have an infinite number of offsets. The here pre-

sented structure is a multilayer structure that will take advantage of this property of the mesh.

This property is much more important in structures where the load bearing beams follow the

directions of their respective edges, because it allows them to meet precisely at the normal

axes of the mesh's vertices. In this case a conical mesh will result in nicer joints in the load bear-

ing structure. The angles at which the beams in the reciprocal structure meet each other would

deviate more from the optimal 90 degrees angle if the mesh wasn't optimized to be conical,

but the joints would be formed nevertheless. This is the reason why I will set less importance

48

Figure 4.18: Step by step creation of the PQ mesh (a) the reference surface, (b) the coarse mesh, (c) optimized mesh after one step of subdivision, (d)

optimized mesh after two steps of subdivision, (e) two groups of vertices set to become planar to reference planes during the optimization, (f ) the

final PQ mesh

(a)

(b)

(c)

(d)

(e)

(f )

on this property than on the above mentioned

ones during the mesh optimization.

Creating the mesh

With the designed surface set as reference in Ev-

oluteTools for Rhino, the first step towards the

discretization with a PQ mesh was the creation

of a coarse mesh that very roughly represents

the reference surface (Figure 4.18 a ). This step

is done manually by creating a simple planar

mesh in the front viewport and adjusting the

positions of its vertices until it takes an overall

satisfying shape (Figure 4.18 b ). The creation of

the coarse mesh is in no way a very precise or

scientific matter. It is rather the stage of the de-

sign where the designer has the most creative

freedom. The number of faces, their size and

aspect ratio will directly influence the outcome

of the whole creation process of the PQ mesh

more than any other step. In the next step (Fig-

ure 4.18 c ), the coarse mesh is subdivided with

the Catmull-Clark algorithm and optimized for

surface and curve closeness, fairness, planarity

and conical mesh. The optimization step moves

the vertices closer to the reference surface, and

the mesh starts looking more like the intended

design, the reference surface. After another

repetition of subdivision with the same algo-

rithm and optimization, the number of faces

quadruples and the mesh's vertices move even

closer to the reference surface (Figure 4.18 d ).

After this step, we can observe how the num-

ber, the size and the aspect ratio of the coarse

mesh's faces influenced the current outcome. It

49

Project: Fair Stand

is possible to fine tune the mesh at this point by adding or removing polylines of vertices from

or to it, but it wasn't necessary in this case since the mesh's overall appearance was satisfying.

When the mesh was almost finished, coplanarity planes in the areas where the supporting

walls would stand had to be inserted and the respective closest vertical polylines of vertices

set to become coplanar[1] to those planes with the next optimization (Figure 4.18 e ). In a final

step of optimization those vertices will move as close as possible to the given planes. The op-

timization algorithm now has to focus besides on the mesh's closeness, fairness, planarity and

conical property also on the coplanarity of the specified vertices. It had to calculate the optimal

positions of all vertices in the mesh that respects all given optimization parameters, while re-

sembling the reference surface as close as possible (Figure 4.18 f ).

1 Geometric objects such as points, lines, and curves are called co-planar if they are contained in the same plane. [Pottmann et al. 2008 p. 713]

50

4.4.2 The Scripting Process

In the introduction to this chapter I stated that the actual structure with all its members is gen-

erated by a script and that without the power of scripting it would not be possible to create

such a complex structure at all. All individual pieces, screws and holes in the structure, as well

as the annotations and production data have been generated within RhinoScript with aid of

the EvoluteTools Scripting Interface for RhinoScript. Evolute's Scripting Interface helps to ac-

cess the mesh's halfedge data structure very precisely. The halfedge data structure can be used

to navigate through the mesh's faces, edges and vertices and to do very precise operations on

those parts. The scripting method has been chosen above more popular tools that provide real

time feedback to every action that is made because of the amount of programming that needs

to be done, which would result in very large and complicated definitions, and because of the

convenience of using the additional commands from the EvoluteTools Scripting Interface that

are not available outside of RhinoScript. In this section I will explain the functionality of the

script without going into details of programming nature, but rather focusing on the logic that

led to its creation, in order to provide the reader with insight to how the structure functions

and how all parts interact with each other.

The construction system that I designed for this project was heavily inspired by the KREOD pa-

vilion for the 2012 Olympic games in London by Pavilion Architecture for which Evolute ration-

alized its complex geometry, designed the panel layout, provided parametric detailing of the

Figure 4.19: Image showing KREOD's geometry and the interlocking reciprocal nodes.

51

Project: Fair Stand

Figure 4.20: Image showing the PQ mesh and the three structural layers based on it. (a) PQ mesh, (b)

inner layer, (c) middle layer, (d) outer layer

(d)

(c)

(b)

(a)

wooden members and production geometry

for fabrication. My goal, inspired by Evolute's

part in the design of KREOD was to automate

the creation process and the process of output-

ting production data.

The construction system of KREOD is based on

a hexagonal mesh and it features nodes where

three wooden members meet to form recipro-

cal nodes. Inside the pavilion there are mem-

branes that are connected to the load bearing

structure in the node points. This makes the

KREOD a double layered structure.

The here presented structure, based on the PQ

mesh that was created previously (Figure 4.20

a) features a reciprocal load bearing structure

on the inside (Figure 4.20 b ) and a panelization

of planar quadrilateral panels on the outside

(Figure 4.20 d )W, with a third layer, which re-

sembles the edges of the mesh, as connection

between the outer two layers (Figure 4.20 c ).

On the following pages the creation process of

each piece in this structure will be described.

52

4.4.2.1 Scripting the Outer Layer

I will start this section with a description of the scripting of the outer layer (Figure 4.20 d ) be-

cause it is the simplest to describe and because things get more and more complicated as we

work our way to the reciprocal structure which forms the inner layer of the wall. The outer layer

consists of a collection of panels that are derived directly from the geometry of the mesh faces,

therefore the panels follow the shape of the faces. This means that flat faces, with all adjacent

vertices lying on a plane or as close as possible to a plane, are translated into flat panels. The

script loops through all mesh faces performing the following actions on all faces one after an-

other. Using the halfedge data structure, the adjacent vertices (v1, v2, v3, v4) and edges (e1, e2, e3,

e4) of the current Face Fn are found and stored. For each vertex the respective normal is calculat-

ed (n1, n2, n3, n4). The four vertex normals are used to calculate one general normal n for the cur-

rent face Fn. The normal n is obtained by adding the four vertex normals and unifying the result.

The normal n will be used to extrude the panel's outline to its intended thickness. Between the

panels, there is a gap supposed to be that makes it impossible to just use the corner vertices to

create the panels. If the corner vertices would be used as corners of the panels then the panels

would touch each other in convex areas of the structure, thereby making any movement in the

structure as wood shrinks and expands impossible. Another even bigger problem are panels in

concave areas which would have to intersect each other in order to assume the desired posi-

tion. Therefore the panel needs to be smaller than the original face. This is achieved by specify-

ing a gap between the panels which is set to five millimetres in this case, but can be adjusted

Figure 4.21: The creation of a panel in a mesh face

d

v1

n1

np

p1

Detail BDetail A

p2

n4

n2n3

v4

v3

v2

de2

e1

e4Fn

m

e3

53

Project: Fair Stand

as one of the two parameters in this script, the other parameter being the panel thickness. The

necessary gap between the panels was achieved by offsetting the adjacent edges towards the

midpoint m of the face. The point m is calculated as the mean value of the coordinates of the

four adjacent vertices. The actual mesh edges are not affected by this process. They are first

redrawn as curves that connect the respective edge's end vertices, and then the curves are

used for the offset. Figure 4.21 - Detali A shows that the edge is offset towards the midpoints of

both adjacent faces. The value d is half of the gap size, 2,5 mm in the case of this structure. After

all four adjacent edges have been offset towards the middle of the current face they should

intersect each other at their ends and form a frame which makes the back face of a panel in

the current face. Unfortunately, the above is only true for completely flat faces, because only if

the edges of the face lie on one plane will their offsets towards the face's midpoint intersect.

In all other cases, when the face is not perfectly flat, which is probably the case for all faces in

a PQ mesh, the offset curves will not intersect but lie slightly above each other. In this case it is

possible to calculate an apparent intersection and return two points, one on each curve, that

are closest to each other (Figure 4.21 - Detail B - p1 and p2). The points p1 and p2 are used to

calculate their mean value p which is arguably the best solution for this problem. This process

is repeated in all four corners of the face and the obtained intersection points are used as the

projection of the panel's corners on the mesh. They are moved two millimetres away from the

mesh along the face normal n to become the panel's corner points on its back face. The panel is

moved away from the mesh to make room for the metal clamps that hold it in place[1]. The four

back face corner points are moved along the face normal n to create four new corner points on

the front face of the panel. The length of the movement along the normal is calculated by scal-

ing the normal with the parameter that specifies the panel's thickness. The eight newly created

points are used to created six surfaces between them. In a final step the surfaces are joined into

a panel.

1 The clamps are all the same, so there was no need to include them into the script. They are visible on the model photographs.

54

4.4.2.2 Scripting the Middle Layer

Introduction

While the panels of the outer layer were created

by an alone standing script which is relatively

simple, the middle layer and the inner layer are

created by one script. This has to do with the

fact that those two structural layers are much

stronger interconnected and they also depend

on each other in a structural sense, while the

panels of the outer layer are self standing and

do not influence the stability of the structure.

The middle layer consists of three groups of

elements, the first group being the wooden

beams that follow the mesh edges (Figure 4.22

a ). Underneath the wooden beams there is a

metal cross in each vertex (Figure 4.22 b) that

holds the four adjacent beams and connects

them to the third group of elements, the metal

cylinders (Figure 4.22 c). The cylinders act as a

connection between the middle layer and the

inner layer so they technically belong to both

layers and they will be mentioned again later

in the description of the inner layer. The cylin-

ders are simple metal rods, six millimetres in di-

ameter, that feature tapped holes for machine

screws on both sides. They are standardized

pieces, all of which have the same dimensions,

therefore there is no need to detail their design

with the script, but it is important to have sim-

ple cylinders in the model since they are used

to make holes in their adjacent pieces.

For the creation of the wooden beams it is

necessary to compute three vectors first. The

vectors will be used to move the points in the

Figure 4.22: Three groups of elements that comprise the middle layer of the structure. (a) the wood beams,

(b) the metal crosses, (c) the metal cylinders; (mesh edges represented in red for reference)

(a)

(b)

(c)

55

Project: Fair Stand

beam's adjacent vertices to the positions of the eight corner points of the initial beam (the

beams are first created as simple boxes and later trimmed at their ends). The detail A of Figure

4.23 shows how the three vectors are used to move the point from the position of the vertex to

the corners of the beam.

Calculating the movement vectors

Similar to what has been done for the computation of the face normal vector in the previous

section, for each beam there is one beam normal n (Figure 4.23). The beam's normal vector is

calculated by unifying the sum of the normals n1 and n2 in the beam's adjacent vertices v1 and

v2. This vector will be used to move the beam's end points in order to calculate its depth. If the

points would be moved on different vectors on both sides of the mesh edge, this would lead

to the creation of a twisted element. It is therefore necessary to use one normal vector for the

edge instead of using the vertex normal vector on each end of the edge to prevent twisting in

the beam completely. The underlying mesh is optimized so that it is conical, meaning that the

deviation between the adjacent vertex normals will be minimal. If there is a small amount of

twisting left in the beam, the material should absorb it during the assembly. Through calculat-

ing one average normal for the edge from its vertex normals I assume that the difference be-

tween the two vertex normals, and thereby the twisting in the beam, are within tolerances and

can be ignored, thus the beam can be produced out of flat material as a flat object.

The second vector that needs to be calculated for each beam is the beam's direction vector ed.

Figure 4.23: The creation of a beam over a mesh edge

v1n1

n = n1 + n2

ex = ed × n

n2

v2

v2

v2,1

v2,2

v2,2,1

v2,1,1

e

Detail Aed

-n

-nex

-ex

56

This is done by subtracting the coordinates of the adjacent vertices v2 and v1. The third vector ex

is the cross product of the edge's direction vector ed and the edge's normal vector n.

Creating the beam

The cross product vector ex is used to move the points in the edge's adjacent vertices away from

the edge to both sides, while forming the corners of a flat front face of the beam (Figure 4.23

Detail A - v2,1 and v2,2). The newly obtained points are then moved along the edge's normal vec-

tor n to obtain corner points on the back side of the beam - v2,1,1 and v2,2,1. After the vectors have

been used to determine the locations of the beam's corner points, the beam is created from six

surfaces the same way that the panels were created.

Trimming the ends of the beam

The ends of the beam are then trimmed with four surfaces, so that the beams would not inter-

sect at the connection points. The trimming surface is the extrusion along the normal vector

n of a line that connects one end vertex to the midpoint of one adjacent face. The trimming

surface is moved two millimetres along the edge's direction vector towards its middle prior to

the actual trimming to create a small space between the beams.

Creating the metal crosses

The first task in order to define the location of the metal cross was to determine whether the

current vertex is in a convex or in a concave region of the mesh. Based on that information

there are two different distance values from which one is chosen for each cross. The crosses

need to be created inside the volumes of the adjacent beams so that they intersect them and

thereby can be produced by cutting from a flat sheet of metal. The distance of the cross' base

point and the plane on which its back face lies in a convex vertex is the same as the thickness

of the beams. This means that the cross lies with its back face on the back faces of the adjacent

beams in the area close to the vertex normal and enters deeper into the beams as the distance

from the vertex normal increases. In concave areas, where the beams are oriented in the op-

posite direction, the distance between a beam and the crosses base plane increases with the

distance from the vertex normal, therefore it is necessary to place the cross' base plane closer to

the vertex initially so that at its ends the cross would still be partly submerged into the adjacent

beams. In this case the cross' base plane has been placed at a distance from the vertex that is

two thirds of the beam's thickness. After the distance is determined by the mesh's convexity or

concavity in the current vertex, the vertex point of the current vertex is moved back along the

vertex normal for that amount. The newly obtained point bp is used as base for the cross' base

plane m with the vertex normal being also the normal of that plane (Figure 4.24 a). The base

57

Project: Fair Stand

n

v

m

n

v

m

bp

n

v

ma1,1

d1,1

e

fg

k

h

d1,2a1,2

b1,1

c1,1

c1,2b1,2

n

v

m

bp

a

d

cb

a1

d1

c1b1

(a)

(b)

(c)

(d)

Figure 4.24: The creation of the metal crosses in the structure's nodes.

e1

e2

e3

e4

point bp is then moved along the edge vectors

of the adjacent edges, four times in four differ-

ent directions, thereby obtaining four points a,

b, c, d (Figure 4.24 b). The length of the move-

ment vector is controlled with a parameter in

the script. It directly influences the size of the

crosses by defining the distance of its ends to

the centre. Depending on whether the current

vertex v is in a convex or a concave area the

points a, b, c, d will be either behind or in front

of the base plane m. In any event, the points are

highly unlikely to be on the base surface, so the

points a1, b1, c1, d1 have to be calculated by pro-

jecting the points a, b, c, d on the base surface

m (Figure 4.24 b). Further the points a1, b1, c1, d1

are moved so that for each of the four points on

the ends of the cross two points on the cross'

corners are calculated. Here the halfedge data

structure of the mesh is very useful because

it makes it possible to precisely control which

point is moved by which vector. For example,

the point a1 was calculated by moving the base

point bp along the direction vector of the edge

e1 and projecting it onto the base plane m. The

point a1 is now duplicated to two sides using

two different vectors. The point a1,1 is obtained

by moving point a1 along the direction vector

of the edge e4, while the point a1,2 is calculat-

ed by moving the point a1 along the direction

vector of edge e2 (Figure 4.24 c). The length of

the movement vector is one half of the above

beam's width. Of course the points a1,1 and

a1,2 will leave the base plane by this move-

58

e2

b2

n

e1

b1

(a)

e2

b2

n

e1

b1

(b)

e2

b2

n

e1

b1

(c)

cross

C

cross

Figure 4.25: (a) and (b) Intersecting the beams with the cross, (c) intersecting the beams and the cross with a

cylinder

ment again and need to be projected back

onto it in order to obtain their final positions.

The four remaining points e, f, g, h (Figure 4.24

c) are calculated by moving the corner points

back towards the middle along their respective

edges and calculating the average value of two

points. For example: The point f is calculated by

moving the point a1,1 back along the direction

vector of edge e1 and by moving the point d1,2

back along the direction vector of edge e4. The

two points are projected back onto the base

plane because they probably left it during the

translation and finally the point f is calculated

as the mean value of the two projected points.

After all points are in place, they are connected

by a closed polyline k which forms the outline

of the cross and lies flat on the base surface. The

polyline is extruded along the vertex normal

vector n, thereby creating a closed solid object.

Finally, a cylinder which uses the normal n as

axis is used to make a hole in centre of the cross

and to cut away one part of the beams where

the screw that attaches to the cylinder will be

hidden.

59

Project: Fair Stand

4.4.2.3 Scripting the Inner Layer

Introduction

While the middle layer provides stability to the structure by connecting adjacent nodes di-

rectly to each other via thin wooden beams, thereby creating a raster of vertical and horizontal

beams, the inner layer of structural elements is actually the load bearing structure. It is made up

of 100 × 20 millimetres spruce beams of varying length, that form reciprocal connections in the

place of the vertices of the original mesh. A typical load bearing structure made of steel would

feature beams that lie directly behind the mesh edges and by following the directions of their

respective edges would join directly behind the vertices in their normals.

The term reciprocal connection is used to describe a type of connection between wooden

members which is made up by beams that are rotated away from the direction of their respec-

tive edges and instead of meeting in one node on the vertices normal, they interlock into each

other to form reciprocal nodes. If we tried to connect four wood beams in the vertices' nodes

we would need to manufacture a complicated metal connection for each node. Metal connec-

tions would increase the price, manufacturing time and weight of the structure.

Reciprocal nodes make complicated metal connections in the nodes obsolete by relying com-

pletely on the wood members to form stable connections on their own. Each beam is manu-

factured with pockets so that it fits perfectly to its adjacent beams and so that it has its unique

place in the structure. Further, the beams are numbered as they are being created, each beam

featuring its own name and the names of its adjacent beams on the respective ends that are

inserted into them. This, together with the fact that there is only one possible way of connect-

ing the beams makes the assembly relatively simple and fast.

Initially I wrote a script that would create a reciprocal structure based on what we saw on the

KREOD pavilions. The nodes of the KREOD pavilion are made up of three beams that are con-

nected to each other with screws. Each beam had to be manufactured with a total of eight

holes, two on the front and two on the back side and four on the sides, to fit the screws. Fur-

ther four pockets had to be milled into the sides of the beams to fit special fasteners for the

screws. There were a total of six screws and six fasteners in each node of the structure, which

translated into eight screws and eight fasteners in the here discussed structure, since it features

four instead of three beams meeting in each node. Two wood boards were inserted between

the four beams initially to hold the cylinder that connects the inner structural layer to the mid

layer. With the introduction of those "node-boards" and trough connecting the two inner layers

the whole structure became an intricate assembly in which all parts interact and communicate

with each other. Once the pieces were in place they would stay there because the middle layer

60

would prevent movement in the inner layer

and vice versa. After this "happy accident" was

discovered it became clear it would be possi-

ble to create a structure that is much simpler in

terms of detailing, manufacturing and assem-

bly than the one I had in mind. All screws in the

beams of the inner layer were eliminated and

so were the numerous holes that would have

to be milled in every beam to make room for

glued connections. The two node-boards were

replaced by one thicker element that would

take exactly the same space and position in the

structure, but eliminating the space between

the boards. The thicker box has a larger surface

contact with the adjacent beams and hence it

provides a more stable connection then two

separate boards. The boards are now in the fi-

nal version of the structure glued to each other,

each beam with its front and end to the sides

of its adjacent beams, and to the node box be-

tween them. By comparing the drawings in Fig-

ure 4.26 and the renderings in Figure 4.27 and

Figure 4.28 we can observe the reduction of

individual pieces and the simplification of the

geometry of the beams caused by the switch

from screws to glue as primary fastener. The im-

ages show all pieces that are necessary to build

one node of the structure.Figure 4.26: The connection detail of the reciprocal structure becomes much simpler when the screws

(above) are replaced with glue (below).

61

Project: Fair Stand

Figure 4.27: Image showing the pieces that are necessary to assemble one node of the reciprocal structure us-ing screws as fasteners

Figure 4.28: Image showing the pieces that are necessary to assemble one node of the reciprocal structure us-ing glue as fastener

62

Calculating the movement vectors

The movement vectors are very simply obtained in this case. Just as before, there are the vertex

normal vectors n1 and n2 for every edge (Figure 4.29 a ), and an edge vector n that is calculated

by adding n1 and n2. The end points of edge e1 are again the points in the vertices v1 and v2.

Offsetting and rotating the edges

The points in v1 and v2 are translated along their vertices' respective normals to create the

desired offset distance between the mesh and the reciprocal structure (points v1,1 and v2,1). The

offset curve e1,1 of e1 is created as a curve between v1,1 and v2,1. Consecutively, the edge e1,1 is

rotated by twenty degrees (α) to form the curve e1,2. The angle of twenty degrees has been

chosen through experimenting with different values. It might be that another reference mesh

would require another rotation angle at this point. Therefore, the rotation angle, along with the

offset difference, can be changed using a parameter at the top of the script. The curve e1,2 that

we now created will be the top axis of a beam that will be created underneath it.

Creating the beam

Using the halfedge data structure, the edges in front and behind the curve e1,2 are found (Figure

4.29 b - edges e2 and e3). The same process of offsetting and rotating is then repeated for the

edges e2 and e3 and curves e2,2 and e3,2 are found. Through intersecting curve e1,2 with its adjacent

curves the beam's end points p1 and p2 are found. Unlike before, when we calculated the average

point of the two points that are the result of a line - line intersection when the lines do not

intersect actually, this time the point that lies on the edge that the script currently works with

(e1) is chosen while the other one is ignored. The points p1 and p2 are moved on the edge normal

n, the edge's direction vector (p2 - p1) and the cross product vector of the former two to locate

the corner points of the beam B. This method was described in detail in the section "Scripting

the Middle Layer" on page 55. After a simple solid box is created for the beam, the adjacent

curves are extruded along their edge normals to create trimming surfaces. The surfaces are

moved one fifth of the beam's width away from the centre and towards the newly created

beam. The surfaces are used to trim off the ends of the beam that are eventually penetrating

too deep into the adjacent beams. They have been moved away from their respective beams'

centre axis to make sure that the beam B would penetrate to less than half of the adjacent beam.

Afterwards, adjacent beams and node boxes are used to create pockets to fit these elements

with boolean differences. The current beam B will also be used for boolean differences when

one of its adjacent beams is created. Therefore, the script needs to keep track of all elements

that it creates. After any element of the structure, not only the beams that are discussed right

63

Project: Fair Stand

now, is created, the script writes user data to

the mesh, storing the information about the

current element. It uses the names of mesh

elements to store the pieces of the structure.

Panels of the outer structure are stored with

the names of their mesh faces, whereas beams

such in this case are stored with a prefix rec

for reciprocal structure and the number of

the adjacent edge. When the script needs to

perform Boolean operations on one element

such as the beam B in Figure 4.29 - c, it first

looks for the names of the adjacent elements

in the mesh's user data. If the adjacent beam

or box already exists it is used for the boolean

operation, or else it is created specially for this

purpose. Figure 4.29 - d shows a part of the

complete reciprocal structure where all parts fit

perfectly into each other.

n

n

n

n

n1

n1

n1

n1

v1

v1

v1

v1e1

e1

e2

e2,2

e3,2e3

e1,1

e1,2

e1,2

B

B1

v1,1

v2,1

v2

v2

v2

v2

rα

α

n2

n2

n2

n2

(d)

(c)

(b)

(a)

Figure 4.29: The creation of the beams and connec-tions in of the reciprocal structure.

p1

p2

64

4.4.3 The Physical Model

The structure that was developed in the previous sections needed to be tested once it was

defined. The question that needed to be answered the most was if it was possible to assemble

the parts the way it was conceived. To prove that it was possible and that the structure will be

rigid enough to carry its own weight I created a 1:2 scale model of a part of the structure. A 1:1

model would surely be better to show one or two connections, but I choose the smaller scale

to be able to create a model of more than just one or two nodes. The model's size was limited

to a piece of the mesh from the Stand that was five faces wide and four faces high. The model

therefore covers twenty faces and twelve nodes of the structure which is enough to prove the

concept. The model also shows that wood as constructive material is very flexible in terms of

allowing for rather large tolerances in the structure. The parts of the model were produced on

a 3-axis milling machine. This meant that the pockets on the sides of the beams, where the

adjacent beams should fit in are always milled in a 90 degrees angle to the beam and the co-

ordinate system of the machine, instead of being aligned with the angle at which the adjacent

beam would later be, which would require the milling tool to rotate. Even with those quite large

shortcomings in the production, the pieces could be assembled precisely

Figure 4.30: Detail of the 1:2 scale model

65

Project: Fair Stand

Figure 4.31: Front side image of the physical model. Only three of the twenty faces are represented in the model as proof of the faces' planarity. The other faces were left out to expose the substructure.

Figure 4.32: Back side image of the physical model. Thanks to the glue that holds the parts together instead of screws, there are almost no disturbing metal pieces visible from the inside. This view shows how the structure follows the curvature of the reference surface and the PQ mesh smoothly.

5. Developable Surfaces

and DStrips

Figure 5.1: Form study for the DStrip model in section 5.4. Top view.

5. Developable Surfaces and DStrips

It is out of question that a freeform architectural shape, due to its size, needs to be subdivided

into smaller parts in order to be produced. One way of achieving this is by manufacturing dou-

ble curved surfaces which follow the surfaces curvature perfectly. This approach has been used

for the Entrance of the Metro station Gare Saint Lazare in Paris by Arte Charpentier with the

technical aid of RFR, which was completed in 2003. While this method results in the best and

the most aesthetically pleasing approximation of the desired reference surface it is also cer-

tainly the most expensive method. In most cases the cost factor of such an approach will not

allow the structure to be realized this way [Vaudeville et al. 2012].

Gehry Technologies used a different approach for the glass "sails" of the Fondation Louis Vuitton

pour la création building in Paris by Frank Gehry (Figure 5.2). They optimized the glass panels

for bending on a bending and tempering machine which produces glass panels in the shape of

circular cylinders. The advantage of using this type of machine is that the panels can be bend

at an angle relative to the axis of the machine. In other words the direction of curvature with

respect to the panel's edges can be adjusted. With the help of large scale prototypes, Gehry

Technologies managed to produce an aesthetically pleasing structure, which is not far away

from the intended design, by hot bending cylindrical glass panels and by applying a minimal

degree of cold bending to the panels [Vaudeville et al. 2012].

Another option is the discretization of the surface with flat panels. Besides the discrete segmen-

tation of freeform shapes with triangular-, quadrilateral-, or hexagonal panels, or a combination

of the three types, it is also possible to subdivide surfaces in a semi-discrete fashion, namely

the segmentation into single curved panels [Schiftner et al. 2008]. The semi-discrete models are

surface parametrizations with a continuous and a discrete parameter and they represent a link

between smooth surfaces and discrete surfaces.[Pottmann et al. 2008].

The rationalization with single curved panels is especially an attractive solution in wooden con-

structions. It enables the production of curved panels with simple manufacturing techniques.

The wooden panels can be cut from flat sheet material with the aid of CNC machines and cold

bent into the desired shape. The same applies for metal, but the manufacturing of such surfaces

in glass is more complicated, although still more efficient than the solution with double curved

panels for which a custom mold needs to be manufactured for every panel. Another advantage

73

Developable Surfaces and DStrips

of single curved panels is that offsets of devel-

opable surfaces are developable as well. This

allows the engineers to use developable box

beams as mullions which allows adopting a sin-

gle continuous detail for fixing the panels. This

application was successfully used by [Schiftner

et al. 2012] in the glass facade for the Eiffel Tow-

er Pavilions (Figure 5.3).

5.1 Developability

Developable surfaces are also known as sin-

gle curved surfaces, meaning that they carry

one family of straight curves, the rulings, mak-

ing them ruled surfaces. Developable surfaces,

by definition are ruled surfaces, although not

all ruled surfaces are developable. Develop-

able surfaces can be unrolled onto a flat plane

so that the in-surface distances remain un-

changed. The rulings of such surfaces have the

special property that all points of a ruling have

the same tangent plane. Some well known de-

velopable surfaces are the cone or the cylinder.

Translational surfaces in which one of the two

generating curves is straight, called extrusional

surfaces are also developable, because the ex-

trusion direction of such surfaces consists of

straight, parallel lines. Another explanation of

developable surfaces is that they are ruled sur-

faces with vanishing Gaussian curvature where

K equals zero at all of their points. Developable

surfaces have the property that they can be

Figure 5.2: The Fondation Louis Vuitton pour la créa-tion building in Paris by Frank Gehry

Figure 5.3: Developable box beams are used as mul-lions for the facade of the Eiffel Tower Pavilions

74

Figure 5.4: Developable paper strips. Surfaces that are modelled with paper by applying pure bending are developable

mapped isometrically into the plane [Pottmann et al. 2007]. Since isometric mapping preserves

the Gaussian curvature, a developable surface has the same Gaussian curvature (K=0) as the

plane, which means that in each surface point at least one of the principal curvature lines is a

straight line with a curvature value of zero. All surfaces which can be modelled from a sheet of

paper, no matter the shape of the sheet, without tearing or stretching the paper, are develop-

able surfaces. This statement is obvious because if the paper is not teared or stretched, it can be

unrolled into its original flat shape and thus the modelled surface is unrollable or developable

(Figure 5.4).

5.2 DStrip Models

Let's assume a thin strip of paper which is not as flexible as the paper in Figure 5.4. For illustra-

tion purposes let the strip be 2 cm wide and 20 cm long. The strip is folded along lines that are

parallel to the shorter edges which are all 2 cm apart. After the folding a strip which consists of

ten planar quadrilateral faces is obtained. The paper strip can easily be bent around the edges

to different shapes, but it can also be unrolled back into the flat state. The strip with ten PQ

faces is equivalent to a coarse mesh with ten faces. If we would take the strip and start adding

folds between the existing ones we would refine the strip into a smoother developable strip.

DStrip models are meshes which consist of such strips of PQ faces. These models are obtained

as limits of PQ meshes under a refinement which operates only on the rows while leaving the

columns unchanged [Schiftner et al. 2008]. An iterative process of subdivision and optimization

for planarity of the quad faces is necessary since the subdivision of a planar quad face does not

75

DStrip Models

(a)

(b)

Left Front

(c)

(d)

Figure 5.5: The connection between the network of principal curvature lines, conical meshes and DStrip

models.

guarantee that the resulting faces will remain

planar. There is a strong connection between

PQ meshes, which were discussed in chapter "4.

PQ Meshes", and DStrip models. Especially coni-

cal PQ meshes which are based on the network

of principal curvature lines of a surface are well

suited as base for a DStrip model.

5.2.1 Principal Strip Models

The best results, when working with DStrip

models on a freeform surface, can be expect-

ed if the mesh is laid out so that the edges of

the strips follow the principal curvature lines

of maximum curvature on the surface and the

rulings are placed so that they follow the other

principal curvature [Schiftner et al. 2008]. In

praxis this can be achieved by creating a coarse

mesh which approximates the reference sur-

face and follows the principal curvature lines.

The mesh is then optimized for planarity and

the conical property. The connection between

the network of principal curvature lines and PQ

meshes has been mentioned in chapter "4.3.5.3

Face Offsets" on page 43. Figure 5.5 shows a

reference surface (a) with its network of princi-

pal curvature lines extracted (b). The optimized

conical PQ mesh is aligned with the principal

curvature lines (c) which results in a smooth

DStrip model (d).

76

5.2.2 DStrips Between Two Curves

Developable strips can also be obtained by cre-

ating them between two curves. The curves can

be extracted from a reference surface, for in-

stance by intersecting the surface with a set of

planar surfaces. This approach is helpful where

it is applicable, because the fact that the edges

of the strips are straight lines simplifies the sub-

structure to a great extend. When the two bor-

der curves are obtained (Figure 5.6 - a), a sim-

ple coarse mesh between them is created (b).

The mesh is optimized for planarity of the faces

and closeness to the reference curves. The op-

timized mesh is further subdivided in only one

direction to create density in the rulings of the

strip. The planarity of the faces is not preserved

during the subdivision, meaning that the mesh

needs to be optimized at least once after the

chosen number of subdivision iterations. There

is a possibility to track changes in the mesh via

an analysis mode which represents values such

as planarity, scale invariant planarity and close-

ness to the reference with colours in the view-

port. The colour coding in Figure 5.6 shows the

planarity of the mesh faces (b and c) and the

distance of the vertices to the reference curves

(d). Finally when a good mesh strip is obtained,

its rulings can be extracted and used to create

a NURBS surface that resembles the DStrip. The

method of creating DStrips manually, while giv-

ing the maximum possible control to the user,

(a)

(b)

(c)

(d)

(e)

Figure 5.6: Manual creation of a DStrip between two input curves.

77

DStrip Models

(a)

(b)

(c)

(d)

(e)

Figure 5.7: The creation of a DStrip between two input curves by the usage of the automated method.

has at least two drawbacks. The first drawback

is that it is time consuming due to the amount

of manual work it takes to create the coarse

mesh for each DStrip. The second drawback is

that the first and last ruling of the strip are con-

nections between the end points of the curves.

This is not ideal because it limits the freedom of

the optimization algorithm to adjust the posi-

tion of vertices. The optimization algorithm will

tend to shrink the mesh because of the optimi-

zation for fairness, but the fairness parameter

is very important because it keeps the angles

between two consecutive edges in the mesh

as straight as possible. Omitting the fairness

parameter would result in a strip with rough

edges that does not resemble a smooth sur-

face. To prevent the shrinking of the mesh strip

the four corners are fixed into their positions at

the ends of the curves, which excludes those

vertices from the optimization and limits the

movement of their neighbours. All the above

results in a less accurate DStrip with less good

planarity values and therefore a DStrip that is

less suitable for unrolling. The rulings of most

DStrips will naturally distort and enter the strip

at a certain angle which is not given by this

method. Figure 5.7 shows another method that

is automated through a script and overcomes

both problems of the previous, manual, meth-

od. The reference curves (a) are duplicated and

78

the copies extended while the original curves are kept

and set as reference for the optimization algorithm (b).

The coarse mesh is completely omitted in this method

and a dense mesh is created between the two longer

curves which were obtained earlier (c). The mesh is then

optimized to the original reference curves. The mesh has

now the freedom to shrink during the optimization and

achieve a generally better developable surface between

the two input curves (d). In the final step of the automat-

ed process the rulings of the mesh DStrip are lofted to

create a developable NURBS surface and the overlapping

ends of the strip are trimmed away (e).

79

DStrip Models

5.3 DStrip Studies

The following two facts were established previously:

1. A DStrip is computed as a PQ mesh with only one face in one direction and many faces

along its opposite direction.

2. PQ meshes do not have absolutely planar faces.

Only if a DStrip mesh would be absolutely planar, the strip would be absolutely developable.

In all other cases one has to search for a developable strip that is as close as possible to the

optimal developable strip, meaning that the PQ mesh will be optimized for the best possible

planarity result while remaining as close as possible to the reference curves. A PQ mesh strip

with better planarity values will be more likely developable then a strip with less good planar-

ity values. It is clear that it is unlikely that a perfect developable strip will be achieved, however

certain tolerances in materials and connecting elements will make it possible to build a not per-

fectly developable surface from flat material nevertheless. There is no specific planarity value

or any other value, that tells if a mesh strip is developable or not. Perfectly developable strips

have the same surface area both in their 3D state and in their unrolled state. It is possible to

unroll not perfectly developable strips such as the strips that are created in this work, but the

difference in surface area between the two surfaces increases as the strip becomes less devel-

opable. The aim of the PQ mesh optimization is therefore to create a strip with minimal possible

surface area difference in its 3D and its unrolled state. Unfortunately, there are no parameters

which would tell which surface area differences are within acceptable tolerances and which are

not. In order to gain insight into the behaviour of DStrips created with the method described in

section 5.2.2 I conducted the following studies.

5.3.1 Input Parameters

Initial face aspect ratio

Figure 5.7 -c in section 5.2.2 showed how a mesh strip is created between two reference curves.

The faces of that mesh have an aspect ratio which is parametrically specified in the script that

automates the process of DStrip creation. While it is the assumption that denser strips (strips

with more faces of smaller aspect ratio) will provide better results in terms of planarity and curve

closeness, simpler or less dense meshes will perform better, especially in large projects. The

DStrip studies involve DStrips with different initial face aspect ratios. The DStrips are grouped

80

according to their initial face aspect ratio in five groups — 0.3, 0.2, 0.15, 0.1 and 0,05.

Planarity

In each group of DStrips there is a number of strips with different planarity values, each strip

being an improvement to the previous strip. The first strip in every group is the strip that is

created between the reference curves without any optimization, whereas the last strip in the

group is an evolution of the first strip that is optimized towards a DStrip as good as possible.

5.3.2 Evaluation Parameters

MaxDistance

The maximum distance between a vertex of the PQ mesh and its closer curve. This value is cal-

culated so that the vertices located outside of the DStrip, i.e. vertices whose closest curve point

is an endpoint on one of the two reference curves, are ignored.

AverageDistance

The average distance between the vertices of the PQ mesh and their respective reference

curves. This value is calculated so that the vertices located outside of the DStrip, i.e. vertices

whose closest curve point is an endpoint on one of the two reference curves, are ignored.

Strip area

The surface area of the DStrip before unrolling.

Area of unrolled strip

The surface area of the DStrip after unrolling.

Area difference after unrolling

The surface area difference between the DStrip in its original state and its unrolled state, ex-

pressed in square centimetres and in percent.

5.3.3 Test With Developable Reference Surface

The first series of tests were done with reference curves that are the edge curves of a perfect-

ly developable strip, meaning that an existing developable surface is recreated with DStrips.

Working with an existing developable surface and its edge curves instead of using two arbitrary

curves in space as references has the advantage that there is a developable reference surface

81

DStrip Studies

against the created DStrips can be compared.

Being that a DStrip is created between refer-

ence curves that are borders of a developable

surface it can be expected that it is possible to

create an almost perfect DStrip between those

curves and that the values that are obtained in

this test will not be matched in practice. Never-

theless, the possibility to compare the DStrips

in their original state and after they have been

unrolled to the reference surface which is by

definition developable makes it worth to con-

duct these tests before moving to a more prac-

tice oriented example with arbitrary reference

curves.

5.3.3.1 Developable Reference Surface

Creation

For DStrip studies with border curves of an ac-

tual developable surface as reference curves

for the DStrips, a developable surface had to be

found first. The simplest way of finding such a

surface is to create two planar parallel curves

and calculate a loft surface between them. For

the purpose of studying the behaviour of DStrip

a more complex example had to be found,

because in practice they would be applied to

more complex designs as well. The developable

surface that is used as basis for this studies is

created by scaling a curve from one point. The

control points of a curve, p0, p1 ... p5, are scaled

from the origin o and the points p0,1, p1,1 ... p5,1

o

p0

p0,1

p1,1

p2,1

p3,1

p4,1

p5,1

p1 p2

p3 p5

K

S

S

S*

p4

(a)

(b)

(c)

(d)

Figure 5.8: The creation process of the developable reference surface

82

are obtained (Figure 5.8 - a). The points p0, p1 ... p5, and the points p0,1, p1,1 ... p5,1 are used as corner

points of the control polygon K from which the surface S is calculated (Figure 5.8 - b and c). Due

to the way how the control points of the surface S have been obtained by scaling the control

points of a curve, all rulings of the surface converge in one point making the surface S a cone

which is a developable surface by definition. The strip on which the studies are conducted S*

is a approximately one metre wide and ten metres long part from the surface S (Figure 5.8 - d).

5.3.3.2 Additional Evaluation Parameters

Since this is a special case in which there is a reference surface that is absolutely developable

and that is tried to be matched by a DStrip there are additional parameters with which the re-

sults can be evaluated:

Area Difference to original unroll

The difference between the surface area of the unrolled reference surface S* and the surface

area of an unrolled DStrip. This parameter is expressed in square centimetres and in percent.

Maximal deviation from ideal outline

The term ideal outline denotes the border curve of the unrolled reference surface S*. The maxi-

mal deviation from ideal outline is the maximum distance between the ideal outline and the

outline of a unrolled DStrip. The two curves are previously registered against each other in or-

der to overlay them as precisely as possible before the measurements are made.

Average deviation from ideal outline

The average deviation from ideal outline is the average distance between the ideal outline and

the outline of a unrolled DStrip. The two curves are previously registered against each other in

order to overlay them as precisely as possible before the measurements are made.

83

DStrip Studies

5.3.3.3 Results

The results of these tests show that it is possible to recreate the developable strip S* from Fig-

ure 5.8. The tables in Appendix 1 show the measured results for all of the fifty five tested strips.

As expected there is a continuous improvement in the closeness of the strip to the reference

curves and in the closeness between the surface area of the unrolled reference surface and the

unrolled DStrips. The measured values tend to stabilize after a certain planarity value in the

DStrip is achieved. The studies showed that the following DStrips are suitable for the discretiza-

tion of the reference surface:

DStrips with an initial aspect ratio of 0.3, optimized to a planarity value of at least 0.068 cm

DStrips with an initial aspect ratio of 0.2, optimized to a planarity value of at least 0.081 cm

DStrips with an initial aspect ratio of 0.15, optimized to a planarity value of at least 0.089 cm

DStrips with an initial aspect ratio of 0.1, optimized to a planarity value of at least 0.03 cm

DStrips with an initial aspect ratio of 0.05, optimized to a planarity value of at least 0.013 cm

The results show that higher resolution DStrips need to be optimized to a smaller planarity

value, but that they provide better results in terms of reference curve closeness than less dense

DStrips.

84

5.3.4 Test With Arbitrary Reference Curves

The previous studies proved that it is possible to create a DStrip that resembles a developable

surface closely. The example reference curves that are used for the studies in section 5.3.3 rep-

resent a special case, because the curves were extracted from a developable surface and used

to create a DStrip. In this section the same method is tested using two arbitrary curves that are

extracted from the model that is presented in section 5.4 (Figure 5.9). Since there is no refer-

ence surface to compare the results against, there are no additional evaluation parameters in

this case, but only the evaluation parameters that are shown in section 5.3.2. It is expected that

the planarity and closeness values that were achieved in the previous test will not be matched

in this tests. One of the parameters that will be observed here is the difference between the sur-

face area of the DStrip and the surface area of its development. As the planarity becomes small-

er it is also expected that the difference in surface area decreases. The surface area difference

is expected to stabilize after a certain planarity value, meaning that after that threshold value

in strip planarity, the area difference will not improve significantly. Since the reference curves

do not belong to a developable surface, the mesh will need to move away from the reference

curves in order to ensure the planarity of the faces. The curve closeness is the second important

value that needs to remain as small as possible, because otherwise the resulting strip may be

developable, but it would fail to bridge the area between the reference curves.

Figure 5.9: Image showing the reference curves of the project in section 5.4 in black. The parts of two reference curves that are used for the DStrip studies in this section are highlighted in red.

85

DStrip Studies

5.3.4.1 Results

The studies showed that it is possible to create DStrips between the chosen reference curves

with all types of initial aspect ratios, but that only DStrips with an initial aspect ratio of 0.05

achieve an average distance between its vertices of the DStrips that is acceptable. The studies

showed that strips with lower initial aspect ratios and lower planarity values are better suited

for the task of creating DStrips between two arbitrary curves than strips with higher values.

Detailed results of the here presented studies can be found in appendix 2. DStrips with the

following values, are considered as good discretizations of developable surfaces between the

given reference curves because they have a small difference in surface area between the strip

and its development:

DStrips with an initial aspect ratio of 0.3, optimized to a planarity value of at least 0.15 cm

DStrips with an initial aspect ratio of 0.2, optimized to a planarity value of at least 0.34 cm

DStrips with an initial aspect ratio of 0.15, optimized to a planarity value of at least 0.05 cm

DStrips with an initial aspect ratio of 0.1, optimized to a planarity value of at least 0.33 cm

DStrips with an initial aspect ratio of 0.05, optimized to a planarity value of at least 0.07 cm

Only the DStrips with an initial aspect ratio of 0.05 that are optimized to a planarity value of

at least 0.07 cm provide acceptable curve closeness values. Tables with detailed results of this

tests can be found in the appendix 2.

86

Figure 5.10: Site plan showing the boulder wall on the shore of the Donaukanal, high-

lighted in red.

5.4 Project: Bouldering Wall

The task in this project was to design a boulder-

ing wall at Pier9, on the shore of the Donauka-

nal in Vienna. There is already an existing fifteen

metres high climbing wall (Figure 5.10 - 1.) and

a bouldering area (2.) developed on the site. The

bouldering area is situated between the climb-

ing wall and the Donaukanal. The two areas for

climbing and bouldering are divided trough a

3,5 metres height difference, the bouldering

area being situated 3,5 metres lower than the

climbing area. Figure 5.11 shows three people

exercising in the bouldering area of Pier9 and

the tall climbing wall in the background. My

proposal for a new bouldering wall is a wooden

structure that will attach to the concrete wall

that is the result of the man made difference

in terrain height. The bouldering wall is a forty

metres long and four metres high structure de-

signed with developable surfaces in mind, so

that it could be completely covered with cold

bent plywood sheets. The plywood that is used

is okoume marine grade plywood. This type of

plywood is one of the finest construction mate-

rials for boats available because it is lightweight

and can be sealed against water, therefore it is

a suitable material for an outdoor structure like

this one. The sport discipline of bouldering was

invented as an outdoor activity on natural for-

mations of stone and rock, namely boulders.

The wonderful flowing lines of the red rocks

Figure 5.11: People exercising in the bouldering area at Pier9.

89

Project: Bouldering Wall

of the Antelope Canyon, Navajo, Arizona, USA

(Figure 5.12) were the inspiration for this boul-

dering wall. My goal was to a smooth and pliant

structure that would have the same aesthetic

qualities as the rocks of the Antelope canyon,

as far as that is possible to achieve with an ar-

tificial structure. Besides the aesthetic qualities,

there are practical qualities that the structure

has to fulfil as well.

Bouldering, was intended to serve as training

and preparation for sports climbers. It grew rap-

idly into an independent sport discipline with

its own community of followers. The main dif-

ference between sport climbing and boulder-

ing is in the short, dynamic and daring routes

that are preferred in bouldering instead of long

and high routes that sport climbers climb. Aver-

age boulder "problems" are just a few meters

long with an average of five to seven moves

between start and end of the route. Boulder-

ers are not protected with ropes and harness-

es, but instead they climb close to the ground,

and rely on bouldering mats in case of a fall. All

these things have been kept in mind during

the design of the bouldering wall. The result is

a wall with a vast spectrum of different kinds

of climbing surfaces from flat vertical surfaces

that are suitable for less experienced climbers,

to areas where the structures surfaces form ex-

treme overhangs that are difficult to climb. The

Figure 5.12: Antelope Canyon in Navajo, Arizona, USA

90

overhang areas have been strategically placed closer to the ground because climbers will be

most likely to fall from those parts of the wall. The seven developable surfaces that joined to-

gether make up the visible structure are not a discretization of a pre-designed freeform surface.

Since I had the chance to embed the geometry of developable surface strips in the design from

the beginning I was able to design the surface by laying out the edge curves of the strip in 3D

space and adjusting their shape and position to alter the appearance of the whole structure.

Grasshopper for Rhino was a helpful tool in this process because it allowed creating a simple

definition that creates lofted surfaces between the curves and updates the shape of the sur-

face automatically and in real time when changes to the edge curves were done. The resulting

surfaces that emerged as a result of the loft with Grasshopper were not developabe, but they

were a good visual representation of the shape that the final structure would have. The small

disadvantage of inaccuracy when using the above described method was outweighed by the

much larger advantage of speed which meant seeing an almost perfect result instantly on the

screen. A series of studies on DStrip meshes, which are discussed in section "5.3 DStrip Stud-

Figure 5.13: Explosion diagram showing the substructure and the panelized developable surfaces of the bouldering wall

91

Project: Bouldering Wall

ies", followed after the exact edge curves of the

DStrips were determined. The DStrip studies

are discussed in the next section.

The studies provided information about the

connection between the aspect ratio of the

DStrip's faces and their planarity values with

the accuracy of the development. The develop-

able surfaces in this project were created using

the methods described in section 5.2.2 with the

information from the studies in mind. The pro-

cess of designing the DStrip meshes involved

small adjustments on the reference curves since

it was not possible to create developable strips

between all designed curves without the strips

moving too far away form the curves. Each strip

has been carefully analysed for the planarity of

it's faces and for the DStrip's closeness to the

reference curves before the rulings were loft-

ed to create developable NURBS surfaces. The

developable surfaces are divided into patches

that could be built out of plywood boards. The

dimensions of the patches in their developed

state is limited to 250 x 125 centimetres, which

is the standard size for plywood panels. Figure

5.13 shows the outer skin of the bouldering

wall consisting of panelized developable strips

and the substructure on which the plywood

panels are mounted. The substructure was ob-

tained by intersecting the DStrip model with

a series of seventy vertical and five horizontal Figure 5.14: (Above) Frontal view of the bouldering wall. (Below) Left-side view of the bouldering wall.

Scale 1:200

92

planes. The Intersection curves were extruded

and intersected with each other to create an or-

thogonal waffle structure on which the DStrips

can be mounted. The same type of substruc-

ture is also used in the scaled physical model

in the next section. The final auter layer would

have to be made of three to five layers of four

millimetres thick plywood. The seems between

the panels in two consecutive layers have to

be displaced because they would weaken the

structure if they would be on top of each other.

The fact that the seams between panels are

displaced and that panels are connected to

each other across several layers means that the

placement of the seams can be done with no

regards for the underlying substructure, which

in turn makes it possible to maximize the size of

the panels and to minimize production waste.

Figure 5.15: (Above) Top view of the bouldering wall. (Below) Section A-A.

Scale 1:200

A A

93

5.4.1 The Physical Model

The physical model in this project has an im-

portant role besides providing a truthful rep-

resentation of the design. The model is also a

proof that the DStrips are in fact developable

and that they can be produced out of flat ply-

wood by only the means of cold bending the

wood into shape. Each of the seven strips were,

for practical reasons, made from one piece of

plywood instead of dividing them into smaller

patches, as it would be necessary if a full sized

structure were to be built (Figure 5.13). This

doesn't change the fact that if the whole strip

can be produced from a flat sheet of plywood,

it can also be divided into smaller patches, each

of which is developable for itself. The model's

scale is 1:33 which makes it one hundred and

fifteen centimetres long and fourteen centime-

tres high. The DStrips were CNC cut out of a one

millimetre thick sheet of plywood. This thick-

ness is proportionally larger than it should be,

but it was chosen so that the strips would pro-

vide more resistance to bending and thus be

less prone to twisting around the stronger axis.

Substructure

Before the strips could be joined into the de-

sired shape, a substructure had to be created

first. The substructure consists of seventy-eight

individual pieces all of which are CNC cut from

a two millimetres thick plywood panel. The

pieces of the substructure are designed in such

(a)

(b)

Figure 5.16: By intersecting the DStrip model with vertical and horizontal planes a robust substructure,

that follows the shape of the DStrips exactly, was calculated. (a) The substructure of the DStrip physical model. (b) The first of the seven DStrips is being glued

onto the substructure

94

a way that each piece has predefined pockets cut into it in which its adjacent pieces fit perfectly

(Figure 5.16 - a). This method of self intersecting vertical and horizontal frames made it possible

to build the complex form of the bouldering wall relatively simply. The substructure is mounted

on a MDF board that represents the concrete wall behind the structure at the shore of the

Donaukanal. The wood strips are bent over the substructure to take the exact shape that was

intended and glued onto the substructure (Figure 5.16 - b). After all strips were in place, and the

glue dried, the strips and the frames of the substructure connected into a rigid structure where

all parts work together in providing structural stability.

Figure 5.17: Close-up side views of the physical model

95

Project: Bouldering Wall

Figure 5.18: Physical model of the boulder-ing wall consisting of seven developable surfaces. (Top) aerial view, (bottom) front

view

5.4.1.1 Model Evaluation

The physical model provided visually satisfying results. Besides proving that the designed de-

velopable surfaces can be produced from flat plywood panels, it also provides a trustworthy

representation of the design. However, it is not given that the physical model accurately resem-

bles the designed surface. Due to the fact that the plywood is a relatively soft material it can be

bent eas ily. Plywood can especially be easily bent when working with thin sheets, as it was the

case in this model, thus it is possible that the wood strips incorporated an amount of twisting

over the stronger axis, besides bending over their week axis, in order to match the shape that

the substructure determined.

Scan and preparation

In order to precisely evaluate the physical model and to determine how accurate it represents

the design the model is scanned with a 3D scanner. The scan returned a collection of scan strips

that are represented by point clouds (Figure 5.19 - a). The individual strips lie close together, but

there are small distances between them due to precision limitations of the scanner. The indi-

vidual scan strips are matched against each other in order to find the average of all overlapping

layers (b). A final point cloud is computer from the scan strips of points. The point cloud (c) is a

low resolution model of all the points from (b). The points for the point cloud (c) are filtered out

because the laser scanner provides far more resolution than is needed. Subsequently, the point

cloud is used to compute a triangular mesh (d) which uses the points from the point cloud as

its vertices. Finally, in order to make it possible to evaluate the physical model's scan, the trian-

gular mesh is registered to the original DStrip surfaces. Registering two shapes means that they

are aligned in 3d space in such a way that they lie as close as possible to each other. Figure 5.19

(e) shows the scanned model in gray and the reference DStrip surfaces in blue.

98

Figure 5.19: A 3D scan of the physical model. (a) Collections of scan strips containing point

clouds are provided by the scanner, (b) The scan strips are

matched against each other to find the average values of

two or more overlaying strips, (c) one final point cloud is

computed, (d) the points from the point cloud are used as

vertices for a triangular mesh, (e) the triangular mesh is reg-istered against the designed

surfaces.

(a)

(b)

(c)

(d)

(e)

99

Scan results and evaluationThe images below show a graphic representing the distances between the designed model and the scanned model with colours. The graphic shows that a major part of the model re-sembles the designed surface precisely with an average distance of 1 - 1.5 mm shown in the colour spectrum between blue and green. An error of this size in a model that is 1150 mm long can be accounted to manufacturing and assembly tolerances.The left side of the graphic shows a problem zone where the distance between the scanned and the designed model is over five millimetres. Behind the gray area there are four wooden poles that connect the substructure to the wood board behind the model. The wooden poles are cut to the precise distance that the substructure should have to the board behind at their respective positions. Unfortunately, the model's substructure pulled away from the poles before the glue that holds them together was completely dry, which results in the DStrips moving away from the designed surfaces in the area of the poles. Due to the fact that the model showed satisfactory results in the major part of its surface area despite having is-sues in a small part of it, the conclusion is made that the model proved the methods used to design and manufacture the DStrips. Further, the assumption is made that the problem with the displaced substructure influenced the precision of the entire model meaning that with a more precise model, the average distance to the designed surfaces would be even lower. The distances between physical model and DStrip design can be accounted to manufacturing and assembly errors and tolerances.

Figure 5.20: Designed DStrip surfaces matched against the scanned physical model. Front view (top) and top view

(bottom)

100

101

6. Geodesic Curves on

Freeform Surfaces

ΨiΨ

si

si+1

x

x’

si+2

Ai(x)≈Bi+2(x)

Bi(x)

A (x)

6. Geodesics Curves on Freeform Surfaces

This chapter investigates how geodesic patterns of curves on freeform surfaces can be used in

architectural applications, in regard to timber cladding and supporting structures.

6.1 Geodesic Curves

A geodesic curve in a surface is a curve that has prin-

cipal normal vectors that are parallel or anti-parallel to

the surface's respective normal vectors at each of the

curve's points. The shortest curve between two points

on a surface, that lies in that surface, is always a geo-

desic. It is simple to determine the geodesic lines on

regular shaped surfaces, e.g. the geodesic lines of a cyl-

inder correspond to helixes and on spheres to the great

circles (intersections of a sphere with planes passing

through the sphere's centre). It is more complex to de-

termine the geodesic lines in freeform surfaces [Pirazzi

and Weinand 2006]. In other words, geodesic lines are,

besides distance minimizers, also curves of zero geo-

desic (sideways) curvature.

6.2 Application in Architecture

Geodesic lines can be used to cover freeform surfaces

with wooden panels and to aid the layout of the sup-

porting structure of such a surface.

Cladding

The absence of geodesic curvature makes patterns of

geodesic curves suitable for dealing with the cladding

of freeform structures with straight wooden panels

which bend only around their weak axis. Such a clad-

ding will mainly be used in interior applications as

shown in the wood ceiling for the office lobby of the

Figure 6.1: NOX architects designed the surfaces of the office space (top and middle) with an experimental approach, by cladding

a model with paper strips (down).

105

Geodesics Curves on Freeform Surfaces

Burj Khalifa in Dubai by Gehry Technologies (Figure 6.4). The

used panels should be close to developable and their devel-

opment should be a rectangle whose length is much larger

than its width, or it should at least be possible to cut out the

panel of such a rectangle. This means that each panel should

follow a geodesic curve. NOX architects have successfully

approached this problem experimentally by designing the

cladding of an office space with paper strips as shown in Fig-

ure 6.1 [Spuybroek 2004]. [Wallner et al. 2010] showed how

it is possible to decompose a surface into regions of which

each can be covered with a family of geodesic lines at nearly

constant distance using computational tools.

Support structure

Geodesic curves are also suitable for the design of load bear-

ing support beams in freeform structures that can be man-

ufactured with less effort, and waste and that have better

static properties than beams that follow arbitrary curves. In

such structures the stress due to the initial curvature is re-

duced because the bending around the strong axis is avoid-

ed. This has positive consequences on the manufacturing of

(a)

(b)

(c)

(d)

Figure 6.2: Centre Pompidou Metz (a) roof plan, (b) the built roof structure, (c) CNC fabrication, (d) manufactured

beams and connections.

Figure 6.3: Centre Pompidou-Metz

106

the beams because laminated beams in which the individual boards are only twisted and bent

around the weak axis, are easier to manufacture [Pirazzi and Weinand 2006]. Some innova-

tive contemporary timber constructions such as the roof of the Centre Pompidou in Metz by

Shigeru Ban could benefit from a computational approach for the layout of the load bearing

structure with geodesic lines (Figure 6.3). In this case the curve network that drives the layout

of the beams is found by projecting a network of straight curves (except for the areas where

the structure touches the ground) from the ground onto the roof's freeform surface, see Figure

6.2 (a) and (b). This approach resulted in heavily double curved beams of which 18,000 running

metres had to be individually CNC fabricated (c) and (d) [Scheurer 2010].

Geodesic patterns

[Pottmann et al. 2010] study geodesic N-patterns[1] on surfaces. They provide efficient ways to

design such patterns on freeform surfaces in form of a computational framework. N=1, N=2 and

N=3 patterns have been emphasised in their work although ways to design geodesic webs with

4-patterns and a further extraction of patterns from such webs have been described as well.

However, this chapter focuses on the geodesic 1-patterns and their application in the cladding

of freeform surfaces. Unfortunately, the results and panelization techniques that [Pottmann et

al. 2010] and [Wallner et al. 2010] present are not accessible in form of a commercially available

software at the moment of writing this work. Despite that, I will discuss those methods in the

following section and compare them with the experimental method that NOX presented in

1 See [Pottmann er al. 2010] for a detailed explanation of the terminology.

Figure 6.4: The wooden ceiling of the Burj Khalifa office lobby in Dubai, by Gehry Technologies. See [Meredith N. and Kotronis J. 2012]

107

Application in Architecture

their work on the cladding solution for an office space (Figure 6.1), which is used to design the

cladding for the project in section 6.4 [Spuybroek 2004].

6.3 Algorithmic Panelization of Surfaces with Geodesic 1-Patterns

Different problems arise when trying to design the panelization of a freeform surface with

rectangular panels. The panelization with rectangular panels should not be mistaken with the

discretization of freeform surfaces with PQ meshes, which has been discussed in chapter 4.

When working with PQ meshes, the goal is to find a quadrilateral mesh with planar faces which

approximates the surface as closely as possible, while in this approach we are looking for a

panelization of the freeform surface with panels whose length is much larger than their width

and the panels are not planar but bend around their weak axis. Several properties are desired

to be present in the resulting patterns, but unfortunately only in rare cases it is possible to have

all of them. In general the panelization will be a compromise between the different properties:

The geodesic property

Long wooden panels easily bend around their weak axis and it may twist a little. The bending

of such panels around their strong axis is not desired. Such a wooden board, if laid on a surface,

follows a geodesic curve. Therefore, when the layout of the panels is driven by geodesic curves

it can be safely assumed that those panels will bend only around their weak axis.

The constant width property

Only developable surfaces, e.g. a cylinder, can be covered with panels whose development is

a true rectangle, while the panelling remains seamless and non-overlapping. In all other sur-

faces it is not possible to have a panelization without gaps or overlaps and panels that have a

rectangular development. However, due to practical reasons, it is important to cut panels out

of rectangular shapes with minimal waste. If all panels in a panelization project can be cut out

of boards which have the same dimensions, or at least a few types of boards, then the cost of

the cladding can be largely reduced and such a cladding tends to be visually pleasing (Figure

6.4). This leads to the mathematical requirement that the geodesic curves, which are used for

the layout of the cladding, must be at approximately constant distance from their respective

adjacent curves.

108

The developable (pure bending) property

When it comes to the developable property a certain amount of twisting in the panels is al-

lowed since wood is a more forgiving material in comparison to other materials. The twisting

must to be held at a minimum because the wooden panels need to be produced by cutting a

2D shape and bending it into shape without much effort. The previous two properties actively

influence all algorithmic approaches presented by [Wallner et al. 2010], while the developable

property is present in only one of them. The process of designing a decent panelization with

geodesic lines can be divided into two equally important steps — the design of patterns of

geodesic curves on a surface (section 6.3.1) and the translation of those curves into actual pan-

els/timber boards (section 6.3.2).

Depending on the method that i used to create the panels based on the geodesic curve pat-

tern and the design intent for the cladding, there are two goals, one of which is pursued in the

process of designing patterns of geodesic curves [Wallner et al. 2010].

Panelization with gaps between panels

If the geodesic curves are intended to be used as guidelines over which rectangular wooden

panels will be bent, then the goal is to find a system of geodesic curves which are at approxi-

mately constant distance from their adjacent curves. It is possible to cover a surface with rec-

tangular panels with this method, but gaps between the panels will be unavoidable [Wallner

et al. 2010].

Gapless panelization

If a gapless panelization is intended then the shapes of the panels need to be altered. It will

not be possible to achieve such a panelization, on most surfaces, with panels whose develop-

ments are true rectangles. For a gapless panelization it is necessary to search for a system of

geodesic curves which represent the edges of wooden panels which should cover the surface

without gaps or overlaps. The wooden panels should have an approximately straight develop-

ment which is as close as possible to a rectangle. With that said, it also must be possible to cut

the boards out of such rectangles.

Before starting to create a project, a compromise between machining time and cost on one

side and the appearance of the final cladding on the other side has to be made. A smooth gap-

109

Algorithmic Panelization of Surfaces with Geodesic 1-Patterns

less cladding is possible, but it will cost more

to achieve such a result because the panels

need to be produced individually instead of us-

ing off-the-shelf wooden boards [Wallner et al.

2010].

6.3.1 Designing 1-Patterns of Geodesic

Curves

There is a number of different approaches to

solving the problem presented by [Pottmann

et al. 2010] and [Wallner et al. 2010]. It is pos-

sible to approach the problem with design by

parallel transport and design by evolution and

segmentation.

Design by parallel transport

The design by parallel transport allows pre-

scribing the points at which either the maxi-

mum or the minimum distance between neigh-

bouring curves occurs. In differential geometry

the notion of parallel transport of a vector V

along a curve c contained in a surface means

that the vector is moved in such a way that it

stays tangent to the surface, while changing

as little as possible. The length of such a vector

remains unchanged. The surface will in most

instances, for computational reasons, be rep-

resented with a dense mesh and a curve as a

polyline between vertices P0, P1, P2 ... Pn. In that

case the vector Vi is found by orthogonally pro-

jecting Vi-1 onto the tangent plane of Pi. The pro-

jected vector is then normalized. For the design

P2

P1

P0

c

V0

V1 V0

V1V2

Figure 6.6: Parallel transport of vector V0 along the polyline P0,P1,P2...

Figure 6.5: Example of designing a sequence of geo-desics. The locus of minimum or maximum distance between adjacent curves has been prescribed with

the red curve. This surface was segmented prior to ap-plying the parallel projection method.

110

of patterns of geodesics, an input curve is sampled at points P0, P1, P2... The parallel transport

results with the vectors V0, V1, V2 .... V2, which are attached to those points. The geodesic rays

which emanate from the point Pi in direction Vi and -Vi make one unbroken geodesic. This way

the extremal distances between neighbouring geodesic curves, or the extremal widths of strips

between two neighbouring curves, will occur near the chosen input curve. The extremal dis-

tances depend on the underlying geometry. In an area of positive Gaussian curvature (K>0), the

distances on the input curve can only be the maximum widths of the strip, whereas in areas of

negative Gaussian curvature (K<0) the distances can only be local minima. Strips with constant

width are only possible on surfaces that have Gaussian curvature that equals zero, meaning

that they are developable surfaces.

Design by evolution

This method starts from a prescribed geodesic curve g on the surface and computes iteratively

the next geodesics g+, on an approximately constant distance to the previous, evolving a pat-

tern of geodesics. The transfer from g to g+ considers only the local neighbourhood of g and can

nicely be governed by Jacobi fields, which are vector fields along a geodesic in a Riemannian

manifold describing the difference between the geodesic and an infinitesimally close geodesic

[do Carmo 1992]. All possible Jacobi fields of a geodesic g are calculated and one of them is se-

lected. The selection of the Jacobi field depends on the design intent, which was mentioned in

section 6.3. The selected Jacobi field is further used to compute the next geodesic. For a deeper

understanding of the algorithmic processes the reader is referred to [Pottmann et al. 2010]. In

areas of positive curvature there is a possibility that we will not find a geodesic g+ of g which

does not intersect g and in some areas of negative Gaussian curvature, the geodesics will drift

too far apart and violate a given distance constraint which is driven by the design intent. In

such cases, when the 1-pattern of geodesics runs into obstacles due to the few degrees of free-

dom ,it is possible to consider the option of broken geodesics. Broken geodesics are achieved

by introducing breakpoints in critical areas. The breakpoints are automatically inserted when-

ever the distance between two adjacent geodesics violates the distance constraint. The paths

of breakpoints are oriented so that they bisect the angle of their adjacent geodesic segments.

This approach makes it possible to cover more complex shapes with 1-patterns of geodesics

than it would be possible to do with continuos lines.

111

Algorithmic Panelization of Surfaces with Geodesic 1-Patterns

Design by segmentation

It becomes more difficult to cover surfaces with a single geodesic pattern as they become more

complicated. The Gaussian curvature of the input surface limits the maximal length of a strip

which is bounded by geodesic curves and has a width which is limited by the design intent.

The approach of designing 1-patterns by evolution solved this problem by introducing broken

geodesics at critical points. In the current approach it is the goal to divide the input surface into

segments which can be covered by a geodesic 1-pattern without violating the distance con-

straint. In order to achieve a segmentation like that, [Pottmann et al. 2010] introduce geodesic

vector fields and piecewise-geodesic vector fields. The workflow with geodesic vector fields

involves three main steps:

1. Design a near-geodesic vector field on the surface

The first step involves designing a vector field on the surface which consist of tangent vectors

of a 1-parameter family of geodesic curves. This type vector field is called a geodesic vector

field. The freeform surface is represented as a triangular mesh for this purpose and the vectors

are unit vectors that are attached to the incenters of the mesh faces. It is possible for the user to

interactively influence the selection of the vector field in real time.

2. Generate a piecewise-geodesic vector field by modifying (sharpening) the original vector field

For the segmentation of a surface we need a piecewise-geodesic vector field. Such a vector field

fulfils the geodesic property in the inside area of certain patches of the surface. The piecewise-

geodesic vector field is obtained with an optimization algorithm from the original vector field

(Figure 6.7). It will be similar to the geodesic vector field, especially in the inside areas of the

surface patches, where the proximity to the geodesic vector field is kept close, whereas the

areas closer to the boarders of the patches are given more freedom during the optimization.

3. Segment the input surface

The surface is consecutively segmented along the edges where the vector field is sharp. The

lines along which the surface is divided are found by measuring the angle between two

consecutive vectors in the vector field and collecting all edges where this value is higher than

a specified threshold value. The edges are then polished to create smooth curves. Those curves

are further used to create a clean segmentation of the surface into parts which can be covered

by a smooth geodesic vector field.

112

6.3.2 Creating Panels from Geodesic

1-Patterns

The final task, after a satisfying network of

geodesic curves has been laid out on the sur-

face, regardless if the surface is segmented, or

which method has been used, is to create pan-

els based on the network of curves. There are

two ways of mathematically representing those

panels presented by [Wallner et al. 2010]. The

first method, the tangent developable method,

creates panel surfaces that are tangentially

circumscribed to the surface along given geo-

desic lines. The second method, the binormal

method, creates panels that are inscribed into

the input surface between two adjacent geo-

desics.

6.3.2.1 The Tangent Developable Method

Conjugate tangents

For this method the notion of conjugate tan-

gents and tangent developables needs to be

explained. Let's assume a tangent plane on a

smooth surface. If the tangent plane is moved

just a small amount by means of parallel trans-

lation and intersected with the surface, the in-

tersection will result in a curve which approxi-

mates a conic section - the Dupin indicatrix

(Figure 6.8). The shape of the Dupin indicatrix

depends on the Gaussian curvature of the un-

derlying surface. In hyperbolic points, that is ar-

eas with negative Gaussian curvature, the inter-

Figure 6.7: Through the process of sharpening a geo-desic vector field (left) becomes piecewise geodesic

(right)

(a)

(b)

Figure 6.8: The Dupin indicatrix in (a) positively curved areas of a surface and (b) in negativaly curved

areas of a surface.

113

Algorithmic Panelization of Surfaces with Geodesic 1-Patterns

section will result in two different hyperbolae.

The asymptotes A1 and A2 of the hyperbola

form the asymptotic directions. Any parallelo-

gram tangentially circumscribed to the Dupin

indicatrix yields two conjugate tangents T and

U. The asymptotic directions A1 and A2 can be

used to find such parallelogram if necessary be-

cause they are known to be its diagonals. [Wall-

ner et al. 2010] By knowing the Dupin indicatrix

and the asymptotic directions it is possible to

find pairs of conjugate tangents in every point

on the surface. If we prescribe one of the tan-

gents it is not difficult to find the other one, its

conjugate tangent, which is the goal of this ap-

proach. [Wallner et al. 2010]

Tangent developables

A tangent developable is a surface which is

tangentially circumscribed to a surface along a

curve. In this case, the given surface is the input

surface and the curves are curves from the

network of the geodesic pattern. A geodesic

curve s on the surface Φ is sampled in a point x.

The tangent T(x) to the curve s in point x is found

and its conjugate tangent U(x) is computed.

The union of all conjugate tangents U(x) is a

tangent developable Ψ on Φ along the curve s

(Figure 6.9). The geodesic curve s is not only a

geodesic to the input surface Φ any more, but

to the tangent developable Ψ as well, which

means when Ψ is unrolled into a plane the geo-

s

x

U(x)Ψ

Φx

T(x)

Figure 6.9: U(x) is the conjugate tangent of T(x) in point x of the geodesic s.

Ψi

Ψi+2

Ψ

si

si+1

x

x’si+2

Ai(x)≈Bi+2(x)

Bi(x)

Ai+2(x)

Figure 6.10: Tangent developable surfaces of geodes-ics with even indices are trimmed by neighbouring

geodesics with odd indices.

s B(t)

L(t)

R(t)

N(t)

Ψ

Φ T(t)P(t)

Figure 6.11: The binormal method. The ruled panels are defined by the Frenet frame T, N, B of a geodesics s.

114

desic curve s becomes a straight line. [Wallner et al. 2010] Given these facts it would be obvi-

ous to calculate a tangent developable Ψi for each geodesic curve si in the pattern and then

trim the tangent developables where they intersect each other. The unfolded surfaces would

yield the flat state of the panels. Unfortunately this does not work well in practice because the

angles between neighbouring tangent developables are very small and thus the intersection is

numerically not robust, so an alternative strategy had to be found. [Wallner et al. 2010]

Algorithm for creating the panels

Instead of considering all geodesic curves in the pattern, only every second geodesic (geodes-

ics si where i is an even number) is used to create the tangent developable as described above.

The rulings Ui(x) of Ψi are then tested and those of them which enclosed an angle with the

tangent smaller than a given threshold value were deleted to clean up the surface. The holes

were filled with an standard interpolation procedure. Instead of searching for an intersection

curve between two adjacent surfaces Ψi and Ψi+1 and trimming the surfaces along those curves

each ruling is investigated separately. The end points Ai(x) and Bi(x) of each ruling are found as

the points on the ruling that are closest to the geodesics si-1 and si+1 respectively. A final step is

the global optimization of the positions of Ai(x) and Bi(x) in such way that the trim curves are

smooth, and that Ai(x) and Bi(x) are close to their closest geodesics and that the ruling segments

Ai(x)Bi(x) lie close to the input surface Φ . This optimization changes the surface slightly and the

developability is compromised a little as well (Figure 6.10). [Wallner et al. 2010]

6.3.2.2 The Binormal Method

The tangent developable method could not ensure pure developable panels. The method de-

scribed in this section aims at fulfilling the pure bending property while not aiming at a gapless

panelization. The binormal method (Figure 6.11) uses the Frenet frame of the curve si on the

input surface Φ to define a ruled panel. The Frenet frame is a coordinate system with its centre

on the curve s which moves with unit speed t, represented by the surface normal N(t) in the cur-

rent point P(t), the velocity vector T(t) and the binormal vector B(t) . In each point P(t) on curve

s there is a ruling represented by the binormal vector B(t). The endpoints of the ruling L(t) and

R(t) are found on B(t) at the distance from P(t) which is half the intended width of the panel.

[Wallner et al. 2010]

115

Algorithmic Panelization of Surfaces with Geodesic 1-Patterns

6.4 Project: 21er Raum

This project involved the design and panelization of the outer surface of a new exhibition space

in the "21er Haus" museum in Vienna. The "21er Haus" originally known as Museum of the 20th

Century is a building designed by Karl Schwanzer for the world Expo 1958 in Brussels. After

the Expo, the building was disassembled and shipped to Vienna, where it was built up again

between 1959 and 1962 to serve as a Museum. The former Expo pavilion was a state of the art

structure for its time with its striking cage structure made from steel profiles that complied to

the DIN norm and the Eternit panels on the facade [Toman 2010]. The museum was recently

renovated by the architect Adolf Krischanitz after it was incorporated as part of the Belvedere

museum in 2002. Currently the 21er Haus is a museum where Austrian art of the twentieth and

twenty-first centuries is exhibited. The museum features an exhibition space, the 21er Raum,

which is dedicated to exhibiting work of young Austrian artists. The 21er Raum is located on

the gallery on the north side of the building opposite to the entrance. The outer walls of the

exhibition space are covered by mirrors that are arranged similar to the pattern of the panels

on the facade, in order to pay homage to the design of the building. The here presented de-

sign proposes to replace the currently present structure with a new one. The shape of the new

21er Raum is designed to draw attention instead of being hidden. It does not try to hide the

fact that it is a strange artefact inside Karl Schwanzer's and Adolf Krischanitz's museum, but

Figure 6.12: The 21er Raum on the gallery of the 21er Haus museum in Vienna

116

rather embraces it by being formally the oppo-

site of the straight lines and orthogonal angles

that dominate the museum's design. The only

formal connection between the room and its

surrounding is its entrance. The entrance to the

21er Raum is a simple orthogonal box that in-

tersects the freeform surface of the room, pos-

ing a allusion to the facade of the museum.

The room is pressed between the floor of the

gallery and the ceiling as if the museum is to

small for it, or for what is inside it. With its curved

design, leaning over the edge of the gallery as if

it was a large mass of viscous substance threat-

ening to roll over the edge, this room creates a

new presence for the young artists that exhibit

inside that can not be ignored.

(a)

(b)

Figure 6.13: The 21er Raum. (a) inside view, (b) detail of the mirrors on the outside walls

117

Project: 21er Raum

Figure 6.14: Section of the museum revealing a side elevation of the 21er Raum. (Above) Elevation

of the 21er Raum and its immediate surrounding — scale 1:100. (Below) Section through the 21er Haus museum revealing the position of the 21er Raum in

regards to its surrounding — scale 1:500.

119

Figure 6.15: Top view of the 21er Raum. The drawing below shows the 21er Raum and its position in regard to the surrounding mu-

seum — scale 1:500. The drawing above shows a closer look of the 21er Raum and its immediate surrounding — scale 1:100.

120

Figure 6.16: Section of the museum and the 21er Raum. (Above) Section of the 21er Raum and its immediate surrounding — scale 1:100. (Below)

Section through the 21er Haus museum and the 21er Raum revealing the position of the 21er

Raum in regards to its surrounding — scale 1:500.

121

6.4.1 The Physical Model

The initial shape of the exhibition space was designed manually with a small plaster model. The

model was scanned into the computer using a 3D scanner and subsequently improved inside

a 3D modeller. After the design was decided upon, a 1:20 scale model was CNC milled out of ex-

panded polystyrene (EPS). The polystyrene model was subsequently covered with multiple lay-

ers of glass fibre sheets glued with an acrylic resin. After drying, the glass fibres and the acrylic

resin bound into a very stable shell from which the polystyrene could be removed. The surface

irregularities on glass fibre shell were levelled out and the shell was painted with acrylic paint.

The finished shell was once more scanned in order to obtain the definite final shape of the

room. The glass fibre shell was at last covered by five millimetres wide strips of 0,6 millimetre

plywood, in a demanding and time consuming process. One advantage of this manual design

process over a computerized process is that there is a direct connection between designer and

object. The first strips on the surface are the ones that define the overall appeal of the entire

panelization, and while laying out the first strips it was very helpful having instant feedback

from the physical model. The manual process showed however, besides being very time con-

suming, also to be not especially precise. If the structure were to be built in full scale only using

manual tools and not the algorithmic knowledge from section 6.3 it would be difficult to create

a visually pleasing and qualitative result. All of the strips on the scaled model, which are surely

not perfect already at this stage of the design, would have to be traced or removed from the

model, scanned or otherwise imported into a CAD application, then scaled and produced in

full size. It is plausible to assume that small errors would add up and multiply during the above

process and that the resulting covering's quality would not be acceptable.

Figure 6.17: The scan results in a pointcloud that is used to create a triangular mesh with the scanned points as its vertices.

122

Figure 6.18: Images of the physical model showing the panelization

with plywood strips that has been found experimentally.

6.5 Conclusion

The physical scale model proved that the manual method of finding a geodesic pattern in order

to cover a freeform surface with long wooden boards offers only the advantage of immediate

feedback to the designer, while being inferior to a possible computerized method in various

other areas:

In a computerized process the reference surface can be completely defined in a CAD pro-

gram.

In a computerized process one does not have to rely on manual craftsmanship.

A computerized method would make an expensive and time consuming model obsolete,

unless it is a presentation model.

A computerized method would enable a straight design to production workflow with small

errors and tolerances.

124

7. References

Bablick, M. (2009). Holz und Holzwerkstoffe. Deutsche Verlags-Anstalt, München, in der Ver-lagsgruppe Random House GmbH.

Deleuze, Gilles (1987), A Thousand Plateaus: Capitalism and Schizophrenia. Minneapolis: University of Minnesota Press.

do Carmo, M. (1992), Riemannian Geometry. Birkhäuser.

Eigensatz, M. and Schiftner, A. (2011), Case Studies in Optimization of Glass-panelized Ar-chitectural Freeform Designs. In Glass Performance days Finland (Proceedings), 2011.

Eigensatz, M., Kilian, M., Schiftner, A., Mitra, N., Pottmann, H., Pauly, M. (2010), Paneling Architectural Freeform Surfaces. ACM Trans. Graphics, 29/4, #45, Proc. SIGGRAPH.

Flöry, S., Nagai, Y., Isvoranu, F., Pottmann, H., Wallner, J. (2012), Ruled Free Forms. In Ad-vances in Architectural Geometry, (Proceedings). p.57-66

Liu, Y., Pottmann, H., Wallner, J., Yang, Y. and Wang, W. (2006), Geometric modeling with conical meshes and developable surfaces. ACM Transactions on Graphics 25,3, 681-689

Lynn, G. (2004), Architectural Curvilinearity - The Folded, the Pliant and the Supple. In Archi-tectural Design - Folding in Architecture revised edition. Wiley-Academy.

McLeod, Virginia (2010), Details - Holzarchitektur. München: Deutsche Verlags-Anstalt.

Meredith, N. and Kotronis, J. (2012) Self-Detailing and Self-Documenting Systems for Wood Fabrication: The Burj Khalifa. In Advances in Architectural Geometry, 2012 (Proceedings). p.185-198.

Natterer, J., Herzog, T., Volz, M. (1996) Holzbau Atlas. R.Müller, Köln.

Pirazzi, C. and Weinand, Y. (2006), Geodesic Lines on Free-Form Surfaces - Optimized Grids for Timber Rib Shells. In World Conference in Timber Engineering WCTE, 2006

Porteous, J. and Kermani, A. (2007), Structural Timber Design to Eurocode 5. Blackwell Pub-lishing Ltd.

Pottmann, H., Asperl, A., Hofer, M., Kilian, A. (2007). Architectural Geometry. Bentley Insti-tute Press.

Pottmann, H., Huang, Q., Deng, B., Schiftner, A., Kilian, M., Guibas, L., Wallner, J. (2010), Geodesic Patterns. ACM Trans. Graphics, 29/3, #43, Proc. SIGGRAPH.

Pottmann, H., Schiftner, A., Bo, P., Schmiedhofer, H., Wang, W., Baldassini, N. and Wall-

ner, J. (2008), Freeform surfaces from single curved panels. ACM Trans. Graphics, 27/3, Proc. SIGGRAPH (2008).

Powell, K. (2004), Unfolding Folding. In Architectural Design - Folding in Architecture revised edition. Wiley-Academy.

Scheurer, F. (2010), Materialising Complexity. Archit Design, 80: 86–93. doi: 10.1002/ad.1111

126

Schiftner, A., Baldassini, N., Bo, P., Pottmann, H. (2008), Architectural freeform structures from single curved panels. In Advances in Architectural Geometry 2008 (Proceedings).

Schiftner, A., Leduc, N., Bompas, P., Baldassini, N., Eigensatz, M. (2012), Architectural Ge-ometry from Research to Practice: The Eiffel Tower Pavilions. In Advances in Architectural Geometry, (Proceedings). p.213-228.

Schmiedhofer, H., Brell-Cokcan, S., Schiftner, A. and Ziegler, R. (2008), Design and Paneli-zation of Architectural Freeform-Surfaces by PQ-Meshes. Poster presentation. Advances in Architectural Geometry, Conference, Akademie der Wissenschaften, Wien, 2008.

Schumacher, P. (2012), The Autopoiesis of Architecture: A New Framework for Architecture, Volume 1. Wiley.

Shelden, D. (2002), Digital Surface representation and the Constructibility of Gehry's Archi-tecture. Ph.D thesis in Department of Architecture, Massachusetts Institute of Technol-ogy, Cambridge MA. - noch nicht aufgetrieben

Spuybroeck, L. (2004), NOX: Machining Architecture. New York: Thames & Hudson.

Toman, R. (2010), Wien Kunst und Architektur. Potsdam: h.f.ullmann publishing

Vaudeville, B., Raynaud, J., King, M., Chalaux, M., Aubry, S., Witt, A. (2012), How Irregular Geometry and Industrial Process Come Together: A Case Study of the "Fondation Louis Vuitton Pour la Création", Paris. In Advances in Architectural Geometry, (Proceedings). p.279-294

Wallner, J., Schiftner, A., Kilian, M., Flöry, S., Höbinger, M., Deng, B., Huang, Q., Pott-

mann, H. (2010), Tiling Freeform Shapes With Straight Panels: Algorithmic Methods. In Advances in Architectural Geometry, (Proceedings).

127

8. Figure Credits

Figure 2.1 on page 8 - underground image cour-tesy of pampalini/123rf

Figure 2.2 on page 10 - courtesy of Weyland, www.weyland.at;

Figure 2.3 on page 11 - source: http://www.ihb.de/madera/srvAuctionView.html?AucTIid=797928;

Figure 2.4 on page 11 - courtesy of Don Schulte;

Figure 2.5 on page 11 - source: http://img.weiku.com//waterpicture/2011/11/13/5/high_quality_plain_mdf_sheet_634658679902609570_1.jpg;

Figure 4.1 on page 30 - courtesy of Zaha Hadid Architects, source: www.evolute.at;

Figure 4.3 on page 32 - (a), (c) and (d) courtesy of Waagner Biro, (b) courtesy of solidform.co.uk;

Figure 4.11 on page 41 - based on Figure 19.23 from Pottman, H., Asperl, A., Hofer, M., Kilian, A. (2007). Architectural Geometry. Bentley Institute Press;

Figure 4.13 on page 42 - based on Figure 19.31 from Pottman, H., Asperl, A., Hofer, M., Kilian, A. (2007). Architectural Geometry. Bentley Institute Press;

Figure 4.14 on page 43 - based on Figure 19.27 from Pottman, H., Asperl, A., Hofer, M., Kilian, A. (2007). Architectural Geometry. Bentley Institute Press;

Figure 4.15 on page 43 - based on Figure 19.26 from Pottman, H., Asperl, A., Hofer, M., Kilian, A. (2007). Architectural Geometry. Bentley Institute Press;

Figure 4.19 on page 51 - courtesy of Evolute;

Figure 5.2 on page 74 - courtesy of fondationlou-isvuitton.fr;

Figure 5.3 on page 74 - courtesy of Evolute;

Figure 5.12 on page 90 - courtesy of Luca Galuzzi;

Figure 6.1 on page 105 - courtesy of NOX;

Figure 6.2 on page 106 - (a) Shigeru Ban Architects, source: Centre Pompidou-Metz ©2008 Prestel Publish-ing, New York; (b) www.thedailytelecraft.com; (c) and (d) courtesy of design to production;

Figure 6.3 on page 106 - courtesy of Roland Halbe;

Figure 6.4 on page 107 - courtesy of Gehry Technol-ogies, source: www.wconline.com;

Figure 6.5 on page 110 - source: Wallner, J., Schiftner, A., Kilian, M., Flöry, S., Höbinger, M., Deng, B., Huang, Q., Pottmann, H. Tiling Freeform Shapes With Straight Pan-els: Algorithmic Methods. In Advances in Architectural Geometry, 2010 (Proceedings).

Figure 6.6 on page 110 - based on Figure 3 from Wallner, J., Schiftner, A., Kilian, M., Flöry, S., Höbinger, M., Deng, B., Huang, Q., Pottmann, H. Tiling Freeform Shapes With Straight Panels: Algorithmic Methods. In Advances in Architectural Geometry, 2010 (Proceed-ings);Figure 6.7 on page 113 - image from Pottmann, H., Huang, Q., Deng, B., Schiftner, A., Kilian, M., Guibas, L., Wallner, J. (2010), Geodesic Patterns. ACM Trans. Graph-ics, 29/3, #43, Proc. SIGGRAPH;

Figure 6.8 on page 113 - image from Wallner, J., Schiftner, A., Kilian, M., Flöry, S., Höbinger, M., Deng, B., Huang, Q., Pottmann, H. Tiling Freeform Shapes With Straight Panels: Algorithmic Methods. In Advances in Architectural Geometry, 2010 (Proceedings);

Figure 6.9 on page 114 - based on Figure 9 from Wallner, J., Schiftner, A., Kilian, M., Flöry, S., Höbinger, M., Deng, B., Huang, Q., Pottmann, H. Tiling Freeform Shapes With Straight Panels: Algorithmic Methods. In Advances in Architectural Geometry, 2010 (Proceed-ings);Figure 6.10 on page 114 - based on Figure 10 from Wallner, J., Schiftner, A., Kilian, M., Flöry, S., Höbinger, M., Deng, B., Huang, Q., Pottmann, H. Tiling Freeform Shapes With Straight Panels: Algorithmic Methods. In Advances in Architectural Geometry, 2010 (Proceed-ings);Figure 6.11 on page 114 - based on Figure 15 from Wallner, J., Schiftner, A., Kilian, M., Flöry, S., Höbinger, M., Deng, B., Huang, Q., Pottmann, H. Tiling Freeform Shapes With Straight Panels: Algorithmic Methods. In Advances in Architectural Geometry, 2010 (Proceed-ings);

129

Figure Credits

Appendix 1

The tables show the results of the studies con-

ducted on DStrips in section 5.3.3. Five groups

of DStrips with different initial aspect ratios of

their quad faces are tested. For each strip the

planarity value, the maximum and the aver-

age curve closeness and the surface area of the

strip in its original state as well as in its unrolled

state are measured. The values are compared

to the same values extracted from a reference

surface strip that provided the reference curves

for the strip creation. A surface strip which is

developable by definition is chosen as refer-

ence surface. For each group of DStrips there is

a table showing the results. The red line marks

an threshold. The strips below the red line are

considered to be suitable for the discretization

of the reference surface.

Asp

ec

t

Ra

tio

Na

me

Pla

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a (

cm

2)

Are

a o

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lle

d

stri

p (

cm

2)

Are

a d

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ce

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er

un

roll

ing

(cm

2)

Are

a d

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ce

aft

er

un

roll

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(%)

Are

a D

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to o

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(cm

2)

Are

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0.3

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fte

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0971

32

0.05

_0.0

0068

30,

0006

7899

0,04

0570

130,

0034

7215

9705

7,78

4554

4097

058,

0451

6031

0,26

0605

910,

0002

6851

160,

8580

2196

0,16

6008

97

0.05

_0.0

0028

20,

0002

9325

0,02

6487

59-0

,000

5222

597

054,

0501

6748

9705

4,55

4829

970,

5046

6249

0,00

0519

9815

7,36

7691

620,

1624

0687

133

Appendix 2

The tables show the results of the studies

conducted on DStrips in section 5.3.4. Five

groups of DStrips with different initial aspect

ratios of their quad faces are tested. For each

strip the planarity value, the maximum and

the average curve closeness and the surface

area of the strip in its original state as well

as in its unrolled state are measured. This ta-

bles provide less information than the tables

in appendix 1 because in this series of stud-

ies there is no reference surface to compare

the DStrip against. The main parameter in

this study is the area difference after unroll-

ing ,which compares the surface area of the

DStrip to the surface area of the unrolled sur-

face. There is a noticeable stagnation of the

values below the respective red lines in each

table.

Aspe

ct

Nam

ePl

anar

itySc

ale

Inva

riant

Pl

anar

ity-

ness

-ne

ssSt

rip a

rea

(cm

2)st

rip (c

m2)

(cm

2)

0.3

0.3_1.99

1,98

5765

290,03

4433

881,74

5764

640,17

1808

2119

1631

,081

0547

819

1843

,231

4088

721

2,15

0354

090,11

0707

700.3_1.13

1,13

2038

410,02

5131

120,94

4668

270,15

9314

8819

1765

,046

9759

319

1866

,246

5908

410

1,19

9614

910,05

2772

710.3_0.77

0,77

2962

670,01

6431

900,98

5305

290,17

8465

8419

1831

,747

2219

419

1888

,938

6043

757

,191

3824

30,02

9813

300.3_0.53

0,53

1520

650,00

2841

451,57

6906

510,35

2792

6919

1780

,496

8509

319

1788

,077

5150

47,58

0664

110,00

3952

780.3_0.15

0,15

2910

300,00

1152

671,53

6188

470,36

8788

7419

1816

,130

4735

219

1818

,732

5171

82,60

2043

670,00

1356

530.3_0.10

0,10

1280

120,00

0892

091,63

2909

600,42

3910

4419

1836

,965

7097

319

1838

,474

7361

51,50

9026

410,00

0786

62

0.2

0.2_1.41

1,40

7554

210,02

4993

470,00

0077

77-0,003

4052

219

1569

,436

4111

919

1777

,530

6962

720

8,09

4285

080,10

8626

040.2_0.71

0,71

4330

050,01

6502

940,12

8903

490,03

4961

7919

1758

,027

0790

819

1809

,302

5313

551

,275

4522

70,02

6739

660.2_0.34

0,34

3403

090,00

4441

330,44

7473

820,10

0312

5019

1819

,513

4447

919

1835

,898

1207

416

,384

6759

40,00

8541

710.2_0.07

0,07

6684

160,00

0496

721,18

6776

170,28

9485

4019

1836

,411

0047

419

1838

,470

4403

2,05

9435

570,00

1073

54

134

Aspe

ct

Nam

ePl

anar

itySc

ale

Inva

riant

Pl

anar

ity-

ness

-ne

ssSt

rip a

rea

(cm

2)st

rip (c

m2)

(cm

2)

0.15

0.15

_1.05

1,05

5894

650,01

9082

380,00

0076

33-0,002

5257

119

1569

,426

5730

719

1777

,686

8750

620

8,26

0301

990,10

8712

700.15

_0.39

0,39

8748

020,00

6200

990,77

5524

410,10

0447

6119

1812

,290

7957

819

1871

,070

3829

958

,779

5872

10,03

0644

330.15

_0.12

0,12

3571

470,00

0906

192,15

7660

150,36

0773

4119

1834

,995

0306

419

1846

,032

3431

711

,037

3125

30,00

5753

540.15

_0.05

0,04

8417

330,00

0946

251,77

7628

050,48

5265

6819

1779

,639

4031

119

1781

,732

5836

82,09

3180

560,00

1091

450.15

_0.02

0,01

6030

790,00

0329

262,13

5404

810,68

6923

2819

1770

,304

8770

719

1770

,702

5045

00,39

7627

430,00

0207

350.15

_0.01

0,01

0010

900,00

0115

932,83

3149

830,86

0037

3819

1768

,147

3228

719

1768

,566

2176

60,41

8894

780,00

0218

44

0.1

0.1_0.7

0,70

4321

570,01

2887

440,00

0080

93-0,001

6842

219

1569

,424

6484

819

1777

,574

9578

720

8,15

0309

390,10

8655

290.1_0.33

0,32

8881

070,00

7788

940,28

3287

470,03

4502

7519

1780

,464

7995

719

1828

,199

9986

247

,735

1990

50,02

4890

540.1_0.17

0,16

9340

450,00

3222

011,26

7100

680,20

1982

3219

1889

,814

6631

219

1915

,732

7289

825

,918

0658

50,01

3506

740.1_0.08

0,07

6097

650,00

1036

880,59

5652

020,18

0045

3919

1795

,135

9852

119

1805

,066

7485

69,93

0763

340,00

5177

80

0.05

0.05

_0.35

0,35

2717

020,00

6503

630,00

0083

52-0,000

8268

919

1569

,427

7363

719

1777

,382

9850

520

7,95

5248

680,10

8553

460.05

_0.16

0,16

3000

300,00

2628

550,30

6400

740,03

8661

5319

1789

,725

9653

219

1858

,168

2512

468

,442

2859

10,03

5686

110.05

_0.07

0,06

9702

860,00

0905

630,86

9665

450,07

7261

2919

1900

,774

6351

819

1910

,123

6182

19,34

8983

030,00

4871

780.05

_0.03

0,03

3516

240,00

0304

220,27

4190

320,06

7214

4719

1881

,368

4700

219

1879

,271

4555

82,09

7014

440,00

1092

870.05

_0.02

0,01

8847

130,00

0138

020,06

5090

600,02

0895

5119

1847

,921

3470

319

1919

,115

0912

371

,193

7442

10,03

7109

47

135