Freeform Geometries in Wood Construction - Marko Tomicic by Marko Tomicic

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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 E104 Institute of Discrete Mathematics and Geometry Geometric Modeling and Industrial Geometry Research Unit

eingereicht an der Technischen Universität Wien Fakultät für Architektur und Raumplanung von 0DUNR 7RPLþLü 1029061

Wien, am 01.10.2013 eigenhändige Unterschrift

For my parents.

Abstract Despite being one of the oldest construction materials on earth and having numerous advantages over modern high tech construction materials, wood has been greatly marginalized in the construction of todayâ&#x20AC;&#x2122;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 freeform architectural structures. For each of the above rationalization techniques a project is designed, rationalized for and detailed for the manufacturing 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 aufgrund ihrer Größe vor dem Bauen rationalisiert werden. Diese Arbeit untersucht Nutzungsmöglichkeiten des Baustoffs Holz beim Bau von Freiformstrukturen. Dies erfolgt durch die Kombination von Rationalisierungen von Freiformflächen mit planaren Vierecksnetzen, abwickelbaren Streifen und geodätischen Mustern mit neuen rechnerischen Werkzeugen und Scripting. Für jede der genannten Rationalisierungsmethoden wurde ein Projekt entworfen, rationalisiert und für die Fertigung 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. Iâ&#x20AC;&#x2122;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. Iâ&#x20AC;&#x2122;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 StraĂ&#x;l, 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

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2

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

1.3

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

1.4

Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5

Disposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Wood as a Construction Material

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

The Geometry of Freeform Architecture 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 BĂŠzier 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

PQ Meshes 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

Developable Surfaces and DStrips

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

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

References

126

Figure Credits

129

1. Introduction 1.1 Background Smooth architectural surfaces The modernist movement began its over five decades long hegemony in the field of architectural 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 Architecture 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 represents 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 surfaces 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 2

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

answer those questions. Smooth surfaces that are part of a designed industrial object like a car or a household appliance or a kidâ&#x20AC;&#x2122;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 division 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 processed. 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, produces oxygen, and stabilises the ground, wood is a lightweight material that has an up to sixteen 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 structural 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 molecular composition allows wood to adopt a curved shape through pure bending. All the properties above make wood a good material for constructing freeform structures.

CNC - Computerized Numerical Control

Aim of the Thesis

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 glasssteel 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 parametric 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 5

Planar Quadrilateral (quad) meshes DStrip - developable strip

1.5 Disposition The thesis is structured as follows: t chapter 2 gives an introduction to wood as a construction material and to the engineered wood products that are used later in the projects of chapters 4, 5 and 6. t chapter 3 gives a profound theoretical background on the geometry that is necessary to follow the work in the later chapters. t chapter 4 explores PQ meshes and the possibilities of using them in wooden freeform structures. t chapter 5 explores DStrips and the possibilities of using them to cover freeform surfaces with panels of plywood by means of bending the panels without tearing or breaking them. t chapter 6 explores geodesic patterns and the possibilities of using them to cover freeform surfaces with wooden planks.

Wood as a Construction Material

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 building 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 century brought a new invention which was going to change architecture once more â&#x20AC;&#x201D; 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 construction 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: t development under environment friendly conditions t production, manufacturing and processing without mentionable use of fossil fuels t good dead weight to load bearing capacity ratio t different wood species with different visual qualities t good isolator and heat accumulator t availability of high quality connection techniques t possibility of prefabrication Wood has proven to be one of the most sustainable building materials. It is a self growing material which is completely recyclable The tree regulates our climate, stabilizes the ground and purifies 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].

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. Chemical 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 treeâ&#x20AC;&#x2122;s growth, the cambium [Bablick 2009]. Underneath the cambium, 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

Sapwood Heartwood

Juvenile wood

Outer bark Inner bark

Pith

Cambium

Rays

Annual rings

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

Types of Timber

and the summerwood or latewood. The springwood forms during a relatively fast growth period 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 raysâ&#x20AC;&#x2122; purpose is to transport minerals 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 available and comparatively cheaper because of their quick growth and speed of felling [Porteous and Kermani 2007].

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 substances. Due to the long growth period hardwoods 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 product is sawn limits the quality and size of the final product. If one wantâ&#x20AC;&#x2122;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 advantages 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]. Figure 2.2: Bending plywood, thickness 7mm, bending radius 25 cm

2.3.1 Plywood Plywood was the first EWP to be invented. It is a flat panel made by bounding together at

Engineered Wood Products (EWPs)

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 number and then bounded under high pressure. The outer laminates (face ply and back ply) are always made of veneer, whilst the inner laminates can be made of veneer as well as of sliced or sawn wood. The inner laminates are the core

Figure 2.3: Glulam member

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 plywoodâ&#x20AC;&#x2122;s thickness, to a radius of 25 cm in perpendicular direction to the slope of grain [Por-

Figure 2.4: Laminated veneer lumber

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-

Figure 2.5: Medium-Density Fibreboard (MDF) panels

tially parallel to the longitudinal axis. This technology enables the production of straight and curved members. Timber boards with thickness 11

of 33 - 50 mm are used as laminates in straight or slightly curved members, whereas much thinner 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 plywood, 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) , tempered hardboard, cement-bonded particleboard etc. are used extensively in housing construction 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 fibreboards is kerfed MDF which is possible to be bent, depending on itâ&#x20AC;&#x2122;s thickness, to a radius of just 25 cm [Porteous and Kermani 2007].

The Geometry of Freeform Architecture

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 surfaces. 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 [Pottmann et al. 2007].

3.1.1 Rotational Surfaces Rotational surfaces (or surfaces of revolution) are surfaces in Euclidean space created by rotating 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 A A M m c p

Sc cp

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

d o

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 k

a translational surface (Figure 3.2). If the path curve d is a straight line, then an extrusion surface is generated [Pottmann et al. 2007].

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

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 [FlĂśry et al. 2012].

Cylinders, cones, one-sheet hyperboloids and hyperbolic paraboloids are some of the simplest ruled surfaces. Generally, ruled surfaces are created by moving a point p of a straight line segment g along a curve c and changing

Figure 3.3: A ruled surface

the lineâ&#x20AC;&#x2122;s direction continuously. [Pottmann 16

Freeform Surfaces

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. r

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 circles with the radius r lying in the normal planes of a spatial curve. [Pottmann et al. 2007] c

3.1.5 Offset Surfaces An offset surface Sd of the surface S is the surFigure 3.4: A pipe surface as (a) an envelope of

face with a constant normal distance d to the spheres, and (b) a collection of circles lying normal to original surface S. The offset surface Sd and the

the curve c.

surface S share their normals. Further, the tangent 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 d

surfaces and surface offsets are not closely sufficient to meet the high demands of todayâ&#x20AC;&#x2122;s

S Sd

state-of-the-art architectural designs. More complex shapes are nowadays designed using freeform surfaces because those surfaces offer more flexibility compared to the traditional

Figure 3.5: The offset surface Sd lies at constant distance d to the surface S.

surfaces. We will discuss three types of surfaces 17

in this chapter: Bézier surfaces, B-Splines and NURBS surfaces and subdivision surfaces. Bézier 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 (a)

Before the invention of computers, the designd0 d2

er had to draw the freeform curve by hand using 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 d3

(b)

in particular difficult because the entire arm had to be moved smoothly. To draw very long

smooth curves designers used a mechanical d0 d2

aid to guide their hands. Such tools were called splines. Splines were usually bendable wooden or metal rods whose shape was controlled by a

d3 Figure 3.6: Image showing a degree 3 B-Spline curve with six control points (a) consisting of three cubic Bézier curves (b).

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 simplicity 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 polygon. [Pottmann et al. 2007]

Freeform Surfaces

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-

(a)

points with the highest possible smoothness. B-Splines can be generated by curve subdivi-

d0 d2

sion. This is a process in which a given coarse polygon is subdivided iteratively with Chaikin’s

algorithm or Lane-Riesenfeld’s algorithm. [Pottd3

mann et al. 2007] (b)

3.2.1.3 NURBS Curves NURBS1 curves are the third and most sophis-

d0 d2

ticated type of freeform curves. They offer further 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-

d3 (c)

Spline curves are special cases of NURBS curves wherein all weights in the control points are

d0 d2

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

d3 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 attracted by that point (c).

Translational Bézier surfaces can be created from Bézier curves. Two curves are needed for

NURBS — NonUniform Rational B-Spline

the creation of a translational Bézier surface, one Bézier curve of degree m (bm) and one Bézier curve of degree n (bn). In order for the curves b22

to be suited for generating a translational sur-

b23

face, they need to share a common endpoint b21

b20

b12 b13

b00. The control points of the Bézier curves are denoted with the double index notation. The

b11 b02 b10

control points of the quadratic curve b2 in the

b03

example shown in Figure 3.8 will be called b00,

b01

b10 and b20. The control points of the cubic curve

b00

b3 are b00, b01, b02 and b03. TThe translational surface of those two Bézier

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 .

curves will carry one family of quadratic Bézier curves b2 and one family of cubic Bézier curves b3. In order to distinguish between the parameters of those two Bézier curves, the parameter b2 is denoted as the u parameter and the parameter along b3 as the v parameter of the sur-

b12 b22

face. [Pottmann et al. 2007]

b20 b12 b11

b10

General Bézier surfaces are a straightforward

b32 b02

b13 b30

General Bézier surfaces

b01

b00

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

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

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 describes the vertex column. The number of con-

Freeform Surfaces

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] p

3.2.3 B-Spline and NURBS Surfaces Since Bézier surfaces are constructed from Bézier curves, they share the same drawbacks. When the degree of the Bézier surface becomes too high it stops representing its control mesh accurately. The other drawback is that Bézier sur-

faces do not feature local control, hence changing the position of one control point will have effect on the whole surface. This makes the design 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 vcurves. Similar to the case with freeform curves where the NURBS curve is the most sophisticated, NURBS surfaces are among the most powerful freeform surfaces. They have all the Figure 3.10: A NURBS surface with different weights in same possibilities as B-Splines, offering in addition the possibility to adjust the weight of each

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 surface with increased weight in control point p.

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 sur21

face wherein all weights in the control points are equal. [Pottmann et al. 2007] 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)

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-, quadrilateral- 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-

Array of faces A = (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)

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

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

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.

15 13

5 0

A 1

E B

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

Meshes

Meshes can represent a coarse discretization of a smooth surface, like on some buildings or very smooth discretizations like those which are used in todayâ&#x20AC;&#x2122;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 valence four, have certain topological restrictions for the modelling 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 dominant mesh most of the vertices would be of valence four. A vertex 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 singularities. 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 subdivision refinement

divided mesh inherits the singularities from itâ&#x20AC;&#x2122;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. According 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 intersection 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 RiviĂ¨re (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

Freeform Surface Rationalization

made famous in his designs (Figure 3.14). Many of Gehryâ&#x20AC;&#x2122;s designs are modelled in paper or thin metal or plastic sheets that are easily bendable, to ensure developability of the final surfaces [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 order 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 geometry 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.

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 meeting in each node. PQ (planar quadrilateral) meshes have, in comparison to triangular meshes, fewer panels, less cutting waste and less total joint length. (a)

Developable strip panelizations Developable strip panelizations (Figure 3.15 - c) are one-directional refinements of a planar quad pan(b)

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

(c)

planar. There are numerous benefits from the design approach with DStrips. Developable strips can be manufactured from planar sheets without the necessity

Figure 3.15: Freeform surface rationalization. (a) for any moulding or advanced bending machines. A freeform surface rationalized with triangular panels. (b) A freeform surface rationalized with planar This leads to a significant reduction of cost. The fact quadrilateral panels. (c) A freeform surface rationalized with developable strips that they are produced by bending material in one

direction makes them a semi-discrete representation of a freeform surface, thus they are perfectly smooth in one direction as opposed to panelizations with PQ panels. Another benefit of developable surfaces is that offsets of a developable surface are developable again, which means that multilayered 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]. 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 panelizations 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 scripting to create a multi-layered freeform wood structure.

4.2 Application in Architecture Upon 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)

Application in Architecture

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 (a) 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 (b) 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

(c)

(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

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

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

can be used in interior applications, where applicable, (a)

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 Dongdaemun Design Plaza building by Zaha Hadid architects in Seoul (Figure 4.1). Evolute created a layout of quadrilateral panels to cover the entire exterior surface of the building. This

(b)

panel layout comes close to a PQ panelization. In some cases it is not possible to achieve a PQ panelization in which all panels are planar, but if the majority of the panels are planar the panelization can be considered a PQ panelization. The panelization of the Dongdaemun Design Plaza is not a true PQ paneliza-

(c)

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 architectural freeform project with a true PQ panelization to show as reference. A very interesting

(d)

structure however, is the steel glass roof for the Department of Islamic Arts at Louvre Museum (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

Figure 4.3: Department of Islamic Arts at Louvre Museum.

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 hybrid structure consists of totally planar quad-

Application in Architecture

rilateral panels and triangles. A spacial optimization 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.

(a)

Starting from the reference surface (Figure 4.4 - a) a fairly coarse quad mesh is constructed first. The size and proportions of the quads in this mesh will have direct influence on the (b)

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 optimized for planarity because the planarity (c) 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(d)

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 algorithm results in a mesh whose edges form polylines that are diagonal to the polylines (e) 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-

Figure 4.4: Geometry of the roof structure of the Department of Islamic Arts at Louvre.

face and allow the first next group of vertices to snap to its boundary (c). This step was necessary because of what happens in the next step, which is a "dual" subdivision. The dual

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 alternating 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.

Geometric Properties of PQ Meshes

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 double 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 â&#x20AC;&#x153;niceâ&#x20AC;&#x153; 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.

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

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 translational 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 B

S c

family A

Figure 4.6: A conjugate curve network on a freeform surface

Geometric Properties of PQ Meshes

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 tangent 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 conjugate 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

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 pleasing 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 result. The best optimization results will generally be obtained if the network of principal curvature 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 roughly 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 different levels of detail and it comes closer to the input shape after each subdivision and optimization 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.

Geometric Properties of PQ Meshes

This makes the optimization algorithm move those vertices as close as possible to the speci-

(a)

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-

(b)

our coding shows the planarity of the faces. The dark blue faces have the lowest planarity value, hence they are close to completely planar, whereas the yellow panels have a slightly higher curvature and planarity value.

(c)

(d)

(e)

(f )

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

4.3.4 Mesh Offsets Mesh offsets play an important role in multilayered 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 structures correspond to the mash faces. The nodes are formed in the mesh vertices and the beams M

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 efficient structure, those meshes should be paral-

lel. Two meshes M, M* are parallel (Figure 4.9) if they are combinatorially equivalent and the

Figure 4.9: The Parallel meshes M and M* are combinatorially equivalent and their corresponding edges are parallel.

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 central planes of incoming beams accommodate

the node axis A in the adjacent vertex. This case M*

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

Figure 4.10: A geometric support structure, formed by connecting corresponding vertices of two parallel meshes M and M*.

can be obtained via a parallel mesh M* by connecting the corresponding vertices. The axes

Geometric Properties of PQ Meshes

should be approximately orthogonal to M, thus they are obtained from a mesh M* lying at a approximately 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 M

mesh Md of a PQ mesh M is parallel to M and lies at a constant distance to M. The way of defining the distance needs to be specified. There are three possibilities for this: vertex offsets, edge offsets and face offsets. The three methods all have different outcomes and utilize different approaches. In order to continue with

the explanation of these concepts a digression to the discrete Gaussian image is in order. S*

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 (Gaussian image mesh) is defined as the scaled difference mesh S = (Md - M)/d of the two parallel meshes M and Md. The resulting scaled differ-

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

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 distances 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 41

distance d between M and Md is constant. The difference of the two meshes (M and Md) was did

vided with d and therefore the Gaussian image S must have the distance 1 to the origin, thus

a c

S* is a unit sphere. Considering the above (PQ mesh M with offset mesh Md at constant distance d and the Gaussian image mesh S = (MMd)/d ) all offset properties are encoded in the Gaussian image mesh S. [Pottmann et al. 2007]

4.3.5.1 Vertex Offsets d1

Vertex offsets result in a special type of meshes, known as circular meshes. Circular PQ meshes c1

share the property that their PQ faces have cir-

cum-circles (Figure 4.12). In vertex offset meshes 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 b1

planes on which the faces lie intersect the unit

Figure 4.12: Two parallel quads with their circumcircles.

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 because all edges of the Gaussian image mesh of a pair of meshes M and Md at constant edge-toedge distance d lie tangent to the unit sphere. Figure 4.13: The edges of a Koebe mesh S are tangent to the unit sphere S*.

They are called Koebe meshes. The Gaussian

Geometric Properties of PQ Meshes

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. Various tools for the computation of a Koebe mesh

n v

S are available. As mentioned above, when the Koebe mesh is obtained it stores all necessary

r3 r2

information from which an endless number of offset meshes can be obtained. A offset mesh

r1 C

Md is calculated as Md = M + d.S. Meshes with edge offsets also have the property that incom-

Figure 4.14: The faces of a conical mesh that meet in

ing edges form the same angle with the node one vertex v are tangent to a cone of revolution C. The vertex normal n is the cone's axis.

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 offsets 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 offFigure 4.15: The faces of the planar Gaussian image

set meshes M and Md are parallel in this case. mesh of a mesh with face offsets (conical mesh) are tangent to the Unit sphere. Further, the Gaussian image mesh S has face 43

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 image 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 corresponding 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 common 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.

Project: Fair Stand

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 exhibited 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 exhibited already at Bau 2013. The stand that they used in 2013 was a 6 meters long and thirteen meters wide end stand, where one of the thirteen meter long sides was blocked by a neighbouring 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 dominated 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 neighbouring 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

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

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 created, 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 following 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 millimetre 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 planar 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 presented 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 bearing 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

Project: Fair Stand

on this property than on the above mentioned

(a)

ones during the mesh optimization. Creating the mesh With the designed surface set as reference in EvoluteTools for Rhino, the first step towards the discretization with a PQ mesh was the creation

(b)

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 (c)

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 design where the designer has the most creative

(d)

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 (Figure 4.18 c ), the coarse mesh is subdivided with

(e)

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

(f )

the mesh starts looking more like the intended design, the reference surface. After another repetition of subdivision with the same algorithm 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 number, the size and the aspect ratio of the coarse mesh's faces influenced the current outcome. It

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

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 optimization 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 resembling 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]

Project: Fair Stand

4.4.2 The Scripting Process In the introduction to this chapter I stated that the actual structure with all its members is generated 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 access 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 pavilion for the 2012 Olympic games in London by Pavilion Architecture for which Evolute rationalized 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.

wooden members and production geometry for fabrication. My goal, inspired by Evolute's part in the design of KREOD was to automate (a)

the creation process and the process of outputting production data. The construction system of KREOD is based on a hexagonal mesh and it features nodes where three wooden members meet to form reciprocal nodes. Inside the pavilion there are mem-

(b)

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

(c)

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 resembles the edges of the mesh, as connection between the outer two layers (Figure 4.20 c ). On the following pages the creation process of

(d)

each piece in this structure will be described.

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

Project: Fair Stand

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 ) because 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 another. 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 calculated (n1, n2, n3, n4). The four vertex normals are used to calculate one general normal n for the current 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 position. Therefore the panel needs to be smaller than the original face. This is achieved by specifying a gap between the panels which is set to five millimetres in this case, but can be adjusted

n4 v4 n1

p2 Fn

n e2

Detail B v3 n2

Detail A

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

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 scaling 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.

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

Project: Fair Stand

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

(a)

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 (b) 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 (c)

inner layer so they technically belong to both layers and they will be mentioned again later in the description of the inner layer. The cylinders are simple metal rods, six millimetres in diameter, that feature tapped holes for machine screws on both sides. They are standardized pieces, all of which have the same dimensions, Figure 4.22: Three groups of elements that comprise therefore there is no need to detail their design with the script, but it is important to have sim-

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)

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 55

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 calculating one average normal for the edge from its vertex normals I assume that the difference between 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.

v2 v2,1

v2,1,1

n = n1 + n2

v2 v1

ed e

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

v2,2 -n v2,2,1

-n

ex = ed Ă&#x2014; n

-ex

Detail A

Project: Fair Stand

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 vector 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 intersect 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 opposite 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

point bp is then moved along the edge vectors n

(a)

of the adjacent edges, four times in four different 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

(b)

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 n

(c)

are moved so that for each of the four points on

e3 c1,1

b1,2

c1,2

h e d1,1

b1,1

f e4

d1,2

a1,2 a1,1

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

(d)

the ends of the cross two points on the cross'

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

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

beam's width. Of course the points a1,1 and a1,2 will leave the base plane by this move-

Project: Fair Stand

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

(a)

back towards the middle along their respective edges and calculating the average value of two e1

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 (b)

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

b2 cross

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.

(c)

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

b2 cross

the screw that attaches to the cylinder will be C

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

4.4.2.3 Scripting the Inner Layer Introduction While the middle layer provides stability to the structure by connecting adjacent nodes directly 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 Ă&#x2014; 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 respective 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 connections would increase the price, manufacturing time and weight of the structure. Reciprocal nodes make complicated metal connections in the nodes obsolete by relying completely on the wood members to form stable connections on their own. Each beam is manufactured 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 connecting 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 connected 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. Further 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

Project: Fair Stand

would prevent movement in the inner layer and vice versa. After this "happy accident" was discovered it became clear it would be possible to create a structure that is much simpler in terms of detailing, manufacturing and assembly 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 final 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 between them. By comparing the drawings in Figure 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 images show all pieces that are necessary to build Figure 4.26: The connection detail of the reciprocal structure becomes much simpler when the screws one node of the structure. (above) are replaced with glue (below).

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

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

Project: Fair Stand

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 (Îą) 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

now, is created, the script writes user data to the mesh, storing the information about the (a)

current element. It uses the names of mesh

n n1

n2 e1

v1 v1,1

v2 e1,2

e1,1

Panels of the outer structure are stored with the names of their mesh faces, whereas beams

Îą v2,1

Îą

elements to store the pieces of the structure.

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

(b)

n n1

e2 e2,2

n2 v2

e1,2 p1

perform Boolean operations on one element

purpose. Figure 4.29 - d shows a part of the n2

n n2 v1

Figure 4.29: The creation of the beams and connections in of the reciprocal structure.

in the mesh's user data. If the adjacent beam

operation, or else it is created specially for this n

looks for the names of the adjacent elements

or box already exists it is used for the boolean

e3,2

(c)

(d)

such as the beam B in Figure 4.29 - c, it first

complete reciprocal structure where all parts fit perfectly into each other.

Project: Fair Stand

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 coordinate 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

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.

Developable Surfaces and DStrips

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 double 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 certainly 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 crĂŠation 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 segmentation 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 constructions. 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

of single curved panels is that offsets of developable surfaces are developable as well. This allows the engineers to use developable box beams as mullions which allows adopting a single continuous detail for fixing the panels. This application was successfully used by [Schiftner et al. 2012] in the glass facade for the Eiffel Tower Pavilions (Figure 5.3).

5.1 Developability Developable surfaces are also known as single curved surfaces, meaning that they carry one family of straight curves, the rulings, making them ruled surfaces. Developable surfaces, by definition are ruled surfaces, although not Figure 5.2: The Fondation Louis Vuitton pour la crĂŠation building in Paris by Frank Gehry

all ruled surfaces are developable. Developable surfaces can be unrolled onto a flat plane so that the in-surface distances remain unchanged. The rulings of such surfaces have the special property that all points of a ruling have the same tangent plane. Some well known developable surfaces are the cone or the cylinder. Translational surfaces in which one of the two

Figure 5.3: Developable box beams are used as mullions for the facade of the Eiffel Tower Pavilions

generating curves is straight, called extrusional surfaces are also developable, because the extrusion direction of such surfaces consists of straight, parallel lines. Another explanation of developable surfaces is that they are ruled surfaces with vanishing Gaussian curvature where K equals zero at all of their points. Developable surfaces have the property that they can be

DStrip Models

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 developable 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 illustration 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

Figure 5.4: Developable paper strips. Surfaces that are modelled with paper by applying pure bending are developable

guarantee that the resulting faces will remain planar. There is a strong connection between

(a)

PQ meshes, which were discussed in chapter "4. PQ Meshes", and DStrip models. Especially conical 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 Left

Front

(b)

The best results, when working with DStrip models on a freeform surface, can be expected 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

(c)

praxis this can be achieved by creating a coarse mesh which approximates the reference surface and follows the principal curvature lines. The mesh is then optimized for planarity and the conical property. The connection between (d)

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 principal curvature lines extracted (b). The optimized conical PQ mesh is aligned with the principal curvature lines (c) which results in a smooth

Figure 5.5: The connection between the network of principal curvature lines, conical meshes and DStrip models.

DStrip model (d).

DStrip Models

5.2.2 DStrips Between Two Curves Developable strips can also be obtained by creating them between two curves. The curves can be extracted from a reference surface, for instance by intersecting the surface with a set of planar surfaces. This approach is helpful where it is applicable, because the fact that the edges (a)

of the strips are straight lines simplifies the substructure to a great extend. When the two border curves are obtained (Figure 5.6 - a), a sim(b)

ple coarse mesh between them is created (b). The mesh is optimized for planarity of the faces and closeness to the reference curves. The optimized mesh is further subdivided in only one

(c)

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

(d)

needs to be optimized at least once after the chosen number of subdivision iterations. There (e)

is a possibility to track changes in the mesh via an analysis mode which represents values such as planarity, scale invariant planarity and close- Figure 5.6: Manual creation of a DStrip between two input curves.

ness to the reference with colours in the viewport. 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 giving the maximum possible control to the user, 77

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 connections between the end points of the curves. (a)

This is not ideal because it limits the freedom of the optimization algorithm to adjust the position of vertices. The optimization algorithm will

(b)

tend to shrink the mesh because of the optimization for fairness, but the fairness parameter is very important because it keeps the angles between two consecutive edges in the mesh

(c)

as straight as possible. Omitting the fairness parameter would result in a strip with rough edges that does not resemble a smooth sur-

(d)

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

(e)

vertices from the optimization and limits the movement of their neighbours. All the above results in a less accurate DStrip with less good

Figure 5.7: The creation of a DStrip between two input planarity curves by the usage of the automated method.

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, method. The reference curves (a) are duplicated and

DStrip Models

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 automated 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).

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 planarity 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 perfectly 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 developable. 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

DStrip Studies

according to their initial face aspect ratio in five groups â&#x20AC;&#x201D; 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 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. 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, expressed 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 perfectly 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

against the created DStrips can be compared. Being that a DStrip is created between refer(a)

p1,1

ence curves that are borders of a developable

p0,1

surface it can be expected that it is possible to

p2,1

create an almost perfect DStrip between those

p4,1 p1 p0

curves and that the values that are obtained in

p3,1

p5,1

this test will not be matched in practice. Nevertheless, the possibility to compare the DStrips

(b)

in their original state and after they have been unrolled to the reference surface which is by K

definition developable makes it worth to conduct these tests before moving to a more practice oriented example with arbitrary reference curves.

(c)

5.3.3.1 Developable Reference Surface Creation S

For DStrip studies with border curves of an actual developable surface as reference curves for the DStrips, a developable surface had to be found first. The simplest way of finding such a

(d)

surface is to create two planar parallel curves S

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 Figure 5.8: The creation process of the developable reference surface

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

DStrip Studies

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 results 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 maximal 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 order 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.

5.3.3.3 Results The results of these tests show that it is possible to recreate the developable strip S* from Figure 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 discretization of the reference surface: t

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.

DStrip Studies

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 represent 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 reference 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 surface area of the DStrip and the surface area of its development. As the planarity becomes smaller 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.

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: t

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.

Figure 5.10: Site plan showing the boulder wall on the shore of the Donaukanal, highlighted in red.

Project: Bouldering Wall

5.4 Project: Bouldering Wall The task in this project was to design a bouldering wall at Pier9, on the shore of the Donaukanal 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 climbing 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 designed 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-

Figure 5.11: People exercising in the bouldering area at Pier9.

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 formations of stone and rock, namely boulders. The wonderful flowing lines of the red rocks 89

of the Antelope Canyon, Navajo, Arizona, USA (Figure 5.12) were the inspiration for this bouldering 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 artificial 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 rapidly into an independent sport discipline with its own community of followers. The main difference between sport climbing and bouldering is in the short, dynamic and daring routes that are preferred in bouldering instead of long and high routes that sport climbers climb. Average boulder "problems" are just a few meters long with an average of five to seven moves between start and end of the route. Boulderers are not protected with ropes and harnesses, but instead they climb close to the ground, Figure 5.12: Antelope Canyon in Navajo, Arizona, USA

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 extreme overhangs that are difficult to climb. The

Project: Bouldering Wall

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 together 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 surface 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

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 developable surfaces in this project were created using the methods described in section 5.2.2 with the information from the studies in mind. The process 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 lofted 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 obtained by intersecting the DStrip model with Figure 5.14: (Above) Frontal view of the bouldering wall. (Below) Left-side view of the bouldering wall. Scale 1:200

a series of seventy vertical and five horizontal

planes. The Intersection curves were extruded and intersected with each other to create an orthogonal waffle structure on which the DStrips can be mounted. The same type of substructure 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 A

the panels and to minimize production waste.

Figure 5.15: (Above) Top view of the bouldering wall. (Below) Section A-A. Scale 1:200

5.4.1 The Physical Model The physical model in this project has an important role besides providing a truthful representation of the design. The model is also a (a)

proof that the DStrips are in fact developable and that they can be produced out of flat plywood 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

(b)

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 centimetres high. The DStrips were CNC cut out of a one millimetre thick sheet of plywood. This thickness is proportionally larger than it should be, but it was chosen so that the strips would provide more resistance to bending and thus be less prone to twisting around the stronger axis. 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

Substructure Before the strips could be joined into the desired 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

Project: Bouldering Wall

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

Figure 5.18: Physical model of the bouldering 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 developable 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 resembles the designed surface. Due to the fact that the plywood is a relatively soft material it can be bent easily. 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 individual 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 triangular 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.

(a)

(b)

(c)

(d)

(e) 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 registered against the designed surfaces.

Scan results and evaluation The 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 resembles 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 issues 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

B i(x) Ψi

x A i(x)≈B i+2(x)

si x’

s i+1 s i+2

Geodesics Curves on Freeform Surfaces

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 principal 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 geodesic. It is simple to determine the geodesic lines on regular shaped surfaces, e.g. the geodesic lines of a cylinder 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 determine the geodesic lines in freeform surfaces [Pirazzi and Weinand 2006]. In other words, geodesic lines are, besides distance minimizers, also curves of zero geodesic (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 supporting structure of such a surface. Cladding The absence of geodesic curvature makes patterns of geodesic curves suitable for dealing with the cladding

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).

of freeform structures with straight wooden panels which bend only around their weak axis. Such a cladding will mainly be used in interior applications as shown in the wood ceiling for the office lobby of the 105

Burj Khalifa in Dubai by Gehry Technologies (Figure 6.4). The

(a)

used panels should be close to developable and their development 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 (b) approached this problem experimentally by designing the cladding of an office space with paper strips as shown in Figure 6.1 [Spuybroek 2004]. [Wallner et al. 2010] showed how it is possible to decompose a surface into regions of which (c) each can be covered with a family of geodesic lines at nearly constant distance using computational tools. Support structure

(d)

Geodesic curves are also suitable for the design of load bearing support beams in freeform structures that can be manufactured 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- Figure 6.2: Centre Pompidou Metz (a) duced because the bending around the strong axis is avoided. This has positive consequences on the manufacturing of

Figure 6.3: Centre Pompidou-Metz

106

roof plan, (b) the built roof structure, (c) CNC fabrication, (d) manufactured beams and connections.

Application in Architecture

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 innovative 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

Figure 6.4: The wooden ceiling of the Burj Khalifa office lobby in Dubai, by Gehry Technologies. See [Meredith N. and Kotronis J. 2012] 1

See [Pottmann er al. 2010] for a detailed explanation of the terminology.

107

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 surfaces 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

Algorithmic Panelization of Surfaces with Geodesic 1-Patterns

The developable (pure bending) property When it comes to the developable property a certain amount of twisting in the panels is allowed 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 â&#x20AC;&#x201D; the design of patterns of geodesic curves on a surface (section 6.3.1) and the translation of those curves into actual panels/timber boards (section 6.3.2). Depending on the method that i used to create the panels based on the geodesic curve pattern 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 approximately constant distance from their adjacent curves. It is possible to cover a surface with rectangular 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 developments 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 development 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

less cladding is possible, but it will cost more to achieve such a result because the panels need to be produced individually instead of using 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 Figure 6.5: Example of designing a sequence of geodesics. The locus of minimum or maximum distance between adjacent curves has been prescribed with the red curve. This surface was segmented prior to applying the parallel projection method.

et al. 2010] and [Wallner et al. 2010]. It is possible 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 prescribing the points at which either the maximum or the minimum distance between neighbouring curves occurs. In differential geometry the notion of parallel transport of a vector V

along a curve c contained in a surface means

P1 c

that the vector is moved in such a way that it

P0 V2 V1

Figure 6.6: Parallel transport of vector V0 along the polyline P0,P1,P2...

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 represented 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 projecting Vi-1 onto the tangent plane of Pi. The projected vector is then normalized. For the design

110

Algorithmic Panelization of Surfaces with Geodesic 1-Patterns

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 distances 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 pattern 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 selected. 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 freedom ,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 whenever 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

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 constraint. 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 piecewisegeodesic 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

Algorithmic Panelization of Surfaces with Geodesic 1-Patterns

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 surface, regardless if the surface is segmented, or which method has been used, is to create panels 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, Figure 6.7: Through the process of sharpening a geocreates panel surfaces that are tangentially

desic vector field (left) becomes piecewise geodesic (right)

circumscribed to the surface along given geodesic lines. The second method, the binormal method, creates panels that are inscribed into the input surface between two adjacent geo-

(a)

desics. 6.3.2.1 The Tangent Developable Method Conjugate tangents For this method the notion of conjugate tangents and tangent developables needs to be explained. Let's assume a tangent plane on a smooth surface. If the tangent plane is moved

(b)

just a small amount by means of parallel translation and intersected with the surface, the intersection will result in a curve which approximates a conic section - the Dupin indicatrix (Figure 6.8). The shape of the Dupin indicatrix depends on the Gaussian curvature of the unFigure 6.8: The Dupin indicatrix in (a) positively

derlying surface. In hyperbolic points, that is ar- curved areas of a surface and (b) in negativaly curved areas of a surface.

eas with negative Gaussian curvature, the inter-

113

section will result in two different hyperbolae. The asymptotes A1 and A2 of the hyperbola form the asymptotic directions. Any paralleloΦ

T(x) x

gram tangentially circumscribed to the Dupin indicatrix yields two conjugate tangents T and

U. The asymptotic directions A1 and A2 can be

U(x)

used to find such parallelogram if necessary beFigure 6.9: U(x) is the conjugate tangent of T(x) in point x of the geodesic s.

cause they are known to be its diagonals. [Wallner et al. 2010] By knowing the Dupin indicatrix and the asymptotic directions it is possible to find pairs of conjugate tangents in every point

Ψi

Bi(x)

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-

Ai(x)≈Bi+2(x)

si si+1

proach. [Wallner et al. 2010]

x’

si+2

Tangent developables

Ai+2(x)

Ψi+2

Figure 6.10: Tangent developable surfaces of geodesics with even indices are trimmed by neighbouring geodesics with odd indices.

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

N(t)

curve s on the surface Φ is sampled in a point x.

R(t) Φ

T(t)

P(t)

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

L(t) s

B(t)

Figure 6.11: The binormal method. The ruled panels are defined by the Frenet frame T, N, B of a geodesics s.

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-

114

Algorithmic Panelization of Surfaces with Geodesic 1-Patterns

desic curve s becomes a straight line. [Wallner et al. 2010] Given these facts it would be obvious 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 (geodesics 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 described 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 current 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

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 design 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

Project: 21er Raum

rather embraces it by being formally the opposite 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 intersects the freeform surface of the room, posing 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

(a)

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 threatening to roll over the edge, this room creates a new presence for the young artists that exhibit inside that can not be ignored.

(b)

Figure 6.13: The 21er Raum. (a) inside view, (b) detail of the mirrors on the outside walls

117

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 â&#x20AC;&#x201D; scale 1:100. (Below) Section through the 21er Haus museum revealing the position of the 21er Raum in regards to its surrounding â&#x20AC;&#x201D; 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 museum â&#x20AC;&#x201D; scale 1:500. The drawing above shows a closer look of the 21er Raum and its immediate surrounding â&#x20AC;&#x201D; 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 â&#x20AC;&#x201D; 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 â&#x20AC;&#x201D; 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 expanded polystyrene (EPS). The polystyrene model was subsequently covered with multiple layers 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 consuming, 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: t

In a computerized process the reference surface can be completely defined in a CAD program.

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 Verlagsgruppe 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 Architectural 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 Advances 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 Architectural 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 Publishing Ltd. Pottmann, H., Asperl, A., Hofer, M., Kilian, A. (2007). Architectural Geometry. Bentley Institute 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 Wallner, 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

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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 Geometry 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 Panelization 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 Architecture. Ph.D thesis in Department of Architecture, Massachusetts Institute of Technology, 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., Pottmann, H. (2010), Tiling Freeform Shapes With Straight Panels: Algorithmic Methods. In Advances in Architectural Geometry, (Proceedings).

127

Figure Credits

8. Figure Credits Figure 2.1 on page 8 - underground image courtesy 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 fondationlouisvuitton.fr; Figure 5.3 on page 74 - courtesy of Evolute;

Figure 6.5 on page 110 - source: Wallner, J., Schiftner, A., Kilian, M., Flรถry, S., Hรถbinger, M., Deng, B., Huang, Q., Pottmann, H. Tiling Freeform Shapes With Straight Panels: 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., Flรถry, S., Hรถbinger, M., Deng, B., Huang, Q., Pottmann, H. Tiling Freeform Shapes With Straight Panels: Algorithmic Methods. In Advances in Architectural Geometry, 2010 (Proceedings); 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. Graphics, 29/3, #43, Proc. SIGGRAPH; Figure 6.8 on page 113 - image from Wallner, J., Schiftner, A., Kilian, M., Flรถry, S., Hรถbinger, 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., Flรถry, S., Hรถbinger, M., Deng, B., Huang, Q., Pottmann, H. Tiling Freeform Shapes With Straight Panels: Algorithmic Methods. In Advances in Architectural Geometry, 2010 (Proceedings); Figure 6.10 on page 114 - based on Figure 10 from Wallner, J., Schiftner, A., Kilian, M., Flรถry, S., Hรถbinger, M., Deng, B., Huang, Q., Pottmann, H. Tiling Freeform Shapes With Straight Panels: Algorithmic Methods. In Advances in Architectural Geometry, 2010 (Proceedings); Figure 6.11 on page 114 - based on Figure 15 from Wallner, J., Schiftner, A., Kilian, M., Flรถry, S., Hรถbinger, M., Deng, B., Huang, Q., Pottmann, H. Tiling Freeform Shapes With Straight Panels: Algorithmic Methods. In Advances in Architectural Geometry, 2010 (Proceedings);

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 ยฉ2008 Prestel Publishing, 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 Technologies, source: www.wconline.com;

129

0.3

0,01714043

0,00599563

0,00153146

0,00069450

0.3_0.0171

0.3_0.00598

0.3_0.00153

0.3_0.000696

0,00001678

0,61883664

0,06835281

0.3_0.068

0.3_0.000016401

0,58444287

0,10678066

0.3_0.107

0,16204845

0,83067178

0,88113381

0,86727456

0,63776937

of the reference surface.

24,10664961

0,01196553

0,10256594

0,10651488

0,11075480

0,12647582

0,12710264

0,12599084

2,93982828

0,80539310

considered to be suitable for the discretization

10,23943929

an threshold. The strips below the red line are

0,51683325

a table showing the results. The red line marks

2,40488815

ence surface. For each group of DStrips there is

0.3_2.4

developable by definition is chosen as refer-

0.3_0.51

for the strip creation. A surface strip which is

-0,02039975

surface strip that provided the reference curves

0,00002021

to the same values extracted from a reference

4,36212730

state are measured. The values are compared

0.3_4.36

strip in its original state as well as in its unrolled

97059,82391109 97059,83746715

97037,55567980 97037,53683586

97030,08237824 97030,07235123

97035,21214625 97035,18491360

97006,87298984 97006,87316492

97032,78248670 97032,79002620

97023,64433505 97023,72994458

97003,14515430 97008,42168998

96763,34909538 96821,73572644

96844,32335503 96994,55939441

0,01355606

0,01884395

0,01002701

0,02723264

0,00017508

0,00753950

0,08560953

5,27653569

58,38663106

150,23603938

0,00001397

0,00001942

0,00001033

0,00002806

0,00000018

0,00000777

0,00008824

0,00543955

0,06033961

0,15513149

162,65032881

140,34969751

132,88521289

137,99777526

109,68602657

135,60288785

126,54280623

111,23455164

75,45141191

97,37225606

0,16785867

0,14484393

0,13714042

0,14241670

0,11319836

0,13994512

0,13059492

0,11479647

0,07786749

0,10049028

Area difference Area difference Area Difference Area DifferArea of unrolled after unrolling after unrolling to original unroll ence to original strip (cm2) (cm2) (%) (cm2) unroll (%)

age curve closeness and the surface area of the

Strip area (cm2)

planarity value, the maximum and the aver-

Average Closeness

their quad faces are tested. For each strip the

MaxCloseness

of DStrips with different initial aspect ratios of

Planarity

ducted on DStrips in section 5.3.3. Five groups

Name

The tables show the results of the studies con-

Aspect Ratio

Appendix 1

131

132

0.15

0.2

Aspect Ratio

0,12669157

0,15015329

0,31677724

0,21999712

2,89613867

1,67383218

0,80472279

0,35724762

0,17290443

0,08114175

0,03683808

0,00645800

0,00225512

0,00091693

0,00009132

2,19574070

1,38440347

0,88912892

0,34518713

0,00967268

0,08949289

0,03930178

0,02050448

0,00300866

0,00163537

0,00065559

0,00034322

0,00018176

0,00007341

0.2_2.896

0.2_1.67

0.2_0.804

0.2_0.35

0.2_0.173

0.2_0.081

0.2_0.0368

0.2_0.00646

0.2_0.00225

0.2_0.000925

0.2_0.0000888

0.15_2.19

0.15_1.38

0.15_0.88

0.15_0.34

0.15_0.0966

0.15_0.089

0.15_0.03929

0.15_0.02

0.15_0.003

0.15_0.001623

0.15_0.00065

0.15_0.00035

0.15_0.00016

0.15_0.0000731

0,17669949

0,15161396

0,17603863

0,17587903

0,32422208

0,19717764

0,34938212

0,20079183

0,09290670

0,07956037

1,28701228

0,00002798

0,27652637

0,69670968

0,27347129

0,25024933

0,32923453

0,22841391

0,19874130

2,16348495

0,00002240

Planarity

Name

0,01363348

0,01300865

0,01417680

0,01416213

0,02689861

0,02095959

0,03192031

0,02002052

0,01777841

0,02771298

0,00435845

-0,00140990

0,36238205

-0,01030136

0,01985888

0,04514487

0,01854680

0,02164804

0,02885018

0,01548053

0,01955574

0,01154702

0,00829137

0,58762742

-0,01369111

97050,34920437

97050,62229214

97050,60348249

97050,59793435

97050,41319786

97061,12156467

97059,16300821

97061,93334127

97047,72039218

97074,15516879

97042,83125743

96999,35927750

97112,08573080

96855,53006856

97050,37205471

97046,32830680

97052,89186473

97049,84951505

97048,46561171

97053,17146143

97045,23231321

97042,43726921

97034,65114685

97195,43488141

96853,98631084

97050,34650515

97050,61622305

97050,59590556

97050,59631818

97050,42090607

97061,11427512

97059,16713797

97061,96032782

97048,02457790

97074,14973178

97047,46078084

97070,55382501

97169,46690648

97007,54613159

97050,38532037

97046,32974741

97052,88791572

97049,86126418

97048,49807162

97053,32141969

97045,95531552

97045,99127297

97056,18246004

97246,67738280

97003,81380612

0,00269922

0,00606909

0,00757692

0,00161617

0,00770821

0,00728955

0,00412976

0,02698656

0,30418572

0,00543701

4,62952340

71,19454750

57,38117568

152,01606303

0,01326566

0,00144061

0,00394901

0,01174912

0,03245991

0,14995826

0,72300231

3,55400376

21,53131319

51,24250139

149,82749528

0,00000278 153,15936680 0,15806379

0,00000625 153,42908470 0,15834215

0,00000781 153,40876722 0,15832118

0,00000167 153,40917984 0,15832160

0,00000794 153,23376773 0,15814057

0,00000751 163,92713677 0,16917636

0,00000425 161,97999963 0,16716687

0,00002780 164,77318948 0,17004951

0,00031344 150,83743955 0,15566751

0,00000560 176,96259343 0,18262924

0,00477060 150,27364249 0,15508566

0,07339693 173,36668666 0,17891818

0,05908757 272,27976813 0,28099863

0,15695135 110,35899324 0,11389288

0,00001367 153,19818202 0,15810385

0,00000148 149,14260906 0,15391841

0,00000407 155,70077737 0,16068658

0,00001211 152,67412583 0,15756301

0,00003345 151,31093327 0,15615617

0,00015451 156,13428134 0,16113397

0,00074502 148,76817717 0,15353199

0,00366232 148,80413462 0,15356910

0,02218930 158,99532169 0,16408662

0,05272110 349,49024445 0,36068152

0,15469420 106,62666777 0,11004104

Area differArea DifArea DifArea difference Average CloseArea of unrolled ence after ference to ference to MaxCloseness Strip area (cm2) after unrolling unrolling original unroll original ness strip (cm2) (cm2) (%) (cm2) unroll (%)

133

0.05

0.1

Aspect Ratio

Planarity

1,46712375

0,91834688

0,26560616

0,13376732

0,03096855

0,00904255

0,00287661

0,00144734

0,00082800

0,00004601

0,74144256

0,39184344

0,20218217

0,06408861

0,01322448

0,00545172

0,00148546

0,00088169

0,00067899

0,00029325

Name

0.1_1.47

0.1_0.92

0.1_0.26

0.1_0.13

0.1_0.03

0.1_0.009

0.1_0.0028

0.1_0.0014

0.1_0.0008

0.1_0.0000429

0.05_0.7414

0.05_0.418144

0.05_0.202181

0.05_0.064090

0.05_0.01323

0.05_0.005461

0.05_0.001487

0.05_0.000870

0.05_0.000683

0.05_0.000282

0,02648759

0,04057013

0,05859017

0,04414810

0,04118255

0,03187372

0,04623425

0,35743768

0,13565780

0,00002530

0,09707738

0,09674684

0,10070531

0,15000959

0,15821444

0,07501470

0,08859446

1,15495374

0,92797695

0,00002240

-0,00052225

0,00347215

0,00061618

0,00117160

0,00088193

-0,00014425

-0,00125639

0,08105174

0,04620227

-0,00347711

0,00388402

0,00466430

0,00434361

0,00704478

0,00842964

0,00507763

0,01861423

0,45412345

0,14379913

-0,00698548

97054,05016748

97057,78455440

97053,09118136

97054,54183591

97054,58816264

97053,89872193

97056,15817257

97069,69988743

97012,34549849

96856,19743418

97053,32803313

97054,58483170

97053,33962782

97052,65386552

97054,89684238

97056,58863920

97061,52585525

97216,92192705

96980,19556609

96856,09149655

97054,55482997

97058,04516031

97053,16382108

97054,57559419

97054,62932380

97053,94981480

97056,29370091

97080,30236152

97057,71502239

97008,17898005

97053,33002024

97054,45170278

97052,71503925

97052,65282937

97054,92975199

97056,57072587

97069,24938267

97200,28412484

97039,98656292

97008,21989832

0,50466249

0,26060591

0,07263972

0,03375827

0,04116116

0,05109287

0,13552835

10,60247408

45,36952390

151,98154587

0,00198711

0,13312893

0,62458857

0,00103615

0,03290962

0,01791333

7,72352742

16,63780221

59,79099683

152,12840176

0,00051998

0,00026851

0,00007485

0,00003478

0,00004241

0,00005264

0,00013964

0,01092254

0,04676675

0,15691463

0,00000205

0,00013717

0,00064355

0,00000107

0,00003391

0,00001846

0,00795735

0,01711410

0,06165279

0,15706643

157,36769162 0,16240687

160,85802196 0,16600897

155,97668273 0,16097132

157,38845584 0,16242830

157,44218545 0,16248375

156,76267645 0,16178248

159,10656257 0,16420143

183,11522317 0,18897888

160,52788404 0,16566826

110,99184170 0,11454599

156,14288189 0,16114284

157,26456443 0,16230044

155,52790090 0,16050817

155,46569102 0,16044397

157,74261365 0,16279380

159,38358752 0,16448732

172,06224432 0,17757197

303,09698649 0,31280267

142,79942457 0,14737211

111,03275997 0,11458822

Area differArea DifArea DifArea difference Average CloseArea of unrolled ence after ference to ference to MaxCloseness Strip area (cm2) after unrolling ness strip (cm2) unrolling original unroll original (cm2) (%) (cm2) unroll (%)

134

0.2

table. 0,00007777 0,12890349 0,44747382 1,18677617

values below the respective red lines in each -0,00340522 0,03496179 0,10031250 0,28948540

191569,43641119 191758,02707908 191819,51344479 191836,41100474

191631,08105478 191765,04697593 191831,74722194 191780,49685093 191816,13047352 191836,96570973

ing ,which compares the surface area of the

0,02499347 0,01650294 0,00444133 0,00049672

face. There is a noticeable stagnation of the

1,40755421 0,71433005 0,34340309 0,07668416

DStrip to the surface area of the unrolled sur-

0.2_1.41 0.2_0.71 0.2_0.34 0.2_0.07

this study is the area difference after unroll-

0.3

the DStrip against. The main parameter in

0,17180821 0,15931488 0,17846584 0,35279269 0,36878874 0,42391044

ies there is no reference surface to compare

1,74576464 0,94466827 0,98530529 1,57690651 1,53618847 1,63290960

in appendix 1 because in this series of stud-

0,03443388 0,02513112 0,01643190 0,00284145 0,00115267 0,00089209

bles provide less information than the tables

1,98576529 1,13203841 0,77296267 0,53152065 0,15291030 0,10128012

as in its unrolled state are measured. This ta-

0.3_1.99 0.3_1.13 0.3_0.77 0.3_0.53 0.3_0.15 0.3_0.10

area of the strip in its original state as well

191777,53069627 191809,30253135 191835,89812074 191838,4704403

191843,23140887 191866,24659084 191888,93860437 191788,07751504 191818,73251718 191838,47473615

ƌĞĂ ŽĨ ƵŶƌŽůůĞĚ strip (cm2)

208,09428508 51,27545227 16,38467594 2,05943557

212,15035409 101,19961491 57,19138243 7,58066411 2,60204367 1,50902641

0,10862604 0,02673966 0,00854171 0,00107354

0,11070770 0,05277271 0,02981330 0,00395278 0,00135653 0,00078662

ƌĞĂ ĚŝīĞƌĞŶĐĞ ƌĞĂ ĚŝīĞƌĞŶĐĞ ĂŌĞƌ ƵŶƌŽůůŝŶŐ ĂŌĞƌ ƵŶƌŽůůŝŶŐ ;йͿ (cm2)

the average curve closeness and the surface

Strip area (cm2)

strip the planarity value, the maximum and

ǀĞƌĂŐĞ ůŽƐĞness

ratios of their quad faces are tested. For each

Scale Invariant DĂǆ ůŽƐĞPlanarity ness

groups of DStrips with different initial aspect

Planarity

conducted on DStrips in section 5.3.4. Five

Name

The tables show the results of the studies

Aspect ZĂƟŽ

Appendix 2

135

Planarity

1,05589465 0,39874802 0,12357147 0,04841733 0,01603079 0,01001090

0,70432157 0,32888107 0,16934045 0,07609765

0,35271702 0,16300030 0,06970286 0,03351624 0,01884713

Name

0.15_1.05 0.15_0.39 0.15_0.12 0.15_0.05 0.15_0.02 0.15_0.01

0.1_0.7 0.1_0.33 0.1_0.17 0.1_0.08

0.05_0.35 0.05_0.16 0.05_0.07 0.05_0.03 0.05_0.02

Aspect ZĂƟŽ

0.15

0.1

0.05

0,00650363 0,00262855 0,00090563 0,00030422 0,00013802

0,01288744 0,00778894 0,00322201 0,00103688

0,01908238 0,00620099 0,00090619 0,00094625 0,00032926 0,00011593

0,00008352 0,30640074 0,86966545 0,27419032 0,06509060

0,00008093 0,28328747 1,26710068 0,59565202

0,00007633 0,77552441 2,15766015 1,77762805 2,13540481 2,83314983

Scale Invariant DĂǆ ůŽƐĞPlanarity ness

-0,00082689 0,03866153 0,07726129 0,06721447 0,02089551

-0,00168422 0,03450275 0,20198232 0,18004539

-0,00252571 0,10044761 0,36077341 0,48526568 0,68692328 0,86003738

ǀĞƌĂŐĞ ůŽƐĞness

191569,42773637 191789,72596532 191900,77463518 191881,36847002 191847,92134703

191569,42464848 191780,46479957 191889,81466312 191795,13598521

191569,42657307 191812,29079578 191834,99503064 191779,63940311 191770,30487707 191768,14732287

Strip area (cm2)

191777,38298505 191858,16825124 191910,12361821 191879,27145558 191919,11509123

191777,57495787 191828,19999862 191915,73272898 191805,06674856

191777,68687506 191871,07038299 191846,03234317 191781,73258368 191770,70250450 191768,56621766

ƌĞĂ ŽĨ ƵŶƌŽůůĞĚ strip (cm2)

207,95524868 68,44228591 9,34898303 2,09701444 71,19374421

208,15030939 47,73519905 25,91806585 9,93076334

208,26030199 58,77958721 11,03731253 2,09318056 0,39762743 0,41889478

0,10855346 0,03568611 0,00487178 0,00109287 0,03710947

0,10865529 0,02489054 0,01350674 0,00517780

0,10871270 0,03064433 0,00575354 0,00109145 0,00020735 0,00021844

ƌĞĂ ĚŝīĞƌĞŶĐĞ ƌĞĂ ĚŝīĞƌĞŶĐĞ ĂŌĞƌ ƵŶƌŽůůŝŶŐ ĂŌĞƌ ƵŶƌŽůůŝŶŐ ;йͿ (cm2)