3. ABSTRACT
This thesis is built around the conception, development and optimization of a helicoidal
skyscraper envelope in order to define a suitable structural choice. The shape is
conceived under geometrical consideration to define a fitting panelization. The aim of
this thesis is to achieve the reduction of costs and the fabricability optimization.
The envelope and the structural system are modelled entirely on Grasshopper, a
Rhino3D plug-in which allows to parametrically design objects. Particular attention was
focused on the studies of the skyscraper’s base shape in order to achieve different
envelopes to examine. Different approaches of geometrical panelization are applied on
defined shells: from the research of the same tangent on curve up to the principal
directions on a surface. This analytical study of the shell is concluded using the software
Evolute Tools PRO, an other Rhino plug-in that allows a complex and advanced analysis
of the considered geometry.
The internal structure is built after the chosen envelope and, because of iterative
optimization process, the best structural performance is found at constant weight.

4. RIASSUNTO
Questo lavoro di tesi si sviluppa intorno all’ideazione, sviluppo e ottimizzazione
dell’involucro esterno di un grattacielo di forma elicoidale per poi individuare una scelta
strutturale consona al progetto. La forma è stata concepita sulla base di considerazioni
geometriche mirate alla pannellizzazione dell’involucro, riducendo costi e ottimizzando
la fabbricabilità.
L’involucro e la struttura portante sono stati interamente modellati con Grasshopper, un
plugin di Rhino 3D che permette di generare delle geometrie in maniera parametrica.
Particolare attenzione è stata dedicata allo studio del piano di base del grattacielo a
partire dal quale sono state raggiunte diverse forme dell’involucro. A queste ultime sono
stati applicati diversi approcci di pannellizzazione geometrica partendo dalla ricerca
della planarità mediante l’individuazione di tangenti alle curve di piano fino all’ideazione
di algoritmi che mostrano le direzioni principali di una superficie. Lo studio analitico
dell’involucro si conclude con l’utilizzo del software Evolute Tools PRO, altro plugin di
Rhino 3D, che permette una definizione avanzata della geometria.
Successivamente alla scelta dell’involucro è stata concepita la struttura interna e, grazie
a processi iterativi di ottimizzazione, si riesce ad avere la maggior performance
strutturale a parità di carichi propri.

5. ACKNOWLEDGEMENTS
This thesis has been developed during my internship at the Technical University in
Vienna.
I would like to express my sincere appreciation to my Supervisors, Prof. M. Froli and
Prof. H. Pottmann, for their support and constructive suggestions during the
development of this research work.
A special thank goes to my whole family, especially my parents, my sister and my
grandmother for being such a huge support through my experiences.
I am deeply grateful to all the people known during my experience in Vienna. I found
amazing colleagues and great friends.
I would like to thank all the people who contribute the development and the support of
this thesis. Every suggestion, critique and help has been useful.
I want to acknowledge all the great people that I met this years of my studies, all of my
old and new friends. Everyone has been essencial for my personal growth.
An especially thank to you, who were present in the last three years of my life. I am here
because of you.
Thanks to the people that I met afterwards, thanks for the support and for all the smiles
during the final period before my graduation.

6. 1
SUMMARY
1 Architectural Geometry ................................................................................9
1.1 Surface Discretization ............................................................................................. 9
1.1.1 Triangle Meshes.............................................................................................. 10
1.1.2 Quadrilateral Meshes.....................................................................................12
1.1.2.1 PQ meshes ...............................................................................................12
1.1.3 Hexagonal Meshes.........................................................................................13
1.1.3.1 P-Hex meshes ..........................................................................................13
2 Differential Geometry.................................................................................. 15
2.1 Parametric search..................................................................................................15
2.2 Definitions ................................................................................................................16
2.3 Curves .....................................................................................................................20
2.4 Surfaces....................................................................................................................21
2.4.1 Surfaces of Revolution .................................................................................. 22
2.4.1.1 Ruled Surfaces........................................................................................ 22
2.4.1.2 Developable surface............................................................................ 25
2.4.1.3 Developable surfaces with a NURBS .................................................. 26
2.4.2 Principal curvature directions......................................................................28
2.4.2.1 Classification of points on a surface................................................... 29
2.4.2.2 Lines of curvature ..................................................................................30
2.4.3 Minimal surfaces ............................................................................................30
3 Case study: twisting skyscraper...................................................................33
3.1 Design of high-rise buildings ................................................................................ 33
3.2 Softwares ................................................................................................................ 33
3.3 Check design......................................................................................................... 36
3.3.1 Twisting and rotating forms.......................................................................... 37
3.3.1.1 My twisting form .....................................................................................38
3.3.1.2 Base shapes............................................................................................ 39
3.4 Panelization............................................................................................................40
3.4.1 Search for the same tangent on curve.....................................................40

7. 2
3.4.2 Developable surface.................................................................................... 45
3.4.3 Panelization with diamonds and triangles ................................................ 52
3.4.3.1 Which triangles can be converted in flat diamonds with cold
bending......................................................................................................... 55
3.4.3.1.1 Panels deviation................................................................................. 59
3.4.3.1.2 Min. Cold Bending radius...................................................................61
3.4.4 Corner modifications of base shapes........................................................68
3.4.5 Principal curvature directions...................................................................... 70
3.4.5.1 Shape with smooth corners ................................................................. 70
3.4.5.2 General B-Spline base with degree 3 ................................................ 74
3.4.6 Paneling Architectural Freeform Surfaces................................................. 79
3.4.6.1 Case studies............................................................................................83
3.4.6.2 Conclusions: Skyscrapers that can be built.......................................88
4 Site ........................................................................................................... 89
5 Actions ...................................................................................................... 91
5.1 Wind load ................................................................................................................91
5.2 Floor system ............................................................................................................96
5.3 Exterior walls ...........................................................................................................96
6 Structural Systems.......................................................................................97
6.1 Frame system .........................................................................................................98
6.2 Shear wall system ..................................................................................................99
6.3 Shear wall and frame system ............................................................................ 100
6.4 Framed tube system ............................................................................................101
6.5 Tube in tube system ............................................................................................ 102
6.6 Bundled – tube system ....................................................................................... 103
6.7 Braced – tube system ......................................................................................... 104
6.8 Outrigger – braced system................................................................................ 105
6.9 Structural system choice .................................................................................... 106
6.9.1 Development of option A.......................................................................... 109
6.9.2 Development of option B............................................................................ 111
6.9.2.1 2D Model................................................................................................ 111
6.9.2.2 3D Model................................................................................................118
6.9.2.3 The Project ............................................................................................ 126
7 Conclusions ............................................................................................. 129

8. 3
LIST OF FIGURES
Figure 1: (left) British Museum Great Court Roof, London, completed in 2000 (right)
DG Bank Court Roof, Berlin, completed in 1998 ..............................................................11
Figure 2: (left) Blob, Eindhoven, completed, (right) Vela Fiera Milano-Rho, Milano,
completed in 2005..............................................................................................................11
Figure 3: (left) a node without an axis. Image of Waagner-Biro Stahlbau AG. (right)
Geometric Torsion in a Node.............................................................................................11
Figure 4: (left) Rotational PQ mesh, (right) Geometry of a conjugate curve network
[PAH07]............................................................................................................................. 12
Figure 5: (left) Mannheim Grid Shell, Mannheim, completed in 1974, (right) Hamburg
History Museum Court Roof, Hamburg, completed in 1989 .......................................... 13
Figure 6: (left) Honeycomb subdivision, (right) Regular triangular tiling [PAH07] ..... 13
Figure 7: P Hex mesh computed using the progressive conjugation method [WLY08] 14
Figure 8: Tangent on a curve [PAH07]............................................................................ 16
Figure 9: Inflection point on a curve ................................................................................ 17
Figure 10: Osculating circles [PAH07] ............................................................................ 17
Figure 11: Bézier curve [PAH07]......................................................................................18
Figure 12: Hyperboloid .....................................................................................................23
Figure 13: Hyperbolic paraboloid .....................................................................................23
Figure 14: Plücker's conoid ...............................................................................................24
Figure 15: Möbius strip .....................................................................................................24
Figure 16: Cylinders, cones, and tangent surfaces of space curves [PAH07].................25
Figure 17: Nurbs curve generated by control points polygon [PAH07] ..........................27
Figure 18: Normal curvatures of a surface S at a point p are the curvatures of the
intersection curves with planes R through the surface normal [PAH07] .......................28
Figure 19: The osculating circle varies depending on curves [Jau11]..............................28
Figure 20: Points change based on the position on torus: elliptic, hyperbolic, parabolic
[PAH07].............................................................................................................................30
Figure 21: Star, lemon and monstar lines of curvature [WIKI].......................................30
Figure 22: Catenary and catenoid..................................................................................... 31

9. 4
Figure 23: Helic and Helicoid........................................................................................... 31
Figure 24: from the right 1. Schwartz minimal surface, 2. Riemann's minimal surface, 3.
Enneper surface, 4. Bour's minimal surface, 5. Gyroid, 6. Chen-Gackstatter surface....32
Figure 25: Development of tall buildings.........................................................................36
Figure 26: (from the left) Burj Kalifa, Millennium Tower, Shard, Sears Tower.............37
Figure 27: Twisting geometry process [PAH07] ............................................................38
Figure 28: (left) Turning Torso in Malmö, Sweden (right) 30 St. Mary Axe in London,
UK ......................................................................................................................................38
Figure 29: Base shapes case study and few stories that define the final building ......... 40
Figure 30: Generic curve with tangents in random points............................................. 40
Figure 31: 74 computed points on curve with a distance of 1.5 m each. ......................... 41
Figure 32: Visualization of the adopted method to find planar meshes between two
consecutive curves............................................................................................................. 41
Figure 33: Algorithm generated to define planar panels with the same tangent on
curves.................................................................................................................................42
Figure 34: Research points with same tangents in the squared curve with smooth
verteces ..............................................................................................................................43
Figure 35: Research points with same tangents in the convex curve..............................43
Figure 36: Research points with same tangents in the curve with inflection points......44
Figure 37: Generic developable surface............................................................................45
Figure 38: (left) Starting control points (right) Generic NURBS curve of degree 2.......45
Figure 39: (left) Control point polygon (right) Control point polygon that intersect the
NURBS curve in inflection points.....................................................................................46
Figure 40: (left) Curvature graph for a generic NURBS curve (right) curvature graphs
for two generic NURBS curves..........................................................................................46
Figure 41: Groups of control points..................................................................................46
Figure 42: Algorithm to generate NURBS curves ............................................................47
Figure 43 Conic sections as special NURBS [PAH07] ....................................................48
Figure 44: (left) intersection points of different NURBS curve in the same control
polygon (right) zoom of vectors tangent to this two curves.............................................48
Figure 45: Planes parallel to vectors tangent on every point selected for one curve and
points projected on the following curve ...........................................................................49
Figure 46: Flat panels connecting two consecutive floors...............................................49
Figure 47: Algorithm to find random weights..................................................................49
Figure 48: Looping algorithm to create random skyscraper with developable surfaces50
Figure 49 Developable skyscrapers with flat panels created by random NURBS curves51

10. 5
Figure 50: Developable skyscrapers with flat panels obtained by a scale alghoritm for
NURBS curves ................................................................................................................... 51
Figure 51: Initial steps to reach flat diamonds.................................................................52
Figure 52: Plane quite tangent to the curves....................................................................52
Figure 53: The intersection of two consecutive planes is a line. Picking the middle point
of every line and connecting these points with points previously found on curves we
found flat diamonds ..........................................................................................................53
Figure 54: Steps to achieve planarity with diamonds and triangles ...............................53
Figure 55: Planarity analysis with Evolute Tools Pro ......................................................54
Figure 56: The algorithm works well for the first floors, then the approximation
becames unacceptable.......................................................................................................54
Figure 57: Mesh with planar diamonds and triangles optimized by Evolute Tools Pro.54
Figure 58: Panels deviation [EPR] ...................................................................................59
Figure 59: Grasshopper definition for evaluating planarity............................................59
Figure 60: Blue flat panels, yellow panels flat with cold bent, red panels with double
curvature........................................................................................................................... 60
Figure 61: Screenshot of grasshopper definition for the analysis of principal curvatures
............................................................................................................................................ 61
Figure 62: Results of mesh analysis and cold bent ..........................................................62
Figure 63: Analysis of cold bending .................................................................................62
Figure 64: Geometry of a laminated glass........................................................................63
Figure 65: Model of a cold bent panel ..............................................................................65
Figure 66: Table from CNR-DT 210/2012 for the analysis of displacement ..................66
Figure 67: Base shapes [Bor13].........................................................................................68
Figure 68: Evaluation of best shape for planarity check .................................................69
Figure 69: Principal curvature lines in the straight part and in the smooth part...........70
Figure 70: Starting mesh for the approximation of principal curvature directions of
smooth corners.................................................................................................................. 71
Figure 71: Planar panels obtained with meshes not weld................................................ 71
Figure 72: Grasshopper definition of principal curvature directions of smooth corners
............................................................................................................................................72
Figure 73: (left) shape non-optimized (right) shape optimized with EVOLUTE Tools..72
Figure 74: Panelization method B ....................................................................................73
Figure 75: (left) zoom of panelization alghoritm with method B (left) zoom of the node
with valence 5 ....................................................................................................................73
Figure 76: Control points for a B-Spline of degree 3 .......................................................74

11. 6
Figure 77: (left) Base shape, (right) Panels achieved from principal curvature directions
............................................................................................................................................76
Figure 78: (left) Base shape, (right) Panels achieved from principal curvature directions
............................................................................................................................................76
Figure 79: (left) Base shape, (right) Panels achieved from principal curvature directions
............................................................................................................................................76
Figure 80: B-Spline with inflection points....................................................................... 77
Figure 81: Principal curvature lines through in two different points of the shape......... 77
Figure 82: Steps to achieve planar panels........................................................................ 77
Figure 83: Shape that follows only one principal direction in every floor......................78
Figure 84: (left) panels completely planar that follow both principal directions (right)
in the red part we have to use non-planar panels ............................................................78
Figure 85: How principal direction lines change the side of the shape ..........................78
Figure 86: Kink angle and divergence between panels [Evo12]......................................81
Figure 87: Panels type used and costs..............................................................................83
Figure 88: Planarity analysis case study 1........................................................................85
Figure 89: Results of panel types for case study 1............................................................85
Figure 90: Planarity analysis case study 2 .......................................................................86
Figure 91: Results of panel types for case study 2............................................................86
Figure 92: Planarity analysis case study 3........................................................................87
Figure 93: Results of panel types for case study 3 ...........................................................87
Figure 94 Analysis with 1 cm of gap, colors define different clusters for panels ........... 88
Figure 95: Analysis of 4 cm of gap and different clusters obtained ............................... 88
Figure 96: Location of the building ..................................................................................89
Figure 97: Donau City (Vienna International Center).....................................................89
Figure 98: DC Tower 1 ..................................................................................................... 90
Figure 99: Site .................................................................................................................. 90
Figure 100: External and internal pressure of the wind..................................................96
Figure 101: Structural systems..........................................................................................97
Figure 102: Frame system.................................................................................................98
Figure 103: Shear wall system ..........................................................................................99
Figure 104: Shear wall and frame system ......................................................................100
Figure 105: Deflection profile .........................................................................................100
Figure 106: Framed tube system .................................................................................... 101
Figure 107: Tube in tube system.....................................................................................102
Figure 108: Bundled-Tube system..................................................................................103

12. 7
Figure 109: Braced-Tube system ....................................................................................104
Figure 110: Outrigger-braced system .............................................................................105
Figure 111: Structural systems ........................................................................................ 107
Figure 112: Utilization importance of structural elements............................................109
Figure 113: Utilization of beams and slabs for the structural option A......................... 110
Figure 114: Left: Top view of the skyscraper with the structural system and the external
shell. Right: System of the structure of one floor............................................................ 111
Figure 115: Left: 2D model created with Karamba3D. Right: zoom of the model.........112
Figure 116: Detail of one floor in the 2D model ..............................................................112
Figure 117: Load acting on floor.......................................................................................112
Figure 118: Grasshopper example of definition for the element that is indicated as
chord inf............................................................................................................................113
Figure 119: Natural vibration: Modal 1, modal 2, modal 3 .............................................117
Figure 120: 2D model's utilization factor of option B.....................................................117
Figure 121: Elements for the 3D model .......................................................................... 118
Figure 122: Method to find an equivalent adequate profil ............................................ 118
Figure 123: Structural system B with analysis result..................................................... 122
Figure 124: Foundamental eigenmodes with SAP2000 ................................................ 125
Figure 125: Vienna International Centre........................................................................ 126
Figure 126: Skyscraper floors.......................................................................................... 126
Figure 127: Skyscraper located in the site of construction ............................................ 127
Figure 128 Maquettes of the Skyscraper Envelope........................................................128

13. 8
LIST OF TABLES
Table 1: Comparing grid shells' topologies main properties..............................................9
Table 2: Relevant material properties of basic soda lime silicate glass according to CEN
EN 572-1 2004 [BIV07] ....................................................................................................55
Table 3: Values of coefficient Ψ for laminated glass beams under different boundary and
load condition....................................................................................................................64
Table 4: The left-hand number represent the floor number, the right-hand number
represent the wind force applied to a specific floor ( / 2 . ..........................................95
Table 5: Modal analysis results with SAP2000.............................................................. 125

14. 9
1 ARCHITECTURAL GEOMETRY
Geometry is the core of the architectural design process and it is present from the initial
form finding to the construction. Free form surfaces represent the emblematic
expression of contemporary architecture, where the façade and the roof tend to merge
into a single element: the skin of the building.
Finding a proper shape by using geometric knowledge helps to ensure a good fabrication.
The complete design and construction process involves many aspects as form finding,
feasible segmentation into panels, functionality, materials, statics and cost. Geometry
alone is not able to provide solutions for the entire process, but a solid geometric
understanding is an important step toward a successful realization of such a project.
1.1 SURFACE DISCRETIZATION
There is a current trend toward architectural freeform shapes based on discrete surfaces,
largely realized as steel/glass structure. Topology is probably the most important
variable when dealing with free forms, and the most common topologies adopted are the
triangular, the quadrilateral one and seldom also the hexagonal one.
Table 1: Comparing grid shells' topologies main properties.
We can introduce some definitions to clarify the Table 1:
 A node (vertex) is a point where more edges converge,
 The valence of a node is the number of edges incident to the node,
Triangular Optimal Intrinsically flat 6 Yes High High
Quadrangular Good Quite easy 4 No Low Low
Hexagonal Quite good Not trivial 3 No Low Very Low
Sensitivity to
Imperfections
Surface
Approx.
Face Planariz.
Complexity
Valence of
Reg. Nodes
Torsion of
Nodes
Overall
Stiffness

15. 10
 The torsion of nodes consists in the twisting of sides of the meshes adopted due
to an applied torque: the tendence of a force to rotate an object about an axis.
This concept will be explain in chapter 1.1.1.
1.1.1 Triangle Meshes
Most of the basic tasks in geometric computing deal with the adaption of triangle meshes
to freeform surfaces. A triangle mesh M can approximate a surface in an aesthetic and
well fitting way, but it has to be noted that we obtain a valence of six using such meshes.
The valence or degree of a vertex is the number of edges incident to a vertex, this means
that in every node of a triangle mesh six edges merge.
To manufacture the mesh at the best possible cost, it is necessary to meet rather tight
constraints on the edge length and the angles in the triangular faces. Designing meshes
with large faces reduces the cost. Triangle meshes are easy to deal with from the
prospective of representing a given surface with the desired accuracy. To achieve
aesthetic aims as well as the proper requirements to statics, we use flat panels, which
provide overall high stiffness.
Howevere there are some disadvantages that we have to consider:
 In a steel/glass or other construction based on a triangle mesh, six beams
meet in a node; this inplies a higher node complexity.
 The cost of triangular glass panels are higher per-area than the cost of
quadrilateral panels.
 More nodes imply more steel and glass, and as a consequence more weight.
 Apart from simple cases triangle meshes do not possess offsets at constant
face-face or edge-edge distance.
 Triangle meshes have high valence as geometric torsion on the nodes.
Excellent examples of triangular grid-shells are shown in the Figure 1 and Figure 2.

16. 11
Figure 1: (left) British Museum Great Court Roof, London, completed in 2000 (right) DG Bank Court
Roof, Berlin, completed in 1998
Figure 2: (left) Blob, Eindhoven, completed, (right) Vela Fiera Milano-Rho, Milano, completed in 2005
A geometric support structure of a connected triangle mesh with torsion-free nodes can
be simply realised if the shape is optimized; instead of a general free form triangle mesh
there is no chance to construct a practically useful support structure with torsion free
nodes. Essentially for the complexity of the nodes, nowadays the triangular topology is
decreasingly used. Instead one uses quadrilateral meshes in most applications.
Figure 3: (left) a node without an axis. Image of Waagner-Biro Stahlbau AG. (right) Geometric Torsion
in a Node

17. 12
1.1.2 Quadrilateral Meshes
Quadrilateral meshes exhibit two remarkable disadvantages: on the one hand their
stiffness is lower and on the other hand we have to consider non-planar panels in general.
Flat panels are of course cheap to produce, but also single curvature panels can be
obtained at little cost through the cold bending technique. In a quad mesh, an interior
vertex of valence four is called a regular vertex. If the valence is different from four, we
talk about an irregular vertex.
1.1.2.1 PQ meshes
Planar quad meshes also known as PQ meshes, can be easily used to represent
translational surfaces which are obtained by traslating a polygon along another polygon.
Also rotational surfaces can be generated by PQ meshes.
In a rotational PQ mesh, the mesh polygons are aligned along parallel circles and
meridian curves. Adjacent mesh polygons of the same family form PQ strips, which can
be seen as discrete versions of developable surfaces tangent to a rotational surface S
along the rotational circles and meridian curves. The network of parallel circles and
meridian curves is an instance of a conjugate curve network, and the PQ mesh can be
seen as a discrete version of it. [PAH07]
The tangents to the curves of one family of a conjugate curve form a developable ruled
surface that can be always represented by PQ meshes.
Figure 4: (left) Rotational PQ mesh, (right) Geometry of a conjugate curve network [PAH07]

18. 13
Examples of structures with quad meshes are the following in Figure 5.
Figure 5: (left) Mannheim Grid Shell, Mannheim, completed in 1974, (right) Hamburg History Museum
Court Roof, Hamburg, completed in 1989
1.1.3 Hexagonal Meshes
Hexagonal meshes might have non-planar panels and exhibit a low overall stiffness
compared to an equivalent triangular grid. Furthermore they are aesthetically pleasing,
most of the times they even resemble organic forms and additionally they have a very low
valence of the nodes which makes their production much easier.
The Honeycomb subdivision algorithm is a remeshing operator which translates a
triangular mesh into an hex-dominant one.
Figure 6: (left) Honeycomb subdivision, (right) Regular triangular tiling [PAH07]
1.1.3.1 P-Hex meshes
Free form meshes with planar hexagonal faces, which are called P-Hex meshes, provide
a useful surface representation in discrete differential geometry and are demanded in
architectural design for representing surfaces built with planar glass/metal panels.
According to Liu’s algorithm [WLY08] the progressive conjugation method is used to
obtain a hexagonal mesh with planar panels. This method ensures that the resulting P-
Hex faces are nearly affine regular or quasi-regular hexagons, since ideal triangles are

19. 14
computed within discretization error. A problem with this approach is that the widths
and orientations of the triangle layers cannot easily be predicted or controlled.
Figure 7: P Hex mesh computed using the progressive conjugation method [WLY08]

20. 15
2 DIFFERENTIAL GEOMETRY
This chapter contains a brief summary of some important concepts and definitions that
will be useful in the remainder of this thesis.
The differential geometry of curves and surfaces has two aspects. One, which may be
called classical differential geometry, is connected with the beginnings of calculus.
Roughly speaking, classical differential geometry is the study of local properties of curves
and surfaces. By local properties we mean those properties which depend only on the
behavior of the curve or surface in the neighborhood of a point. In this thesis curves and
surfaces will be defined by functions which can be differentiated a certain number of
times.
The other aspect is the so-called global differential geometry. Here one studies the
influence of the local properties on the behavior of the entire curve or surface.
2.1 PARAMETRIC SEARCH
Parametric search is a technique that can sometimes be used to solve an optimization
problem when there is an efficient algorithm for the related decision problem.
The parametric search technique was invented by Megiddo as a technique to solve certain
optimization problems. It is particulary effective if the optimization problem can be
phrased as a monotonic root-finding problem and if an efficient algorithm for the
corresponding fixpoint problem can be constructed.
More specificall a root- finding problem consists of finding the largest value ∗
of with
the property that ∗
0. Let be a monotonic function with a root and let be
an algorithm that computes , written in the form of a binary decision tree whose
nodes correspond to inequalities 0. The parametric search technique evaluates
∗
, and in the process discovers ∗
, by evaluating the sign of at some of the roots
of .

21. 16
Suppose that the optimization problem has inputs. Then the decision problem has
1 inputs where the additional input is for the parameter . [dC76]
2.2 DEFINITIONS
A Curve indicates any path, whether actually curved or straight, closed or open. A curve
can be on a plane or in three-dimensional space. Lines, circles, arcs, parabolas, polygons,
and helices are all types of curves
A Curve tangent is a line that touches a curve at a point without crossing over.
Formally, it is a line which intersects a differentiable curve at a point where the slope of
the curve equals the slope of the line.
Figure 8: Tangent on a curve [PAH07]
Curvature is the amount by which a geometric object deviates from being flat,
or straight in the case of a line, but this is defined in different ways depending on the
context. There is a key distinction between extrinsic curvature, which is defined for
objects embedded in another space (usually a Euclidean space) in a way that relates to
the radius of curvature of circles that touch the object, and intrinsic curvature.
A vertex V is a point with locally extremal curvature. At a generic vertex, the osculating
circle remains locally on the same side of the curve.
The inflection point is a point on a curve at which the sign of the curvature (i.e.,
the concavity) changes. Inflection points may be stationary points, but are not local
maxima or local minima.

22. 17
Figure 9: Inflection point on a curve
The osculating circle o of a curve at a given point is the circle that has the
same tangent and curvature as the curve at point . Similar, as the tangent is the best
linear approximation of a curve at a point , the osculating circle is the best circle that
approximates the curve at . Let , , denote the circle passing through three
points on a curve , with . Then the osculating circle is given by
lim
⟶
, , .
Figure 10: Osculating circles [PAH07]
An osculating paraboloid p is the counterpart of an osculating parabola that
approximates a surface at a point . We use a special coordinate system at to get the
equation:
∶
2 2
(1)
Here the -plane is the tangent plane and the z-axis is the surface normal of at . Then
the -plane and the -plane are symmetry planes of and has locally the same
curvature behaviour as the surface . The two numbers and are called principal
curvatures of , whereas the -axis and -axis are called principal curvature directions.
Given a set of 1 control points P , … , P the corresponding Bézier curve (or
Bernstein-Bézier curve) is given by

23. 18
c t B , t P, (2)
where B , t 1 and t ∈ 0,1 .
A "rational" Bézier curve [MWW] is defined by
C t B , t w P / B , t w , (3)
where is the order, B , is defined as in (2), are control points, and the
weight of is the last ordinate of the homogeneous point. These curves
are closed under perspective transformations, and can represent conic sections exactly.
In the plane every Bézier curve passes through the first and last control point and lies
within the convex hull of the control points. The curve is tangent to and
at the endpoints. The "variation diminishing property" of these curves tells that no
line can have more intersections with a Bézier curve than with the curve obtained by
joining consecutive points with straight line segments. An other desirable property of
these curves is that the curve can be translated and rotated by performing these
operations on the control points only.
Figure 11: Bézier curve [PAH07]
A B-spline is a generalization of a Bézier curve. Therefore we define the knot vector
T t , t , … , t , (4)
where T is a nondecreasing sequence with t ∈ 0,1 . Furthermore we define control
points P , … , P and the degree as

24. 19
p ≡ m n 1. (5)
The "knots" t , … , t are called internal knots.
We define the basis spline functions via
N , t
1, if t t and t t
0, otherwise
(6)
N , t
t t
t t
N ,
t t
t t
N , t , (7)
where j 0,1, … , p. Then the curve
c t N , t P, (8)
is a so-called B-spline.
Specific types include the nonperiodic B-spline ( 1 knots equal 0 and where the
last 1 knots equal to 1) and the uniform B-spline (all internal knots are equally
spaced). A B-spline with no internal knots is a Bézier curve.
A B-spline curve is p k times differentiable at a point, where k duplicate knot values
occur. The knot values determine the extent of the control of the control points.
A nonuniform rational B-spline curve (NURBS) is defined by
, / , , (9)
where is the order, , are the B-spline basis functions, are control points, and
the weight of is the last ordinate of the homogeneous point. These curves are
again closed under perspect ive transformations and can represent conic sections
exactly.

25. 20
2.3 CURVES
The world of Euclidean geometry is inhabited by lines and planes. If we wish to go beyond
this flat world to a universe of curvature, we need to understand more general types of
curves and surfaces.
A curve [Opr07] in 3-space is a continuous mapping : I → where Ι is some type
of interval on the real line . Because the range of is , ’s output has three
coordinates. We then write, for ∈ , a parametrization for ,
, , (10)
where the are themselves functions : I → . We say is differentiable if each
coordinate function is differentiable as an ordinary real-valued function of .
In order to define curvature and torsion, we will need each to be at least 3-times
differentiable.
The velocity vector of at is defined to be
| , | , | (11)
Where / is the ordinary derivative and | denotes the evaluation of the
derivative at .
Parametric search could be concretized in parametric representation of a parametric
curve that is expressed as functions of a variable . This means that the spatial curve
can be represented by , , , where is some parameter assuming all
values in an interval . We could consider a curve as the result of a continuous mapping
of an interval into a plane or three-dimensional space. Thereby, every parameter is
mapped onto a curve point . The functions , and are called the
coordinate functions and is a parametrization of .
 Helics in parametric representation: Given the center , and the radius
, the points , of the circle are described as
(12)

26. 21
2.4 SURFACES
Surfaces are 2-dimensional objects and should be describable by two coordinates. We
should try to spread part of the plane around a surface and, in terms of the required
twisting and stretching, understand how the surface curves in space.
Let denote an open set in the plane . The open set will typically be an open disk or
an open rectangle. Let:
: D →
, → , , , , , , (13)
denote a mapping of into 3-space. The , are the component functions of the
mapping . We can perform calculus on time depending variables by partial
differentiation. Fix and let vary. Then , depends on one parameter and
is, therefore, a curve. It is called a u-parameter curve. Simillary, if we fix then the
curve is , is a v-parameter curve. Both curves pass through , in .
Tangent vectors for the u-parameter and v-parameter curves are given by differentiating
the component functions of with respect to and respectively.
We write
, , (14)
, ,
(15)
We can evaluate this partial derivatives at , to obtain the tangent, or velocity,
vectors of the parameter curves at that point, , and , . [Opr07]

27. 22
2.4.1 Surfaces of Revolution
Suppose is a curve in the -plane and is parametrized by , ,0 .
Revolve about the -axis. The coordinates of a typical point may be found as follows.
As it is mentioned in [Opr07], the -coordinate is that of the curve itself since we rotate
about the -axes. If denotes the angle of rotation from the -plane, then the -
coordinate is shortened to cos v and the -coordinate is given by
sin v . The function may be defined by:
, , cos v , h u sin v (16)
Examples of surfaces of revolution are:
Catenoid: obtained by revolving the catenary cosh about the -axis.
Torus: obtained by revolving the circle of radius about the -axis.
, cos , sin , (17)
Torus has been analised better in the chapter 2.4.2.1 and Catenoind in chapter
2.4.3.
2.4.1.1 Ruled Surfaces
A surface is ruled if it has a parametrization
, (18)
where and are space curves. The entire surface is covered by this one patch, which
consists of lines emanating from a curve going in the direction . The curve
is called the directrix of the surface and a line having as direction vector is called a
ruling.
Examples of ruling surfaces are:
 Cones: , where is a fixed point.
 Cylinders: , where is a fixed direction vector.
 Helicoid: take a helix acos , asin , and draw a line through
0,0, and acos , asin , . The surface sweept out by this rising and
rotating line is a helicoid. A patch for the helicoid is given by

28. 23
, , , (19)
Hyperboloid (a doubly ruled surface):
Figure 12: Hyperboloid
cos ∓ sin
sin cos
cos
sin
0
sin
cos (20)
Hyperbolic paraboloid (a doubly ruled surface):
Figure 13: Hyperbolic paraboloid
2
0
2
(21)

29. 24
Plücker's conoid:
Figure 14: Plücker's conoid
cos
sin
2 cos sin
0
0
2 cos sin
cos
sin
0
(22)
Möbius strip:
Figure 15: Möbius strip
cos cos
1
2
cos
sin cos
1
2
sin
sin
1
2
cos
sin
0
cos
1
2
cos
cos
1
2
sin
sin
1
2
(23)

30. 25
2.4.1.2 Developable surface
A developable surface is a ruled surface with global Gaussian curvature 0. Gaussian
curvature will be introduced in section 2.4.2. Developable surfaces include
the cone, cylinder, and plane.
A developable surface has the property that it can be made out of a sheet, since such a
surface must be obtainable by transformation from a plane (which has Gaussian
curvature zero) and every point on such a surface lies on at least one straight line. There
are three basic types of developable surfaces: cylinders, cones, and tangent surfaces of
space curves.
Figure 16: Cylinders, cones, and tangent surfaces of space curves [PAH07]

31. 26
2.4.1.3 Developable surfaces with a NURBS
One of the advantages of NURBS curves [PT97] is that they offer a way to represent
arbitrary shapes while maintaining mathematical exactness and resolution
independence.
Among their useful properties are the following:
 They can represent virtually any desired shape, from points, straight lines, and
polylines to conic sections (circles, ellipses, parabolas, and hyperbolas) to free-
form curves with arbitrary shapes.
 They give great control over the shape of a curve. A set of control points and knots,
which determine the curve's shape, can be directly manipulated to control its
smoothness and curvature.
 They can represent very complex shapes with remarkably little data. For instance,
approximating a circle three feet across with a sequence of line segments would
require tens of thousands of segments to make it look like a circle instead of a
polygon. Defining the same circle with a NURBS representation takes only seven
control points.
In addition to draw NURBS curves directly as graphical items; we can use them
as a tool to design and control the shapes of three-dimensional surfaces, for
purposes such as:
- surfaces of revolution (rotating a two-dimensional curve around an axis
in three-dimensional space)
- extruding (translating a curve along a curved path)
- trimming (cutting away part of a NURBS surface, using NURBS curves to
specify the cut)
One very important motivation for using NURBS curves is the ability to control
smoothness. The NURBS model allows you to define curves with no kinks or sudden
changes of direction or with precise control over where kinks and bends occur.
One of the key characteristics of NURBS curves is that their shape is determined by the
positions of a set of points called control points. As in the Figure 17, the control points
are often joined with connecting lines to make them easier to see and to clarify their
relationship to the curve. These connecting lines form is known as control polygon.

32. 27
Figure 17: Nurbs curve generated by control points polygon [PAH07]
The second curve in Figure 17 is the same curve, but with the weight increased in one of
the control points. Notice that the curve's shape isn't changed throughout its entire
length, but only in a small neighborhood near the changed control point. This is a very
desirable property, since it allows us to make local changes by moving individual control
points, without affecting the overall shape of the curve. Each control point influences the
part of the curve nearest to it but has little or no effect on parts of the curve that are
farther away.
One way to think about this is to consider how much influence each of the control points
has over the path of our moving particle at each instant of time.

33. 28
2.4.2 Principal curvature directions
Before we can introduce principal curvature directions we have to introduce the concept
of normal curvature [MWW]. Let be a surface in that is given by the graph of
a smooth function , . Assume that passes through the origin and its
tangent plane in is represented by the 0 plane. Let 0, 0, 1 be a unit
normal to at the origin.
Figure 18: Normal curvatures of a surface S at a point p are the curvatures of the intersection curves
with planes R through the surface normal [PAH07]
Furthermore we denote , , 0 a unit vector in . Let be the parameterized
curve given by slicing through the plane spanned by and . We obtain,
, , , (24)
For a plane curve we introduce the concept of signed curvature at with respect to
the unit normal : is the reciprocal of the radius of the osculating circle to at , taken
with sign as in the examples below:
Figure 19: The osculating circle varies depending on curves [Jau11]
More rigorously, is defined by the formula:
| (25)

34. 29
where represents arc length with 0 0 (i.e., | . For
instance, one can readily verify that a circle of radius has signed curvature 1/ at each
point with respect to the inward-pointing unit normal.
[Jau11] The normal curvature has a maximum value and a minimum value . These
two quantities are called the principal curvatures and the corresponding directions are
orthogonal and are called principal directions. This was shown by Euler in 1760.
The quantity
(26)
is called the Gaussian curvature and the quantity
2
(27)
is the so-called the Mean curvature, which both play a very important role in the
theory of surfaces.
2.4.2.1 Classification of points on a surface
A point of a surface is called:
Elliptical, both principal curvatures , have the same sign, and the surface is locally
convex.
0. (28)
Umbilic, the principal curvatures , are equal and every tangent vector is a principal
direction. These typically occurs in isolated points.
Hyperbolic, the principal curvatures , have opposite signs, and the surface will be
locally saddle shaped.
0. (29)
Parabolic, one of the principal curvatures is zero. Parabolic points generally lie in a
curve separating elliptical and hyperbolic regions.
Flat umbilic, both principal curvatures are zero. A generic surface will not contain flat
umbilic points.

35. 30
Torus:
Figure 20: Points change based on the position on torus: elliptic, hyperbolic, parabolic [PAH07]
2.4.2.2 Lines of curvature
A line of curvature of a regular surface is a regular connected curve ⊂ , such that for
all the tangent line of is a principal curvature direction at .
Typically forms of lines of curvature near umbilics are are star, lemon and monstar
(derived from lemon-star).
Figure 21: Star, lemon and monstar lines of curvature [WIKI]
2.4.3 Minimal surfaces
A surface that locally minimized its area is called a minimal surface. Equivalently one
can define a minimal surface as a surface whose mean curvature vanishes. We may
observe minimal surfaces as the shape of a soap membrane through a closed wire .
Neglecting gravity, surface tension implies that the soap membrane attains the shape of
the surface with minimal surface area. A minimal surface has vanishing mean curvature
in each of its points.

36. 31
Examples of minimal surfaces are:
Catenoid: [Opr07]
Figure 22: Catenary and catenoid
A catenoid is a surface in 3-dimensional Euclidean space arising by rotating
a catenary curve about its directrix. Not counting the plane, it is the first minimal
surface to be discovered. It was found and proved to be minimal by Leonhard Euler in
1744. Apart from the plane, the catenoid is the only minimal surface of revolution.
The catenoid may be defined by the following parametric equations:
cos ,
sin ,
,
(30)
where ∈ , and ∈ and is a non-zero real constant.
In cylindrical coordinates:
, (31)
where is a real constant.
Helicoid: [Opr07]
Figure 23: Helic and Helicoid

37. 32
Its name derives from its similarity to the helix: for every point on the helicoid, there is
a helix contained in the helicoid which passes through that point. It can be described by
the following parametric equations in Cartesian coordinates:
cos ,
sin ,
,
(32)
where and range from negative infinity to positive infinity, while is a constant.
If is positive, then the helicoid is right-handed, if negative then left-handed.
The helicoid has principal curvatures 1/ 1 .
Other minimal surfaces from the 19th century are: Schwartz minimal surfaces, Riemann's
minimal surface, Enneper surface, Bour's minimal surface, Gyroid, Chen–Gackstatter
surface.
Figure 24: from the right 1. Schwartz minimal surface, 2. Riemann's minimal surface, 3. Enneper surface,
4. Bour's minimal surface, 5. Gyroid, 6. Chen-Gackstatter surface

38. 33
3 CASE STUDY: TWISTING SKYSCRAPER
After this short introduction on architectural geometry and differential geometry, we
want to consider a particular case study: a skyscraper. The goal of this research is to find
a good shape that can be easily built under the constraint of optimising materials and
costs. The work starts with different design processes in order to be aware of a good
choice.
The choice of the shape is furthermore influenced by:
 the geometrical aspect,
 the secondary structure exposed by wind loads.
3.1 DESIGN OF HIGH-RISE BUILDINGS
Technology and engineering of high-rise buildings have become far better and much
more sophisticated, but most, if not all of the skyscrapers constructed today remain
fundamentally the same in built configuration: in particular the basic planning remains
the same.
Whether built of concrete or steel, most are still nothing more than a series of stacked
trays piled homogeneously and vertically one on top of the other. [Yea02]
The shape of a skyscraper is mostly influenced by wind loads, which contribute to
aerodynamic modifications of the shape and different structural reinforcement. Thus
aerodynamics of the tower’s shape need to be considered as a critical parameter from the
first stage of design.
3.2 SOFTWARES
Complex surfaces, freeform geometry and relative structures are difficult to draw and
commonly used tools, such as AutoCAD or Revit, are not suited enough. Furthermore, it
is well known that projects change hundreds of times during their conception and their
realization: updating it with CAD and subsequenteky with all the softwares involved (for
structural analysis, energetics, fabrication, costs, etc…) it is unfeasible.

39. 34
One of the software that allow to satisfy this demand is Grasshopper, a plug-in for the
highly advanced 3D modeller Rhinoceros. This allows the user to see and exploit all the
possible solutions inside the domain space (the imput parameters vary in a defined
domain).
Grasshopper hosts a lot of different plug-ins, each one specialized on some aspect of
design workflow. Numerous plug-ins have been used throughout the entire project
development. In the following, a list of them is presented along with a short introduction.
Kangaroo is probably the most known and used plug-in of all. What it does is to add a
physical engine and number of physical forces and interactions. A physical engine may
have a lot of useful uses. In this thesis it has been used to find planar quad mesh of the
case-studies.
Hoopsnake came to solve one of the biggest flaw of Grasshopper: the impossibility to
perform a so called for loop. Since Grasshopper has a linear workflow, meaning that the
flow of data goes in one and one only direction. Hoopsnake comes to change this habit.
There one can imput the initial data, at step 0, run them through the script, and re-input
the updated data from a different input plug.
Python Script is a plug-in that include Python inside Grasshopper. Python is a modern
programming language that is used to automate a repetitive task in Rhino much faster
than a manually way. Perform task that are not accessible in the standard set of Rhino
commands or Grasshopper components are available in Python.
WeaverBird is a powerful mesh editing tool. It can perform various mesh subdivisions
(e.g. Catmull-Clark, Sierpinsky, midedge, Loop).
LunchBox is a plug-in that include unusual geometry, panels and structures.
Furthermore with this component it is possible to write/read a .xls file.
Karamba3D is a commercial plug-in by Bollinger+Grohmann ingenieur, a German
structural engineering firm. It allows to perform structural analysis, right inside
grasshopper. Although it is not as powerful as other stand-alone well-known softares,
first of all SAP2000 and GSA, its being directly connected to the design workflow
increase the time performance.

40. 35
Moreover, since it is inside grasshopper, it can perform various operations at any step of
the workflow, something which would be quite difficult to achieve with a stand-alone
software. Here’s a list of some interesting operations Karamba is able to do:
 Actively operate on the design geometry (topology);
 Modify any set of data, according to certain structural output;
 Allowing the creation of an optimization loop, with the aim of minimize
(maximize) of some structural output.
Geometry Gym is another plugin for anyone interested in structural
analysis/optimization, keeping linked with Grasshopper. This plug-in enables
reading/writing of files from/to any structural software (e.g. SAP2000, GSA, Robot
structural Analysis). Projects realized inside Grasshopper can be exported, analysed and
re-imported into it, giving the designer an important edge over the workflow process.
Evolue Tools PRO is a plug-in for Rhino, not for Grasshopper, differently from the
others. This software is an advanced geometry optimization tool for freeform surfaces
with a user friendly interface. Established computational tools from Evolute's core
software library as well as ground breaking technology from our cutting edge research
results provide you with optimization functionality not offered by any other CAD system.
This software offers:
 multi-resolution mesh modelling;
 global and local subdivision rules;
 mesh editing tools and mesh optimization for various goals (closeness, planarity,
fairness, coplanarity, edge length repetition, ballpacking);
 specification of vertices as anchor/corner points, constraints (floor slabs, generai
co-planarity constraints, reference curves, fine grained fairing);
 specific analysis such as closeness, planarity, edge length, principal curvature and
asymptotic line analysis;
 pattern mapping;
 NURBS fitting.
Evolute Tools is used at the end of the research process based on knowledges of
differential geometry.

41. 36
3.3 CHECK DESIGN
Development of tall buildings has been changing from year to year starting with the
Home Insurance Building in Chicago (1885) up to the tallest building the Burj Khalifa
realised in Dubai in 2010. In the Figure 26 the evolution of the building design of
skyscrapers is depicted; examples include the WTC, the Sears (Willis) Tower built in the
1907 and Taipei 101 in 2004.
Figure 25: Development of tall buildings
Figure 25 shows how reducing the section area gradually toward the top is a good
strategy to enhance lateral performance of a tall building.
Examples of tapering buildings are the Burj Kalifa in Dubai (828 m height), the
Millennium Tower in Tokyo with his 840 meters (not yet built), the Shard in London
(319 m height) and the Sears Tower in Chicago (527 m), where tapering is very often
associated with the changing of the cross section.

42. 37
Figure 26: (from the left) Burj Kalifa, Millennium Tower, Shard, Sears Tower
3.3.1 Twisting and rotating forms
An interesting approach in contemporary tall building design is a twisted form. In
general, twisting and rotating forms are effective in reducing vortex-shedding induced
dynamic response of tall buildings by disturbing vortex creation.
The twisting of buildings minimises the wind loads from prevailing directions and avoids
the simultaneous vortex shedding along the height of the building. Rotating the building
can also be very effective because its least favourable aspect does not coincide with the
strongest wind direction.
To define a twist deformation, we introduce a fixed bottom plane and a straight line ,
which is called the twist axis, orthogonal to the plane . The layers of the object in the
planes orthogonal to the axes are rotated about as follows (Figure 27). The bottom
plane remains fixed and the rotational angle of the top plane is prescribed. The
distance between the bottom and top planes is , the height of the object to be deformed.
The rotational angle / .
This is a linear variation of the rotational angle with respect to the distance. For the
bottom plane, we have 0 and thus 0 0 which means that the bottom slice
remains fixed. As desired, the top plane is rotated by an angle . The plane
at bottom distance /2 is rotated by /2, and so on. [PAH07]

43. 38
Figure 27: Twisting geometry process [PAH07]
Examples of twisting towers are:
 The Turning Torso (190 m height) designed by Calatrava, which has a twist of 90
degrees from the bottom plane. It is composed by a central concrete core that is
able to take wind loads even without a secondary structure in the façade.
 30 st. Mary Axe in London (180 m) with a triangulated perimeter steel structure
to eliminate extra reinforcement.
Figure 28: (left) Turning Torso in Malmö, Sweden (right) 30 St. Mary Axe in London, UK
3.3.1.1 My twisting form
In this thesis we study the design of a twisted 74 high building, where each story is
assumed to have an heigh of four meters. We use a twisting algorithm for the rotational
angle based on an exponential function ∗
1 where 0.011 and ∑ . The
goals of the project are the following:
 To optimize the panels for the secondary structure in a geometrical way

44. 39
 To give a substance to the secondary structure and analyse the actions of wind
loads.
Modern architecture employs different kinds of geometric primitives when segmenting
a freeform shape into simpler parts for the purpose of building construction. For most
materials used (glass panels, wooden panels, metal sheets,…), it is very expensive to
produce general double-curved shapes. A popular way aims to use approximation by flat
panels, which most of the time are triangular. A third way, less expensive than the first
and capable of better approximation than the second, is segmentation into single curved
panels. The decision for a certain type of segmentation depends on the costs, but also on
aesthetics. The visual appearance of an architectural design formed by curved panels is
different from a design represented as a polyhedral surface.
The planarity constraint on the faces of a quad mesh however is not so easy to fulfill, and
infact there is only little computational work on this topic. So far, architecture has been
mainly concentrating on shapes of simple genesis, where planarity of faces is
automatically achieved. For example, translational meshes, generated by the translation
of a polygon along another polygon, have this property: all faces are parallelograms and
therefore planar.
3.3.1.2 Base shapes
After fixing the twisting form, the next step deals with the analysis of different kind of
basis shapes in order to find the best solution for a given task.
Three different solutions have been proposed whose initial shape is a square in each case.
The analysis gives differents results depending on the shape analysed.
The first shape is simply a square with smooth vertices; the second shape is completely
curved and convex with no inflection points; the third one is composed of eight inflection
points and it is a NURBS curve. As mentioned in Section 3.3.1.1 the task consists of
constructing a twisted building with 74 stories. We will perform this task for the three
basic shapes described above.

45. 40
Figure 29: Base shapes case study and few stories that define the final building
3.4 PANELIZATION
In the next chapters there will be different approaches to check the best base shape and
the best pannelization type. The target is to achieve flat panels and reduce costs of
fabrication with clusters of panels.
3.4.1 Search for the same tangent on curve
Figure 30: Generic curve with tangents in random points
One way to find planar panels aims to have the same tangent from one floor to the next
floor.
For finding planar trapezoids, a given number of points are fixed on the first curve in
order to achieve 74 panels of a length of around 1.5 meters each (Figure 31).

46. 41
Figure 31: 74 computed points on curve with a distance of 1.5 m each.
For every point on the curves the ortogonal plane tangent has been identified using a
parametric algorithm (Figure 33). To define a strictly planar mesh, the point projected
to the curve above has the same tangent of the point to the curve below (Figure 32).
Figure 32: Visualization of the adopted method to find planar meshes between two consecutive curves.

47. 42
Figure 33: Algorithm generated to define planar panels with the same tangent on curves

48. 43
The behaviour of the base shapes is completely differente for each of the studied cases:
1. The shape with straight edges and smooth vertices is the worst shape for this
algorithm. In the straight area, one can never find a corresponding point
projected from the floor below that has the same tangent on the curve: lines are
always oblique and all the points in the curve below converge in the same point
on the consecutive curve.
Only in the convex part of the shape, which corresponds to the smooth vertices,
it is possible to find points with the same tangent for two consecutive curves and
thus obtain complete planarity of the panels.
Figure 34: Research points with same tangents in the squared curve with smooth verteces
2. For a basis curve of the second type we obtain much better results. Here, it is easy
to find the same tangent from two consecutive floors and the result is satisfying;
every panel is completely planar.
Figure 35: Research points with same tangents in the convex curve

49. 44
3. For the third type of basis curve we obtained mixed results. The results are not
satisfying near the inflection points but the results in the other regions are
acceptable.
Figure 36: Research points with same tangents in the curve with inflection points

50. 45
3.4.2 Developable surface
Another way to find planar panels is to use a developable surface. In mathematics, a
developable surface is a surface with zero Gaussian curvature. Such a surface can be
flattened onto a plane without distortion. Therefore, it is always possible to find planar
panels for a developable surface.
Figure 37: Generic developable surface
It is however not possible to find a developable surface for a twisting shape. Instead, we
will consider a slightly simpler problem namely a simple translation in the z-direction
for every curve. For basis curves of the first and the second type, it is easy to see that this
construction always yields a developable surface. Therefore, we will perform the analysis
in this section only for curves of the third basis type, i.e., for a NURBS curve of degree 2
with 8 control points.
For a curve with degree 2 it is easy to find the inflection points since they are the
intersection points between the spline and the control point polygon.
Figure 38: (left) Starting control points (right) Generic NURBS curve of degree 2

51. 46
Figure 39: (left) Control point polygon (right) Control point polygon that intersect the NURBS curve in
inflection points
Figure 40: (left) Curvature graph for a generic NURBS curve (right) curvature graphs for two generic
NURBS curves
In order to create a free form skyscraper, it has been decided to assign different random
weights to the control points based on a algorithm that considers two groups of control
points: these external and these internal to the curve (Figure 41). Every group of control
points has the same weight for every floor (this means that there are two different values
for the weight of every floor, for example a value of 1,2 for external control points and a
value of 0.1 for internal control points) (Figure 42).
Figure 41: Groups of control points

52. 47
Figure 42: Algorithm to generate NURBS curves

53. 48
The values of the weights are connected from one curve to the consecutive with a function
sin where 0.5 1 for internal control points and 1 2 for
external control points. Figure 49 explains the results of this choice: we obtain a shape
with a sinusoidal motion in z-direction.
We can obtain also different special NURBS curves by changing the weights as arcs of a
parabola, hyperbola, ellipse or circle according to the following table.
Figure 43 Conic sections as special NURBS [PAH07]
By changing the weights of the control points of a NURB curve of degree 2 we obtain a
different shape. However, the curve intersect the polygon in the inflection points.
Therefore, one can compute easily tangent vectors at the inflection points, which are
parallel to every curve.
Figure 44: (left) intersection points of different NURBS curve in the same control polygon (right) zoom of
vectors tangent to this two curves
The curve is divided into 8 segments, where the separation points are choosen to be the
8 inflection points. Note that the tangents in the inflection points are parallel to the curve.
74 panels are created again for each floor and moreover found the pairs of corresponding
points and segments within the 74 points. Every segment has its own curvature and if we

54. 49
look at the convex part of each segment we can find easily the associated point in the
curves above with the same tangent vector.
For finding the associated point from one floor to the next we create a vertical plane with
inclination based on the tangent vector and find where this plane intersect the plane of
the consecutive curve. Making intersections between planes, we project the point on this
line and find the closest point in the curve above. With this procedure, two points are
found in two different curves with the same tangent vector. Planar panels can be achieve
for every floor (Figure 46) and we are able to find different envelopes changing the
weights of control points. To make an example, a sinusoidal function algorithm has been
created to modify every NURBS curve (Figure 47).
Figure 45: Planes parallel to vectors tangent on every point selected for one curve and points projected
on the following curve
Figure 46: Flat panels connecting two consecutive floors
Figure 47: Algorithm to find random weights

55. 50
With a looping algorithm, as shown in Figure 48, it is possible to find different
skyscraper’s shells with planar panels changing the weight of the control points.
Figure 48: Looping algorithm to create random skyscraper with developable surfaces

56. 51
Figure 49 Developable skyscrapers with flat panels created by random NURBS curves
In addition, a simple way to find a developable surface consist in scaling the curve:
parallel tangent vectors can be easily obtained from one curve to the consecutive curve
and thus all panels are completely planar. Examples are in Figure 50.
Figure 50: Developable skyscrapers with flat panels obtained by a scale alghoritm for NURBS curves

57. 52
3.4.3 Panelization with diamonds and triangles
Another way of pannelization is to use planar quads and triangles for every floor. The
algorithm that has been created works for all the three types of curves. As an example we
present the algorithm for the convex curve. This consists in finding a mesh, which is
completely planar. We started with 74 arbitrary points in the first curve and we projected
every point to the curve above; the corresponding points have been connected with a line
and we pick the middle point as an additional point. Now we have 3 points allineated and
a plane can be approximated throught this three points. Obviously, this input does not
uniquely define the plane (Figure 51).
Figure 51: Initial steps to reach flat diamonds
To construct the desired plane, an additional point is add on the above curve, namely we
consider a point very close to the original point (the approximation is 10-13). This
construction yields planes that are quite tangent to this two curves (Figure 52).
Figure 52: Plane quite tangent to the curves
A family of lines is obtained from the intersection of consecutive planes. Afterwards we
pick the middle points of these lines to find flat diamonds (Figure 53).

58. 53
Figure 53: The intersection of two consecutive planes is a line. Picking the middle point of every line and
connecting these points with points previously found on curves we found flat diamonds
Reorganizing the points and connecting them, the algorithm achieves flat quad diamond
panels with triangles. In Figure 54, the whole process to achieve planar diamonds and
triangles is depicted.
Figure 54: Steps to achieve planarity with diamonds and triangles

59. 54
Figure 55: Planarity analysis with Evolute Tools Pro
Repeating the same algorithm for all 74 floors, this works sufficiently well for the first
loops, but the approximation error for the construction of the planes grows from floor to
floor and at a certain point becomes unacceptable. (Figure 56)
Figure 56: The algorithm works well for the first floors, then the approximation becames unacceptable
One way to find good planar quad diamonds associated to triangles is to use the software
Evolute Tools, which achieves the task without an approximation error. Using triangles
and diamonds and the Evolute Tools optimization, all the meshes are strictly planar as it
is shown in Figure 57.
Figure 57: Mesh with planar diamonds and triangles optimized by Evolute Tools Pro

60. 55
3.4.3.1 Which triangles can be converted in flat diamonds with
cold bending
We have to think that every facette in a mesh is a glass panel that has its own property as
a real material. For architectural applications, glass is generally considered to be a
homogeneous and isotropic material. At temperatures below the deformation point
(which is 520° C for basic soda lime silicate glass), it is generally accepted that glass can
be assumed to be a linear elastic material. This behaviour abruptly endes when the failure
strength is reached: glass is brittle. Glass is usually employed as a shelter, or envelop for
the building. It guarantees solar lighting, whilst at the same time protection for external
adverse conditions. Due to recent technological advancements, its mechanical properties
can be exploited. Glass panels can be colored, multi-layered with films in between panels,
so as to five protection from UV rays, or as to change transmissivity with heat.
Table 2: Relevant material properties of basic soda lime silicate glass according to CEN EN 572-1 2004
[BIV07]
Using well-controlled residual stress, a toughened glass, which can be very useful for
structural applications, can be obtained. In this way, one can cause an overall
prestressing effect on the glass element, which increases its resistance against tensile
(bending) stresses: it virtually becomes stronger. Most prestressed glass is made by
means of a temperature treatment, but also chemical processes exist. Depending on the
level of prestress, the glass is called toughened (fully tempered) or heatstrengthened. The
strength of glass is a very complex characteristic which depends on external factors like
humidity (corrosion), ageing, surface flaws and scratches, loading history, loading speed,
and so on. The strength value corresponding to a fully tempered glass, according to CEN
EN 572-1 2004, is 120 / . [BIV07]
Curved glass can be applied in an interesting way in e.g. facades and canopies.
Traditionally, curved glass is manufactured from float glass that is heated above the
weakening point and formed in a heavy curving mould. However, this technique is time-
and energy consuming and consequently relatively expensive. For this reason, a more
affordable alternative has been developed. The technique is called a “cold bending

61. 56
process” because it is used to bend glass plates on the building site at room temperature.
In this process, toughened float glass laminates are gradually bent on a curved frame.
Finally, the newly curved panel is mechanically fixed to the frame, which implies that the
glass is continuously subjected to bending stresses during its lifetime. In this
contribution, time dependent loading-deformation interaction during the bending
process as well as relaxation after the bending process are closely examined [BIV07].
Only FLOAT or tempered glass is acceptable for structural use. Nowadays, the fail-safe
method is employed, which consists in using a multilayer panel, so that in case of a layer
failure, the others will support the load.
Float Glass Uses common glass-making raw materials, tipically consisting of sand,
soda ash (sodium carbonate), dolomite, limestone, and salt cake (sodium sulfate) etc.
Other materials may be used as colorants, refining agents or do adjust the physical and
chemical properties of the glass.
1. The raw materials are mixed in a batch mixing process, then fed together with
suitable cullet (waste glass), in a controlled ratio. The mixture is wet, so that it
will not realise dust. The whole process is computer controlled.
2. The production phase is subdivided into three main parts:
a. The mixture is brought into a furnace where it is heated to approximately
1500 °C. Common flat glass furnaces are 9 m wide, 45 m long, and contain
more than 1200 tons of glass.
b. The homogenization process, where gas bubbles are eliminated.
c. The cooling process at low viscosity, where the temperature of the glass is
stabilized to approximately 1200 °C to ensure a homogeneous specific
gravity.
3. The molten glass is fed into a “tin bath”, a bath of molten tin (about 3-4 m wide,
50 m long, 4 cm deep) at about 1100°C, from a delivery canal and is poured into
the tin bath by a ceramic lip known as the spout lip. The amount of glass allowed
to pour onto the molten tin is controlled by a gate called Tweel. Tin is suitable
immiscible into the molten glass. Tin, however, oxidixes in a natural atmosphere
to form Tin dioxide (SnO2). Known in the production process as dross, the tin
dioxide adheres to the glass. To prevent oxidation, the tin bath is provided with a
positive pressure protective atmosphere consisting of a mixture of nitrogen and
hydrogen. The glass flows onto the tin surface forming a floating ribbon with
perfectly smooth surface on both sides and an even thickness. As the glass flows

62. 57
along the tin bath, the temperature is gradually reduced from 1100 °C until the
sheet can be lifted from the tin onto rollers at approximately 600 °C. The glass
ribbon is pulled of the bath by rollers at a controlled speed. Variation in the flow
speed and roller speed enables glass sheets if varying thickness to be formed. Top
rollers positioned above the molten tin may be used to control both thickness and
the width of the glass ribbon.
4. A pyrolitic layer may be added, to add extra properties to the glass, such as low
emissivity, different transparency, etc.
5. Glass can now be heated again (heat tempering), to increase mechanical
properties.
6. Once off the bath, the glass sheet passes throught a so called étenderie, a cooling
tunnel, for approximately 100 m, where it is further cooled gradually from the
change in temperature. A slow air cooling comes right after, in order to eliminate
internal stress. On exting the “cold end” of the kiln, the glass is cut by machines.
Tempered glass With this method, the external layer is precompressed, so that
when bended, the external part won’t fissure. Moreover, when the glass breaks, only
small fragments with no sharp edges are generated.
For glass to be considered toughened, this compressive stress on the surface of the glass
should be a minimum of 69 MPa. For it to be considered safety glass, the surface
compressive stress should exceed 100 MPa. The greater the surface stress, the smaller
the glass particles will be when broken. It is this compressive stress that gives the
toughtened glass increased strength. This is because any surface flaws tend to be pressed
closed by the retained compressive forces, while the core layer remains relatively free of
the defects which could cause a crack to begin.
Any cutting or grinding must be done prior to tempering. Cutting, grinding, sharp
impacts and sometimes even scratches after tempering will cause the glass to fracture.
The glass solidified by dropping into water, know as “Prince Rupert’s Drops”, which will
shatter when their “tails” are broken, are extreme examples of the effects of internal
tension. The strain pattern resulting from tempering can be observed with polarized light
or by using a pair of polarizing sunglasses.
Toughtened glass must be cut to size or pressed to shape before toughening and cannot
be re-worked once toughening. Polishing the edges or drilling holes in the glass is carried
out before the toughening process starts. Because of the balanced stresses in the glass,
damnage to the glass will eventually result in the glass shattering into thumbnail-sized
pieces. The glass is most susceptible to breakage due to damnage to the edge of the glass

63. 58
where the tensile stress is the greatest, but shattering can also occur in the event of a hard
impact in the middle of the glass pane or if the impact is concentrated. Using toughened
glass can pose a security risk in some situations because of the tendency of the glass to
shatter completely upon hard impact rather than leaving shards in the window frame.
The surface of tempered glass does exhibit surface waves caused by contact with
flattening rollers, if it has been formed using this process. This waviness is a significant
problem in manufacturing of thin film solar cells.
Laminated glass is a type of safety glass that hold together when shattered. In the
event of breaking it is held in place by an interlayer, typically of polyvinyl butyral (PBV),
between its two or more layers of glass. The interlayer keeps the layers of glass bonded
even wen broken, ant its high strength prevents the glass from breaking up into the large
sharp pieces. This produces a characteristic “spider web” cracking pattern when the
impact is not enough to completely pierce the glass. Laminated glass is normally used
when there is a possibility of human impact or where the glass could fall if shattered. The
PVB interlayer gives the glass a much higher sound insulation rating, due to the damping
effect, and also blocks 99% of incoming UV radiation.
A typical laminated glass is composed as as follows:
 Glass
 Transparent thermoplastic material like TPU, PVB or EVA
 LED (led emitting diodes) on transparent conductive Polymer
 Transparent thermoplastic material like TPU, PVB or EVA
 Glass

64. 59
3.4.3.1.1 Panels deviation
Panel’s deviation is the distance between diagonals of a mesh (Figure 58) and the
planarity of a mesh face is measured by the shortest distance between its diagonals.
Figure 58: Panels deviation [EPR]
This distance is measurable and quantifiable. The function for the total planarity, ,
of a mesh with some number of faces, , can be represented as the sum of this distance
for each face in the mesh. [EPR]
… (33)
If triangle meshes are replaced with diamonds, it is possible to analyse with simple initial
geometrical considerations which panels are planar and which are not planar:
 6008 meshes are strictly planar,
 2847 meshes have a deviation smaller than 0,001 (maximum deviation should be
1/250 0,004 that for now is reduced to 0,001 for safety reason)
 2391 meshes have a deviation higher than 0,001.
Figure 59: Grasshopper definition for evaluating planarity

65. 60
In the following pictures we can see in blue planar panels, in yellow panels with cold
bending and in red panels with higher curvature.
Figure 60: Blue flat panels, yellow panels flat with cold bent, red panels with double curvature

66. 61
3.4.3.1.2 Min. Cold Bending radius
The minimum cold bend radius has been calculated from some companies depending on
the structure of the glass and other properties. In this thesis we chose the numbers
provided by SEDAK, a company specialized in structural glass and cold bending.
The minimum radius depends on the thickness of the glass panel:
R 1500 ∗ s (34)
Where:
The thickness adopted for the panel to verify is composed by three layers:
20
8
0.38
With this thickness the minimum radius is R 42.57 m .
In a next step we found the radius of the principal curvature in the point ; in
the main direction of every panel to analyse which of the panels respect this limits.
Figure 61: Screenshot of grasshopper definition for the analysis of principal curvatures
With this analysis the results are:
 6008 meshes are strictly planar (results from the first analysis)
 3486 meshes have a radius higher than 42.57
 1752 meshes have a radius smaller than 42.57

67. 62
Figure 62: Results of mesh analysis and cold bent
In the following pictures we can see in blue planar panels, in yellow panels with cold
bending and in red panels with higher curvature.
Figure 63: Analysis of cold bending
In Figure 63, it is shown how panels on the top of the skyscraper could be planar with
cold bending. The deviation for panels approximated planar with cold bending goes from
0.025 to 5.6 ; therefore the most high deviation is 5.6 .
The other panels have a deviation between 11.6 and 67.5 .
To verify the structure of the panel under the given thickness, one glass panel has been
created as a model using the software SAP2000 and it has been used to confirm that the
choosen thickness satisfies the wind loads.

68. 63
The thickness adopted is evaluated with the Enhanced Effective Thickness method
[EET] that is extended to the case of laminated glass beams composed by three layers of
glass of arbitrary thickness.
Figure 64: Geometry of a laminated glass
The deflection-effective thickness is:
1
12
1 (35)
And stress-effective thickness is:
,
1
2 ,
12
(36)
,
1
2 ,
12
(37)
Where:
 , thickness of the glass
 , , , , , ,

∙ ∙
is the effective thickness of the laminated glass beam.
where:
is PVB thickness

69. 64
Young’s module
Shear’s module
Poisson’s module
Bending stiffness of the glass where 1,2
the value of Ψ depends upon the geometry, boundary and loading
conditions of the beam, and it is reported in Table 3 for the most common
cases.
Table 3: Values of coefficient Ψ for laminated glass beams under different boundary
and load condition
In this specific case, the thickness evaluated with this method is:
Method EET
Enhaunced Effective Thickness
L 1diamond 1400 mm
L 2diamond 3740 mm
hint 0.38 mm d1 10.38 mm
h1 20 mm d2 4.38 mm
h2 8 mm dTOT 28.38 mm
D1 666.66 mm3
d 14.76 mm
D2 42.66 mm3
hs,1 10.54 mm
DTOT 1829.33 mm3
hs,2 4.22 mm
E 70000 MPa A1 28000 mm2
ν 0.22 A2 11200 mm2
ψ 0.068 mm‐2
A* 8000 mm2
Gint 0.44 MPa Is 1244.9
η 6.02831E‐08

70. 65
hw 20.42 mm
h1,σ 20.63 mm
h2,σ 32.62 mm
Using this thinkness, two models have been created in the software SAP2000; one with
the deflection effective thickness, the other with the stress effective thickness. The model
is created using the high deviation for the cold bending (5,2 mm).
Figure 65: Model of a cold bent panel
The panel is analysed using the following loads:
 Distributed wind on the shell (wind load is evaluate at the top of the
skyscraper with h = 300 m).
 Tangential wind dir.x, dir.y.
 Crowd loads :
‐ Distributed
‐ Distributed on a line at h = 1.20 m
‐ Concentrated load.
The action force use different coefficients for the analysis of ultimate limit state and
serviceability limit state. The results obtained from the analysis with the action force
used for ultimate limit state is compared with the project resistance. Meanwhile, with the
action force calculated for the serviceability limit state, we compare the displacement.
The resistance at the ultimate limit state using the CNR-DT 210/2012 is calculated as:

71. 66
,
∙ ∙ ∙ ∙ ∙
(38)
Where every coefficient is explained in the chapter 7.4 of the CNR-DT 210/2012.
In the specific case:
Results from the analysis with SAP2000 are:
The displacement is verified with the values shown in the Table 7.12 of the CNR-DT
210/2012 (Figure 65).
Figure 66: Table from CNR-DT 210/2012 for the analysis of displacement
kmod 0.88 k'ed 1
ked 1 kv 0.95
ksf 1 RM;v 0.9
fg;k 45 γM;v 1.35
RM 0.7 λgl 1
γM 2.55 k 0.145
fb;k 150 λgA 0.85
fg;d 100.92 MPa
Stress
Smax Top Smax Bot
h1,σ 26.53 40.26
h2,σ 10.78 16.36
h1,σ 7.47 9.26
h2,σ 4.27 5.74
h1,σ TOT 34.00 49.52 < fg;d 100.9
h2,σ TOT 15.05 22.10 < fg;d 100.9
Stress Dead + Wind Pressure + Tangential Wind Pressure + Crowd loads
Stress cold bending

72. 67
Conclusion One panel stressed with cold bending is verified at the top of the
skyscraper with the thickness supposed.
1/60*Lmin < 30 mm
wmax 23.33 mm w 56 mm
1/200*Lmin < 12 mm
wmax 7.00 mm wwind 0.689 mm
Displacement
Cold Bending

73. 68
3.4.4 Corner modifications of base shapes
Corner modifications of the shapes, as we have already seen in some of the approaches
analysed before, are really relevant to check planarity. This aspect is important also for
the structural part and the analysis of wind loads.
Investigations have established that corner modifications such as slotted corners,
chamfered corners/corner cut, corner recession are in general effective in causing
significant reductions in both the along-wind and across-wind responses compared to
basic building plan shape. [Bor13]
Figure 67: Base shapes [Bor13]
The most common cross-sectional shape modifications are shown for a rectangular
shape, such as slotting, chamfering, rounding corners and corner cutting.
The modification of windward corners is very effective to reduce the drag and fluctuating
lift through changing the characteristics of the separated shear layers to promote their
reattachment and narrow the width of wake.
Experiments have proved that chamfers of the order of 10% of the building width
produce up to 40% reduction in the along-wind response and 30% reduction in the
across-wind response. [Bor13]
After these considerations, it is interesting to evaluate which form is geometrically more
efficient using EVOLUTE Tools.
For the analysis, the three different shapes, already mentioned in Figure 29, have been
evaluated.
Using as an example the same algorithm of panelization (diamonds and triangles) we can
analyse the planarity of three different skyscrapers with the curves illustrate in Figure 29
as base shapes. The results achieve with Evolute Tools show that the shape with round
corners is the best of the three (
Figure 68) . That is, as we already quoted, also the best solution for decrease wind loads.

74. 69
Figure 68: Evaluation of best shape for planarity check

75. 70
3.4.5 Principal curvature directions
To obtain an overview of the principal directions, one can use principal curvature lines.
A principal curvature line is a curve on a surface whose tangents are in principal
direction. Thus, through each general point of a surface there are two principal curvature
lines that intersect at a right angle and touch the principal directions.
Different alghoritms has been created depending on the behaviour of the initial shape.
In particular, the principal curvature line of the shape with round corner has been
analysed because that results the more efficient in many ways. Afterwards, two general
alghoritm have been created in order to find planar panels that follows principal
curvature lines according to convex shapes or shapes with inflection points.
3.4.5.1 Shape with smooth corners
As we mentioned before, there are two principal curvature lines for every point computed
in a surface. The shape with smooth corners presents two different principal directions
depending on which part of the skyscraper we want to analyse: the linear part or the
curve part.
Figure 69: Principal curvature lines in the straight part and in the smooth part
As it is shown in the Figure 69, principal curve directions are completely different. We
can present two different methods:
 Method A
In order to have a unique direction for panels, the directions present in the surface
of the corners have been chosen as principal directions for the whole shape because

76. 71
this is the area with more bending. The other parts can be easily optimized with cold
bending because have low curvature.
The starting mesh could be the following in Figure 70.
Figure 70: Starting mesh for the approximation of principal curvature directions of smooth corners
One way to find planar panels consist in find flat meshes not entirely weld as it is
shown in Figure 71. The verteces have a gap from one to the other.
Figure 71: Planar panels obtained with meshes not weld

77. 72
Welting verteces, panels planarity get enhance, and conditions for cold bending
are satisfied. After the optimizations with Evolute Tools, the results are even more
satisfied.
Figure 72: Grasshopper definition of principal curvature directions of smooth corners
Figure 73: (left) shape non-optimized (right) shape optimized with EVOLUTE Tools
Figure 73 shows on the left the shape before optimization where panels deviation
is between 1.63 and 4.13 . After optimization steps, the Figure 73 shows
panel deviations between 0 and 3.22 .

78. 73
 Method B
Another solution aims to keep going both principal directions. This takes more
planarity but cause problems in the fabrication of the panels in the corners surface
where the principal directions change (Figure 74).
Figure 74: Panelization method B
After this optimization steps, the planarity scale invariant is between 0 and
2.42 , the best solution we could find so far but the intersection knot is not
easy to fabricate (zoom in Figure 75).
Figure 75: (left) zoom of panelization alghoritm with method B (left) zoom of the node with
valence 5

79. 74
3.4.5.2 General B-Spline base with degree 3
The goal of the next analysis aim to find a general algorithm for a generic B-Spline that
follow principal curvature directions. We have to work differently if the shape is convex
or presents inflection points.
Defined 8 control points, different B-Spline curves have been created starting with
convex shapes up to shapes with inflection points. As we can notice with the study
completed so far, if the B-spline has a low convexity or has inflection points, principal
curvature lines converge in the same points and it is impossible to find a good
panelization.
Figure 76: Control points for a B-Spline of degree 3
The algorithm that shows principal directions in every point evaluated in a surface is
created with a script:
Dim p As Point3d = uv
Dim U As Interval = srf.Domain(0)
Dim V As Interval = srf.Domain(1)
Dim samples As New List(Of Point3d)
Do
'Abort if we've added more than 10,000 samples.
If (samples.Count > 10000) Then Exit Do
'Get the curvature at the current point.
Dim crv As SurfaceCurvature = srf.CurvatureAt(p.X, p.Y)
If (crv Is Nothing) Then Exit Do
'Add the current point.
samples.Add(srf.PointAt(p.X, p.Y))
'Get the maximum principal direction.
Dim dir As Vector3d
If (crv.Kappa(0) > crv.Kappa(1)) Then
If (max) Then
dir = crv.Direction(0)
Else

80. 75
dir = crv.Direction(1)
End If
Else
If (max) Then
dir = crv.Direction(1)
Else
dir = crv.Direction(0)
End If
End If
dir.Rotate(angle, crv.Normal)
If (Not dir.IsValid) Then Exit Do
If (Not dir.Unitize()) Then Exit Do
'Scale the direction vector to match our accuracy.
dir *= accuracy
'Flip the direction 180 degrees if it seems to be going backwards
Dim N As Integer = samples.Count
If (N > 1) Then
If (dir.IsParallelTo(samples(N - 1) - samples(N - 2), 0.5 *
Math.PI) < 0) Then
dir.Reverse()
End If
End If
'Move the last point in the list along the curvature direction.
Dim pt As Point3d = samples(samples.Count - 1) + dir
Dim s, t As Double
If (Not srf.ClosestPoint(pt, s, t)) Then Exit Do
'Abort if we've wandered beyond the surface edge.
If (Not U.IncludesParameter(s, True)) Then Exit Do
If (Not V.IncludesParameter(t, True)) Then Exit Do
'Abort if the new point is basically the same as the old point.
If (Math.Abs(p.X - s) < 1e-12) AndAlso (Math.Abs(p.Y - t) < 1e-
Then Exit Do
p.X = s
p.Y = t
Loop
Return samples
End Function.
We select a certain number of points on the surface in order to show principal directions.
Afterwards meshes that follow these directions have been created. As usual, the number
of subdivision is 74 and in the Figure 77-78-79, we can understand how principal
curvature directions change. All the panels obtained are strictly planar.

81. 76
1. Convexity extreme : Circle
Figure 77: (left) Base shape, (right) Panels achieved from principal curvature directions
2. Convexity
Figure 78: (left) Base shape, (right) Panels achieved from principal curvature directions
3. Convexity
Figure 79: (left) Base shape, (right) Panels achieved from principal curvature directions