Architectural Geometry

750 Pages·2007·45.88 MB·English

by  Helmut Pottmann, Andreas Asperl, Michael Hofer, Axel Kilian

ARCHITECTURAL
GEOMETRY

F i r s t E d i t i o n

Authors
Helmut Pottman
Andreas Asperl
Michael Hofer
Axel Kilian

Editor
Daril Bentley

Formatters
Elisabeth Kasiz-Hitz and Eva Reimer

Exton, Pennsylvania USA

Architectural Geometry

First Edition

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ISBN Number: 978-0-934493-04-5

Library of Congress Control Number: 2007935192

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Printed in the U.S.A.

I

Geometry lies at the core of the architectural design process. It is omnipresent,
from initial form-fi nding stages to actual construction. It also underlies the main
communication medium; namely, graphical representations obtained by precise
geometric rules. Whereas the variety of shapes that could be treated by traditional
geometric methods has been rather limited, modern computing technologies have led
to a real geometry revolution.

Th ese days we are facing dramatic changes, the tools at our disposal becoming
seemingly unlimited. However, the increase in possibilities did not go along with an
increase in the depth of geometry education. In fact, the opposite is true. Th us, it is the
most important task of this book to close the gap between the technical possibilities
and an eff ective working knowledge of the new methods of geometric design.

Modern architecture takes advantage of the greatly increasing design possibilities, yet
architects are not just a new group of CAD users. Scale and construction technologies
pose new challenges to engineering and design. We are convinced that such challenges
can be met more eff ectively with a solid understanding of geometry.

Geometric computing is a broad area with many branches. An interdisciplinary fi eld
such as architecture benefi ts from such variety. We have tried to provide views into this
larger scientifi c context. In fact, we believe that a new research area—which might be
termed architectural geometry—is currently evolving and we hope that our book will
help to promote its future development.

To advance this emerging fi eld, a close cooperation between geometers and architects
is of highest importance. Th is book may be seen as an example. Th e stimulating
cooperation of three geometers and an architect (Axel Kilian) developed the book’s
geometric content, the discussion of its marriage with modern architecture, and the
presentation of both. Moreover, it led to the identifi cation of challenging research
problems—some of which are described in the later chapters of the book.

Intended Use

Our book has been written as a textbook for students of architecture or industrial
design. It comprises material for a basic course in geometry at the undergraduate
level. Roughly starting with parts of Chapter 11, there is plenty of material for a more
advanced geometry course at the undergraduate or graduate level. Th is fi nally leads
us to the cutting edge of research in architectural geometry, which starts to appear in

Preface

II

Chapter 15 and leads to a presentation of our own most recent research on discrete
freeform structures in the fi nal chapter.

Traditionally, the constructive geometry curriculum has been largely based on descriptive
geometry. Computers are now changing geometry education in many ways. Th is book
may also provide a path to constructive geometry education in the digital era beyond the
specifi c application toward architecture.

Th is book is also intended as a geometry consultant for architects, construction engineers,
and industrial designers. Hopefully, scientists interested in geometry processing with
applications in architecture and art may benefi t from it and derive some inspiration. Th ere
is a lot of room for exciting research in architectural geometry. We hope that this book
stimulates research in this challenging and largely unexplored direction.

Prerequisites

Our target audience is generally not well trained mathematically, and we therefore
only assume some basic high school knowledge of mathematics and geometry. In later
chapters, knowledge of basic linear algebra and calculus is of great advantage, but
not absolutely necessary. We have collected some high school geometry in Appendix
A: Geometry Primer, but are aware of the fact that it is both incomplete and a very
subjective collection of some essential background material.

Th roughout the text, some math is collected whose reading is recommended but may
not be necessary to further proceed with the study of the book. Sections marked by an
asterix * are more advanced and may be skipped in a fi rst reading.

Features

Th is book is intended to fi ll a gap in the geometry literature. Geometry books written
for architecture mainly discuss elementary material or classical descriptive geometry.
We explain concepts of descriptive geometry only very briefl y. Instead, we focus on
effi cient CAD construction methods and use CAD to support geometry teaching and
understanding.

III

Most books about the geometric aspects of CAD require a reasonably high level of
math knowledge, which most students of architecture will not have. Th erefore, we
have tried to explain new geometries available in CAD systems (such as freeform
curves and surfaces) without much math. We explain the material via geometric
considerations and support those with numerous fi gures.

Wherever possible, we replace the use of calculus with discrete models and then obtain
properties of smooth analogues by a limiting process. For example, when studying
curves we fi rst investigate the geometry of polygons. Th e transition to curves is
performed by a refi nement process that generates in the limit a smooth curve. Th is
is not a new approach, but it receives increasing interest due to the fact that many
computations are actually based on discrete representations.

Fortunately, the math used in the limiting process can be omitted without a signifi cant loss
of insight. As pointed out by the modern fi eld of discrete diff erential geometry, this approach
can actually be benefi cial in showing which elementary geometric facts serve as kernels of
certain geometric results about curves or surfaces. Calculus may hide such an insight.

Th ere are many threads from basics to research. An example is depicted in the following
fi gure: polyhedra and polyhedral surfaces (Chapter 3) → subdivision curves (Chapter 8)
→ subdivision surfaces and meshes (Chapter 11) → planar quad strips as models of
developable surfaces (Chapter 15) → discrete freeform structures (Chapter 19).

Chapter 3
Chapter 8
Chapter 11
Discrete concepts appear frequently in
this book because they are easier to
grasp than methods from mathematical
analysis. They are even more powerful
than mathematical methods, especially
in view of applications in architecture.
This illustration shows an example of a
thread of discrete ideas from basics to
research.

Chapter 15
Chapter 19

IV

Of course, our presentation of geometry is accompanied by examples of built architecture,
architectural projects, and artwork. However, the selections are for purposes of example, are
very subjective, and are not the main intent of the book. We even included some interesting
and hopefully inspiring geometric insights that may not yet have been used in architecture
but may carry some potential of being useful in the future.

Teaching Support and Feedback

A web site located at www.architecturalgeometry.at will serve as a discussion forum for
anyone interested in architectural geometry. Your feedback will help us tailor our work even
more toward the actual needs in geometry teaching and architectural design. Th e web page
currently contains some teaching material, exercises and solutions to lab sessions, and results
of student projects.We hope that this material will see a signifi cant growth in the near future.

Acknowledgments

Th is book is the result of a large eff ort by many people to whom we would like to
express our deepest gratitude. Scott Lofgren and Jeff Kelly of Bentley have supported
us in the best possible way during the entire process, and our
r Daril Bentley did
a great job in smoothing our texts.

We are very grateful for Bentley’s fi nancial support, which has been used as an
additional motivation for the students involved in the design of fi gures. In fact, the
work on the fi gures has been the most time-consuming part of the entire project.

Th e research performed in connection with the book has been supported by grants
S9201 and P19214-N18 from the Austrian Science Fund FWF. We would like
to thank the members of the research group Geometric Modeling and Industrial
Geometry at the Vienna University of Technology, who have helped in proofreading:
Simon Flöry, Martin Peternell, Niloy Mitra, and Peter Paukowitsch.

Special thanks go to the many people who helped us with the fi gures: Miriam Zotter,
Christian Leeb, Markus Forstner, Heinz Schmiedhofer, Benjamin Schneider, Boris
Odehnal, Philip Grohs, and Michael Wischounig. Martin Reis did a great job of
fi nalizing the fi gures, achieving color harmony and helping in the layout process. We
enjoyed working with our graphical designers Elisabeth Kaziz-Hitz and Eva Riemer
and admire the result of their work.

Many friends from the scientifi c community have provided results of their work to
illustrate the state of the art in geometric computing and we want to thank them
for their help. We are very grateful to Georg Franck for his continuing support of
geometry in the architectural curriculum at the Vienna University of Technology
and Johannes Wallner, whose deep geometric insight and unbelievable working speed
provided great help in the critical fi nal phase of this project.

Last but not least, our sincerest thanks belong to our families—who supported us with
their love and understanding throughout the entire book project. Our families know
best how much time it took us to write and illustrate this book on architectural geometry.

Edito

V

Chapter 1: Creating a Digital 3D Model ............................................................................1
Modeling the Winton Guest House ........................................................................................3
Spheres, Spherical Coordinates, and Extrusion Surfaces .................................................. 17

Chapter 2: Projections............................................................................................................ 23
Projections .................................................................................................................................. 25
Perspective Projection .............................................................................................................. 35
Light, Shadow, and Rendering ................................................................................................ 49
Orthogonal and Oblique Axonometric Projections ......................................................... 57
Nonlinear Projections .............................................................................................................. 65

Chapter 3: Polyhedra and Polyhedral Surfaces ............................................................. 71
Polyhedra and Polyhedral Surfaces ....................................................................................... 73
Pyramids and Prisms ................................................................................................................ 75
Platonic Solids ........................................................................................................................... 79
Properties of Platonic Solids ................................................................................................... 85
Th e Golden Section .................................................................................................................. 87
Archimedean Solids ................................................................................................................. 91
Geodesic Spheres ...................................................................................................................... 95
Space Filling Polyhedra ..........................................................................................................101
Polyhedral Surfaces .................................................................................................................103

Chapter 4: Boolean Operations ........................................................................................109
Boolean Operations ................................................................................................................111
Union, Diff erence, and Intersection ...................................................................................113
Trim and Split ..........................................................................................................................117
Feature-Based Modeling: An Effi cient Approach to Shape Variations .......................125

Chapter 5: Planar Transformations .................................................................................137
Planar Transformations ..........................................................................................................139
Translation, Rotation, and Refl ection in the Plane ..........................................................141
Scaling and Shear Transformation ......................................................................................149
Tilings .......................................................................................................................................151
*Nonlinear Transformations in 2D .....................................................................................159

Chapter 6: Spatial Transformations ................................................................................167
Spatial Transformations .........................................................................................................169
Translation, Rotation, and Refl ection in Space ................................................................171
Helical Transformation ..........................................................................................................181
Smooth Motions and Animation ........................................................................................189
Affi ne Transformations ..........................................................................................................195
*Projective Transformations .................................................................................................201

Content

VI

Chapter 7: Curves and Surfaces ........................................................................................211
Curves and Surfaces ................................................................................................................213
Curves .......................................................................................................................................217
Conic Sections .........................................................................................................................231
Surfaces ......................................................................................................................................237
Intersection Curves of Surfaces ............................................................................................245

Chapter 8: Freeform Curves ..............................................................................................253
Freeform Curves ......................................................................................................................255
Bézier Curves ...........................................................................................................................259
B-Spline Curves ......................................................................................................................269
NURBS Curves .......................................................................................................................275
Subdivision Curves .................................................................................................................279

Chapter 9: Traditional Surface Classes ..........................................................................285
Traditional Surface Classes ...................................................................................................287
Rotational Surfaces .................................................................................................................289
Translational Surfaces ............................................................................................................305
Ruled Surfaces .........................................................................................................................311
Helical Surfaces .......................................................................................................................323
Pipe Surfaces ............................................................................................................................329

Chapter 10: Off sets ..............................................................................................................331
Off sets .......................................................................................................................................333
Off set Curves ...........................................................................................................................335
Off set Surfaces .........................................................................................................................341
Trimming of Off sets ...............................................................................................................347
Application of Off sets ............................................................................................................351

Chapter 11: Freeform Surfaces .........................................................................................359
Freeform Surfaces ....................................................................................................................361
Bézier Surfaces .........................................................................................................................365
B-Spline Surfaces and NURBS Surfaces ............................................................................377
Meshes .......................................................................................................................................381
Subdivision Surfaces ...............................................................................................................397

Chapter 12: Motions, Sweeping, and Shape Evolution .............................................411
Motions, Sweeping, and Shape Evolution .........................................................................413
Motions in the Plane ..............................................................................................................415
Spatial Motions .......................................................................................................................423
Sweeping and Skinning ..........................................................................................................431
Curve Evolution ......................................................................................................................437
Metaballs and Modeling with Implicit Surfaces ...............................................................441

Chapter 13: Deformations .................................................................................................449
Deformations ...........................................................................................................................451
Th ree-Dimensional Transformations ..................................................................................453
Twisting .....................................................................................................................................455
Tapering ....................................................................................................................................459
Shear Deformations ................................................................................................................463
Bending .....................................................................................................................................467
Freeform Deformations .........................................................................................................469
Inversions ..................................................................................................................................475
Th ree-Dimensional Textures ................................................................................................479

VII

Chapter 14: Visualization and Analysis of Shapes ......................................................483
Visualization and Analysis of Shapes ..................................................................................485
Curvature of Surfaces .............................................................................................................487
Optical Lines for Quality Control ......................................................................................503
Texture Mapping .....................................................................................................................509
Digital Elevation Models .......................................................................................................513
Geometric Topology and Knots ..........................................................................................517

Chapter 15: Developable Surfaces and Unfoldings ....................................................531
Developable Surfaces and Unfoldings ................................................................................533
Surfaces Th at Can Be Built from Paper ..............................................................................535
Unfolding a Polyhedron ........................................................................................................561
References and Further Reading ..........................................................................................565

Chapter 16: Digital Prototyping and Fabrication ......................................................567
Model Making and Architecture .........................................................................................569
Fabrication Techniques ..........................................................................................................579
Cutting-Based Processes ........................................................................................................581
Additive Processes: Layered Fabrication ............................................................................583
Subtractive Techniques ..........................................................................................................587
Geometric Challenges Related to Machining and Rapid Prototyping .......................591
Assembly ...................................................................................................................................595

Chapter 17: Geometry for Digital Reconstruction ....................................................599
Geometry for Digital Reconstruction ................................................................................601
Data Acquisition and Registration ......................................................................................607
Th e Polygon Phase ..................................................................................................................615
Segmentation ...........................................................................................................................625
Surface Fitting ..........................................................................................................................629
Th e Surfaces Need to Be Built ..............................................................................................633
References and Further Reading ..........................................................................................637

Chapter 18: Shape Optimization Problems ..................................................................639
Shape Optimization Problems .............................................................................................641
Remarks on Mathematical Optimization ..........................................................................643
Geometric Optimization ......................................................................................................647
Functional Optimization ......................................................................................................665
References and Further Reading ..........................................................................................667

Chapter 19: Discrete Freeform Structures ....................................................................669
Discrete Freeform Structures ................................................................................................671
Triangle Meshes .......................................................................................................................675
Quadrilateral Meshes with Planar Faces ............................................................................677
Parallel Meshes, Off sets, and Supporting Beam Layout .................................................687
Off set Meshes ..........................................................................................................................691
Optimal Discrete Surfaces ....................................................................................................701
Future Research .......................................................................................................................707
References and Further Reading ..........................................................................................709

Appendix A: Geometry Primer ...........................................................................................711
List of Symbols ........................................................................................................................720
Index ..........................................................................................................................................721

Chapter 1
Creating a Digital
3D Model

We have all seen digital architectural models of great complexity in various forms of
visualization. But how do we get started? How do we communicate our ideas with
the help of a computer? What are the geometric fundamentals that enable us to create
digital three-dimensional (3D) models? Many tools and procedures are provided to
us by modern computer-aided design (CAD) systems for creating such models. To
effi ciently employ the existing soft ware—and to go beyond—a good knowledge of
geometry is essential.

It goes without saying that the architect’s design work starts before geometric mod-
eling. According to Frank O. Gehry, his inspiration for the Winton guest house in
Wayzata, Minnesota, came from the still-life paintings by Giorgio Morandi. When
asked to build a guest house for a client in the 1980s, he set a counterpoint to the main
house—designed by Philip Johnson in 1952.

Modeling the
Winton Guest House

4

Gehry conceived the guest house as a large outdoor sculpture in which each room
would constitute its own mini-building (Figure 1.1). Based on sketches, scaled physical
3D models and plan drawings were created manually. In this chapter we learn how to
create a digital 3D model of this structure.

Fig. 1.1
The Winton guest house by Frank O.
Gehry. Sketches (top), scaled physical
models (bottom left), photo of the
building (bottom right).

5

Cartesian coordinates. Geometric objects can be described as a collection of points
that delineate the shape of the object. To describe the position of a point p in 3D
space, we use an ordered triplet of numbers called coordinates. Th ese coordinates are
measured with respect to a chosen coordinate system. A Cartesian coordinate system
(Figure 1.2) is given by three mutually perpendicular oriented axes called the x-, y-,
and z-axis.

Th e three axes are labeled x, y, and z and pass through a common point o called the
origin. On each coordinate axis we use the same unit length. With respect to a chosen
system, a point p in 3D space has the three Cartesian coordinates (xp,yp,zp). Th ey are
called the x-coordinate xp, y-coordinate yp, and z-coordinate zp. Th e positive coordinates
always lie on the ray starting at the origin and pointing in axis direction.

To get from the origin o with coordinates (0,0,0) to a point p with coordinates
(xp,yp,zp) there are six diff erent coordinate paths—all of which lie on a coordinate
cuboid of length xp, width yp, and height zp. Th e eight vertices of the coordinate cuboid
have respectively the coordinates (0,0,0), (xp,0,0), (0,yp,0), (0,0,zp), (xp,yp,0), (xp,0,zp),
(0,yp,zp),and (xp,yp,zp). Each pair of coordinate axes spans a plane called a coordinate
plane. We have the xy-plane, the yz-plane, and the zx-plane. Note that in each coordi-
nate plane we have a 2D Cartesian coordinate system given by the two coordinate axes
that span the respective plane.

Right- and left -handed coordinate system. Let us use a Cartesian coordinate system
(Figure 1.3). When we look along negative z-direction into the xy-plane, a counter-
clockwise turn of 90 degrees aligns the x-axis with the y-axis. Such a right-handed
Cartesian coordinate system can easily be visualized with the fi rst three fi ngers of the
right hand.

Starting with the right hand as a fi st, we open the thumb in the direction of the x-axis
and the forefi nger in direction of the y-axis. Th en we can open the middle fi nger such
that it points in the direction of the z-axis of a right-handed Cartesian coordinate

Fig. 1.2
A Cartesian coordinate system with the
three coordinates (xp,yp,zp) of a point
p in 3D space. A coordinate path that
connects the origin o to the point p lies
on a coordinate cuboid with length xp,
width yp, and height zp.

Fig. 1.3
Right-handed Cartesian coordinate
system.