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Architectural Geometry PDF

750 Pages·2007·45.88 MB·English
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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 Copyright © 2007 Bentley Systems, Incorporated. All Rights Reserved. Bentley, “B” Bentley logo, Bentley Institute Press, and MicroStation are either registered or unregis- tered trademarks or servicemarks of Bentley Systems, Incorporated or one of its direct or indirect wholly-owned subsidiaries. Other brands and product names are trademarks of their respective owners. Publisher does not warrant or guarantee any of the products described herein or perform any inde- pendent analysis in connection with any of the product information contained herein. Publisher does not assume, and expressly disclaims, any obligation to obtain and include information other than that provided to it by the manufacturer. The reader is expressly warned to consider and adopt all safety precautions that might be indicated by the activities herein and to avoid all potential hazards. By following the instructions contained herein, the reader willingly assumes all risks in connection with such instructions. The publisher makes no representation or warranties of any kind, including but not limited to, the warranties of fitness for particular purpose of merchantability, nor are any such representations im- plied with respect to the material set forth herein, and the publisher takes no responsibility with re- spect to such material. The publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or part, from the readers’ use of, or reliance upon, this material. ISBN Number: 978-0-934493-04-5 Library of Congress Control Number: 2007935192 Published by: Bentley Institute Press Bentley Systems, Incorporated 685 Stockton Drive Exton, PA 19341 www.bentley.com www.bentley.com/books 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.

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