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<br /> ARCHITECTURAL<br /> GEOMETRY<br /> FirstEdition<br /> Authors<br /> Helmut Pottman<br /> Andreas Asperl<br /> Michael Hofer<br /> Axel Kilian<br /> Editor<br /> Daril Bentley<br /> Formatters<br /> Elisabeth Kasiz-Hitz and Eva Reimer<br /> Exton, Pennsylvania USA<br /> Architectural Geometry<br /> First Edition<br /> Copyright © 2007 Bentley Systems, Incorporated. All Rights Reserved.<br /> Bentley, “B” Bentley logo, Bentley Institute Press, and MicroStation are either registered or unregis-<br /> tered trademarks or servicemarks of Bentley Systems, Incorporated or one of its direct or indirect<br /> wholly-owned subsidiaries. Other brands and product names are trademarks of their respective<br /> owners.<br /> Publisher does not warrant or guarantee any of the products described herein or perform any inde-<br /> pendent analysis in connection with any of the product information contained herein. Publisher<br /> does not assume, and expressly <a title="arch geo free preview.pdf page 1" href="http://viewer.zmags.com/publication/701112e2?page=1"> ARCHITECTURAL GEOMETRY FirstEdition Authors H</a> <a title="arch geo free preview.pdf page 2" href="http://viewer.zmags.com/publication/701112e2?page=2"> Architectural Geometry First Edition Copyright</a> <a title="arch geo free preview.pdf page 3" href="http://viewer.zmags.com/publication/701112e2?page=3"> Preface Geometry lies at the core of the architec</a> <a title="arch geo free preview.pdf page 4" href="http://viewer.zmags.com/publication/701112e2?page=4"> Chapter 15 and leads to a presentation of our own</a> <a title="arch geo free preview.pdf page 5" href="http://viewer.zmags.com/publication/701112e2?page=5"> Discrete concepts appear frequently in this book </a> <a title="arch geo free preview.pdf page 6" href="http://viewer.zmags.com/publication/701112e2?page=6"> Of course, our presentation of geometry is accomp</a> <a title="arch geo free preview.pdf page 7" href="http://viewer.zmags.com/publication/701112e2?page=7"> Content Chapter 1: Creating a Digital 3D Model .</a> <a title="arch geo free preview.pdf page 8" href="http://viewer.zmags.com/publication/701112e2?page=8"> Chapter 7: Cur ves and Surfaces .................</a> <a title="arch geo free preview.pdf page 9" href="http://viewer.zmags.com/publication/701112e2?page=9"> Chapter 14: Visualization and Analysis of Shapes </a> <a title="arch geo free preview.pdf page 10" href="http://viewer.zmags.com/publication/701112e2?page=10"> </a> <a title="arch geo free preview.pdf page 11" href="http://viewer.zmags.com/publication/701112e2?page=11"> Chapter 1 Creating a Digital 3D Model </a> <a title="arch geo free preview.pdf page 12" href="http://viewer.zmags.com/publication/701112e2?page=12"> </a> <a title="arch geo free preview.pdf page 13" href="http://viewer.zmags.com/publication/701112e2?page=13"> Modeling the Winton Guest House We have all seen</a> <a title="arch geo free preview.pdf page 14" href="http://viewer.zmags.com/publication/701112e2?page=14"> Gehry conceived the guest house as a large outdoo</a> <a title="arch geo free preview.pdf page 15" href="http://viewer.zmags.com/publication/701112e2?page=15"> Fig. 1.2 A Cartesian coordinate system with the t</a> <a title="arch geo free preview.pdf page 16" href="http://viewer.zmags.com/publication/701112e2?page=16"> system. If we change the direction of the z-axis,</a> <a title="arch geo free preview.pdf page 17" href="http://viewer.zmags.com/publication/701112e2?page=17"> Surface and solid models. A geometric model with </a> <a title="arch geo free preview.pdf page 18" href="http://viewer.zmags.com/publication/701112e2?page=18"> A related tool is Central Extrusion. Thereby, the</a> <a title="arch geo free preview.pdf page 19" href="http://viewer.zmags.com/publication/701112e2?page=19"> The top part of the living room in the Winton gue</a> <a title="arch geo free preview.pdf page 20" href="http://viewer.zmags.com/publication/701112e2?page=20"> Polar coordinates. In addition to planar Cartesia</a> <a title="arch geo free preview.pdf page 21" href="http://viewer.zmags.com/publication/701112e2?page=21"> Fig. 1.12 Cylindrical coordinates (r,ϕ,z) are pol</a> <a title="arch geo free preview.pdf page 22" href="http://viewer.zmags.com/publication/701112e2?page=22"> Rotational cylinders are fundamental shapes conta</a> <a title="arch geo free preview.pdf page 23" href="http://viewer.zmags.com/publication/701112e2?page=23"> Handles. Geometric objects in a CAD system usuall</a> <a title="arch geo free preview.pdf page 24" href="http://viewer.zmags.com/publication/701112e2?page=24"> Modeling the second bedroom. Now we will model th</a> <a title="arch geo free preview.pdf page 25" href="http://viewer.zmags.com/publication/701112e2?page=25"> Fig. 1.18 (left) Different representations of the</a> <a title="arch geo free preview.pdf page 26" href="http://viewer.zmags.com/publication/701112e2?page=26"> Fig. 1.19 A rendering of the Winton guest house. </a> <a title="arch geo free preview.pdf page 27" href="http://viewer.zmags.com/publication/701112e2?page=27"> Fig. 1.20 A sphere is the bounding surface of a b</a> <a title="arch geo free preview.pdf page 28" href="http://viewer.zmags.com/publication/701112e2?page=28"> Spherical coordinates. In addition to Cartesian a</a> <a title="arch geo free preview.pdf page 29" href="http://viewer.zmags.com/publication/701112e2?page=29"> Fig. 1.22 Geographic coordinates of the Winton gu</a> <a title="arch geo free preview.pdf page 30" href="http://viewer.zmags.com/publication/701112e2?page=30"> The axis of a circle is the line through the cent</a> <a title="arch geo free preview.pdf page 31" href="http://viewer.zmags.com/publication/701112e2?page=31"> Fig. 1.26 The IKMZ (1998–2004) in Cottbus by Herz</a> <a title="arch geo free preview.pdf page 32" href="http://viewer.zmags.com/publication/701112e2?page=32"> </a> <a title="arch geo free preview.pdf page 33" href="http://viewer.zmags.com/publication/701112e2?page=33"> Chapter 2 Projections 23</a> <a title="arch geo free preview.pdf page 34" href="http://viewer.zmags.com/publication/701112e2?page=34"> </a> <a title="arch geo free preview.pdf page 35" href="http://viewer.zmags.com/publication/701112e2?page=35"> Fig. 2.1 A woodcarving by Albrecht Dürer (1471–15</a> <a title="arch geo free preview.pdf page 36" href="http://viewer.zmags.com/publication/701112e2?page=36"> Geometric models of light and shadow are actually</a> <a title="arch geo free preview.pdf page 37" href="http://viewer.zmags.com/publication/701112e2?page=37"> Fig. 2.4 Essential properties of parallel project</a> <a title="arch geo free preview.pdf page 38" href="http://viewer.zmags.com/publication/701112e2?page=38"> Most of the illustrations in this book, made by h</a> <a title="arch geo free preview.pdf page 39" href="http://viewer.zmags.com/publication/701112e2?page=39"> As we have seen, the parallel projection of paral</a> <a title="arch geo free preview.pdf page 40" href="http://viewer.zmags.com/publication/701112e2?page=40"> As a consequence, the parallel projection of a ci</a> <a title="arch geo free preview.pdf page 41" href="http://viewer.zmags.com/publication/701112e2?page=41"> Fig. 2.10 The projection rays of a normal project</a> <a title="arch geo free preview.pdf page 42" href="http://viewer.zmags.com/publication/701112e2?page=42"> All lines and objects in planes perpendicular to </a> <a title="arch geo free preview.pdf page 43" href="http://viewer.zmags.com/publication/701112e2?page=43"> In technical drawings, often the main views are p</a> <a title="arch geo free preview.pdf page 44" href="http://viewer.zmags.com/publication/701112e2?page=44"> Most CAD systems allow the user to arrange the ma</a> <a title="arch geo free preview.pdf page 45" href="http://viewer.zmags.com/publication/701112e2?page=45"> Perspective Projection So far we have dealt with </a> <a title="arch geo free preview.pdf page 46" href="http://viewer.zmags.com/publication/701112e2?page=46"> By abstracting this very practicable idea, we ded</a> <a title="arch geo free preview.pdf page 47" href="http://viewer.zmags.com/publication/701112e2?page=47"> There is one projection ray through the eye point</a> <a title="arch geo free preview.pdf page 48" href="http://viewer.zmags.com/publication/701112e2?page=48"> Figure 2.17 illustrates the proof of this fact. T</a> <a title="arch geo free preview.pdf page 49" href="http://viewer.zmags.com/publication/701112e2?page=49"> The distance between the eye point e and the prin</a> <a title="arch geo free preview.pdf page 50" href="http://viewer.zmags.com/publication/701112e2?page=50"> Example: Constructing a perspective view of a ho</a> <a title="arch geo free preview.pdf page 51" href="http://viewer.zmags.com/publication/701112e2?page=51"> Example: Creation of perspective views </a> <a title="arch geo free preview.pdf page 52" href="http://viewer.zmags.com/publication/701112e2?page=52"> Fig. 2.21 Fig. 2.21 Note that these construction </a> <a title="arch geo free preview.pdf page 53" href="http://viewer.zmags.com/publication/701112e2?page=53"> Fig. 2.23 Renderings of a scene with different vi</a> <a title="arch geo free preview.pdf page 54" href="http://viewer.zmags.com/publication/701112e2?page=54"> Generation of optical illusions. So far we have d</a> <a title="arch geo free preview.pdf page 55" href="http://viewer.zmags.com/publication/701112e2?page=55"> Fig. 2.24 Artistic use of perspective projection </a> <a title="arch geo free preview.pdf page 56" href="http://viewer.zmags.com/publication/701112e2?page=56"> Let us take a cube and a polyhedron (polyhedra ar</a> <a title="arch geo free preview.pdf page 57" href="http://viewer.zmags.com/publication/701112e2?page=57"> P 1 q 1 </a> <a title="arch geo free preview.pdf page 58" href="http://viewer.zmags.com/publication/701112e2?page=58"> Fig. 2.26 Lighting with distant light, point ligh</a> <a title="arch geo free preview.pdf page 59" href="http://viewer.zmags.com/publication/701112e2?page=59"> Fig. 2.27 Spotlight. Light, Shadow, and Renderin</a> <a title="arch geo free preview.pdf page 60" href="http://viewer.zmags.com/publication/701112e2?page=60"> Boundaries of shadows—cast by a single distant li</a> <a title="arch geo free preview.pdf page 61" href="http://viewer.zmags.com/publication/701112e2?page=61"> Fig. 2.29 Linear and area lights are largely used</a> <a title="arch geo free preview.pdf page 62" href="http://viewer.zmags.com/publication/701112e2?page=62"> Rendering methods. To achieve high-quality images</a> <a title="arch geo free preview.pdf page 63" href="http://viewer.zmags.com/publication/701112e2?page=63"> Fig. 2.32 Polyhedral model of a sphere and the no</a> <a title="arch geo free preview.pdf page 64" href="http://viewer.zmags.com/publication/701112e2?page=64"> Computing the color of each point (pixel) in the </a> <a title="arch geo free preview.pdf page 65" href="http://viewer.zmags.com/publication/701112e2?page=65"> Fig. 2.36 Ray tracing: a ray r is tracked back fr</a> <a title="arch geo free preview.pdf page 66" href="http://viewer.zmags.com/publication/701112e2?page=66"> These new arising rays are handled like the origi</a> <a title="arch geo free preview.pdf page 67" href="http://viewer.zmags.com/publication/701112e2?page=67"> Orthogonal and Oblique Axonometric Projections T</a> <a title="arch geo free preview.pdf page 68" href="http://viewer.zmags.com/publication/701112e2?page=68"> coordinate axes through the unit points, we obtai</a> <a title="arch geo free preview.pdf page 69" href="http://viewer.zmags.com/publication/701112e2?page=69"> Assuming that we can select arbitrary distortion </a> <a title="arch geo free preview.pdf page 70" href="http://viewer.zmags.com/publication/701112e2?page=70"> Construction of shadows. Sometimes hand-construct</a> <a title="arch geo free preview.pdf page 71" href="http://viewer.zmags.com/publication/701112e2?page=71"> Sectional views. Occasionally, important parts of</a> <a title="arch geo free preview.pdf page 72" href="http://viewer.zmags.com/publication/701112e2?page=72"> Sketching images of curves and circles. Sketching</a> <a title="arch geo free preview.pdf page 73" href="http://viewer.zmags.com/publication/701112e2?page=73"> Example: Sketching of a cubical clock. The p</a> <a title="arch geo free preview.pdf page 74" href="http://viewer.zmags.com/publication/701112e2?page=74"> Note that in general the images of the points whe</a> <a title="arch geo free preview.pdf page 75" href="http://viewer.zmags.com/publication/701112e2?page=75"> Nonlinear Projections Parallel and perspective pr</a> <a title="arch geo free preview.pdf page 76" href="http://viewer.zmags.com/publication/701112e2?page=76"> Nonlinear projections arise from perspective proj</a> <a title="arch geo free preview.pdf page 77" href="http://viewer.zmags.com/publication/701112e2?page=77"> Spherical projection. Now we exchange the project</a> <a title="arch geo free preview.pdf page 78" href="http://viewer.zmags.com/publication/701112e2?page=78"> It is impossible to unfold a sphere. Thus, we int</a> <a title="arch geo free preview.pdf page 79" href="http://viewer.zmags.com/publication/701112e2?page=79"> Fig. 2.54 (a) Pieter Neffs the Younger used a non</a> <a title="arch geo free preview.pdf page 80" href="http://viewer.zmags.com/publication/701112e2?page=80"> </a>