Architectural geometry

Geometry lies at the core of the architectural design process. It is omnipresent, from the initial form-finding stages to the actual construction. Modern constructive geometry provides a variety of tools for the efficient design, analysis, and manufacture of complex shapes. This results in new challenges for architecture. However, the architectural application also poses new problems to geometry. Architectural geometry is therefore an entire research area, currently emerging at the border between applied geometry and architecture. This book has been written as a textbook for students of architecture or industrial design. It comprises material at all levels, from the basics of geometric modeling to the cutting edge of research. During the architectural journey through geometry, topics typically reserved for a mathematically well-trained audience are addressed in an easily understandable way. These include central concepts on freeform curves and surfaces, differential geometry, kinematic geometry, mesh processing, digital reconstruction, and optimization of shapes. This book is also intended as a geometry consultant for architects, construction engineers, and industrial designers and as a source of inspiration for scientists interested in applications of geometry processing in architecture and art. Front Matter 2 Preface 4 Table of Contents 8 1. Creating a Digital 3D Model 11 1.1 Modeling the Winton Guest House 13 1.1.1 Cartesian Coordinates 15 1.1.2 Right- and Left-Handed Coordinate System 15 1.1.3 Cuboids 16 1.1.4 Surface and Solid Models 17 1.1.5 Extrusion 17 1.1.6 Global and Local Coordinate Systems 18 1.1.7 Polar Coordinates 20 1.1.8 Cylindrical Coordinates 21 1.1.9 Rotational Cylinder 21 1.1.10 Snapping 22 1.1.11 Handles 23 1.1.12 Modeling the Second Bedroom 24 1.1.13 Layers 25 1.1.14 Color, Texture, and Material 25 1.2 Spheres, Spherical Coordinates and Extrusion Surfaces 27 1.2.1 Sphere 27 1.2.2 Spherical Coordinates 28 1.2.3 Geographic Coordinate System 29 1.2.4 Extrusion Revisited: Cylinder and Cone Surfaces 29 1.2.5 Outlook 31 2. Projections 33 2.1 Projections 35 2.1.1 Projections 36 2.1.2 Parallel Projection 37 2.1.3 Normal Projection 41 2.2 Perspective Projection 45 2.2.1 Example: Constructing a Perspective View of a House by Hand 50 2.2.2 Example: Creation of Perspective Views Using Vanishing Points 51 2.2.3 Hints for Creating Realistic Images 53 2.2.4 Generation of Optical Illusions 54 2.2.5 History 54 2.3 Light, Shadow, and Rendering 59 2.3.1 Light Sources 59 2.3.2 Rendering Methods 62 2.4 Orthogonal and Oblique Axonometric Projections 67 2.4.1 Example: Parallel Projection of a Canopy 68 2.4.2 Visibility of Objects 69 2.4.3 Construction of Shadows 70 2.4.4 Sectional Views 71 2.4.5 Example: Constructing Shadows by Hand 71 2.4.6 Sketching Images of Curves and Circles 72 2.4.7 Sketching Spheres 72 2.4.8 Example: Sketching of a Cubical Clock 73 2.5 Nonlinear Projections 75 2.5.1 Cylindrical Projection 76 2.5.2 Spherical Projection 77 2.5.3 Creating a Planar Image 77 3. Polyhedra and Polyhedral Surfaces 81 3.1 Polyhedra and Polyhedral Surfaces 83 3.1.1 Polyhedra and Polyhedral Surfaces 84 3.2 Pyramids and Prisms 85 3.2.1 Prisms 86 3.2.2 Example: Modeling Pyramids and Prisms Using Extrusion 86 3.3 Platonic Solids 89 3.3.1 Math: Convex Sets 89 3.3.2 Paper Models of the Platonic Solids 90 3.3.3 How to Find the Five Platonic Solids? 91 3.3.4 History: Platonic Solids in Higher Dimensions 93 3.3.5 Convex Polyhedra Whose Faces are Equilateral Triangles 94 3.4 Properties of Platonic Solids 95 3.4.1 The Euler Formula 95 3.4.2 Platonic Solids and Their Duals 96 3.4.3 Spheres Associated with Platonic Solids 96 3.4.4 Symmetry Properties 96 3.5 The Golden Section 97 3.5.1 The Golden Section 97 3.5.2 The Golden Rectangle 97 3.5.3 The Fibonacci Spiral 97 3.5.4 History: The Golden Section in Art and Architecture 98 3.5.5 Modeling the Platonic Solids 99 3.6 Archimedean Solids 101 3.6.1 Corner Cutting of Platonic Solids 101 3.6.2 Corner Cuts of Type 1 102 3.6.3 Corner Cuts of Type 2 103 3.6.4 Remark 104 3.6.5 Example: Prisms and Anti-Prisms That are Archimedean Solids 104 3.7 Geodesic Spheres 105 3.7.1 History 105 3.7.2 Geodesic Domes Derived from an Icosahedron: Alternative 1 107 3.7.3 Geodesic Domes Derived from an Icosahedron: Alternative 2 108 3.7.4 Remark 109 3.8 Space Filling Polyhedra 111 3.9 Polyhedral Surfaces 113 3.9.1 Approximation of Cylinders and Cones by Polyhedral Surfaces 113 3.9.2 Example: A Model of the Courtyard Roof of the Abbey in Neumunster 114 4. Boolean Operations 119 4.1 Boolean Operations 121 4.2 Union, Difference, and Intersection 123 4.2.1 History 125 4.3 Trim and Split 127 4.3.1 Orthogonal Projection onto a Surface 131 4.3.2 Example: Punched Rotational Surfaces 132 4.4 Feature-Based Modeling: An Efficient Approach to Shape Variations 135 4.4.1 Example: Hole Feature 138 4.4.2 Example: Handrail 141 4.4.3 Blending Surfaces 144 4.4.4 Example: Variations of a Simple Dome Model 145 5. Planar Transformations 147 5.1 Planar Transformations 149 5.2 Translation, Rotation, and Reflection in the Plane 151 5.2.1 Translation 153 5.2.2 Rotation 153 5.2.3 Reflection 154 5.2.4 Math 154 5.2.5 Glide Reflection 155 5.2.6 Composition of Congruence Transformations 156 5.3 Scaling and Shear Transformation 159 5.3.1 Uniform Scaling 159 5.3.2 Shear Transformation 160 5.4 Tilings 161 5.4.1 Regular and Semi-Regular Tessellations 162 5.4.2 Tessellations and Congruence Transformations 165 5.4.3 How to Design Nontrivial Tiles 166 5.5 Nonlinear Transformations in 2D 169 5.5.1 Reflection along a Circle: Inversion 169 5.5.2 Planar Transformations and Complex Numbers 172 5.5.3 Möbius Transformations 174 5.5.4 Conformal Mappings 175 6. Spatial Transformations 177 6.1 Spatial Transformations 179 6.2 Translation, Rotation, and Reflection in Space 181 6.2.1 Translation 182 6.2.2 Rotation 182 6.2.3 Example: Reconstruction of an Arbitrary Rotational Axis 184 6.2.4 Reflections 184 6.2.5 Glide Reflection 186 6.2.6 Example: Dodecahedron with Rhombic Faces 187 6.2.7 Composition of Transformations 188 6.3 Helical Transformation 191 6.3.1 Example: Repositioning of an Object 192 6.3.2 Example: Reconstruction of the Helical Axis 193 6.3.3 Helical Motion 194 6.3.4 Mathematical Description 196 6.4 Smooth Motions and Animation 199 6.4.1 Working with Key Frames 200 6.4.2 Animations with Paths 201 6.4.3 Animation Scripts 202 6.4.4 Example: Motion of an Advertising Cube 203 6.5 Affine Transformations 205 6.5.1 Scaling 206 6.5.2 Shear Transformation 208 6.5.3 Spiral Transformation 209 6.6 Projective Transformations 211 6.6.1 Projective Extension of the Plane 212 6.6.2 Homogeneous Coordinates 213 6.6.3 Example: Homogeneous Coordinates of Points 214 6.6.4 Planar Projective Transformations 214 6.6.5 Example: A General Projective Transformation 216 6.6.6 Projective Images of Circles 217 6.6.7 Conic Sections 218 6.6.8 Homogeneous Coordinates and Projective Transformations in Three Dimensions 218 6.6.9 Example: Relief Perspective 219 7. Curves and Surfaces 221 7.1 Curves and Surfaces 223 7.2 Curves 227 7.2.1 Parametric Representation 228 7.2.2 Example: Circle in Parametric Representation 229 7.2.3 Example: Parabola and Spatial Cubic Curve 230 7.2.4 Explicit Representation: Graphs 231 7.2.5 Example: Rational Cubic Curve 231 7.2.6 Implicit Representation 232 7.2.7 Example: The Circle in Implicit Representation 232 7.2.8 Example: Implicit and Parametric Representation of a Circle 233 7.2.9 Curve Tangent 234 7.2.10 Example: Tangents of a Helix 235 7.2.11 Example: Singular Points 235 7.2.12 Discrete Curves 236 7.2.13 Osculating Plane and Osculating Circle 236 7.2.14 Example: Computation of the Curvature 237 7.2.15 Inflection Point and Vertex 238 7.2.16 Evolute 238 7.2.17 Frenet Frame of a Space Curve 239 7.2.18 Example: Frenet Frame and Orthogonal Projections 240 7.3 Conic Sections 241 7.3.1 Implicit Representation 241 7.3.2 Ellipse 241 7.3.3 Hyperbola 244 7.3.4 Parabola 245 7.3.5 The Thread Construction of a Parabola 246 7.4 Surfaces 247 7.4.1 Parametric Representation 247 7.4.2 Example: Parametric Representation of a Sphere 249 7.4.3 Example: Cylinder 250 7.4.4 Example: Tangent Surface 250 7.4.5 Explicit and Implicit Representation 251 7.4.6 Tangent Plane and Surface Normal 251 7.4.7 Example: Whitney Umbrella 252 7.4.8 Example: Surface with Many Singularities 252 7.4.9 Contour and Apparent Contour 253 7.5 Intersection Curves of Surfaces 255 7.5.1 Constructing Points via Auxiliary Planes 255 7.5.2 Use of Auxiliary Spheres 256 7.5.3 Tangents of Intersection Curves 257 7.5.4 Conic Sections as Intersection Curves 258 7.5.5 Example: Rotational Cylinders with Intersecting Axes and Equal Radius 259 7.5.6 Space Curves as Intersection Curves of General Cylinders 260 8. Freeform Curves 263 8.1 Freeform Curves 265 8.1.1 How Do We Design Freeform Curves? 266 8.1.2 Interpolation 267 8.1.3 Approximation 267 8.2 Bézier Curves 269 8.2.1 History: Invention of Bézier Curves 269 8.2.2 Algorithm of de Casteljau 270 8.2.3 Remark 270 8.2.4 Tangents of Bézier Curves 271 8.2.5 Math: Calculation of the Curve Tangent 271 8.2.6 The Meaning of the Four Control Points 272 8.2.7 Subdivision of Bézier Curves 272 8.2.8 Loops and Cusps 272 8.2.9 Convex Hull Property of Bézier Curves 274 8.2.10 Math: Convex Hull 274 8.2.11 Parabolas are Quadratic Bézier Curves 275 8.2.12 Math: Affine Invariance 275 8.2.13 Example: Design a Parabolic Arc with Given Axis Direction and Vertex Position 276 8.2.14 Limitations of Bézier Curves 277 8.2.15 Example: Piecewise Bézier Curves 278 8.3 B-Spline Curves 279 8.3.1 History: Spline 279 8.3.2 B-Spline 279 8.3.3 Defining a B-Spline Curve 280 8.3.4 Example: The Influence of the Degree on a B-Spline Curve 281 8.3.5 Example: Sketching Quadratic Spline Curves for Given Control Polygons 281 8.3.6 Local Control of B-Spline Curves 282 8.3.7 Open and Closed B-Spline Curves 282 8.3.8 Example: Bézier Control Points of Cubic B-Spline Curves 283 8.3.9 Why are Cubic B-Spline Curves So Popular? 284 8.3.10 Why Do We Still Want More? 284 8.4 NURBS Curves 285 8.4.1 Geometric Derivation of NURBS Curves 285 8.4.2 Weights 286 8.4.3 Remark 286 8.4.4 B-Spline Curves are Special NURBS Curves 286 8.4.5 Design Handles 287 8.4.6 Conic Sections as Special NURBS Curves 288 8.5 Subdivision Curves 289 8.5.1 What is a Subdivision Curve? 290 8.5.2 History: Corner Cutting 290 8.5.3 Chaikin's Algorithm 291 8.5.4 Lane-Riesenfeld Algorithm 292 8.5.5 The Four-Point Scheme 293 9. Traditional Surface Classes 295 9.1 Traditional Surface Classes 297 9.2 Rotational Surfaces 299 9.2.1 Mathematical Description 302 9.2.2 Discrete Rotational Surfaces 302 9.2.3 Special Rotational Surfaces 303 9.2.4 Example: Parametric Representation of a Torus 305 9.2.5 Example: Villarceau's Circles of a Torus 305 9.2.6 Rotational Quadrics 309 9.2.7 History 309 9.2.8 Intersection Curves of Quadrics 313 9.3 Translational Surfaces 315 9.3.1 Special Translational Surfaces 317 9.3.2 Example: A Rotational Paraboloid Generated as Translational Surface 318 9.3.3 Elliptic Paraboloids 318 9.3.4 Hyperbolic Paraboloid 319 9.4 Ruled Surfaces 321 9.4.1 Ruled Surfaces by Moving a Straight Line along a Directrix Curve 322 9.4.2 Example: Conoid 323 9.4.3 Example: Möbius Strip 324 9.4.4 Ruled Surfaces by Connecting Corresponding Points of Two Generating Curves 325 9.4.5 HP Surfaces 326 9.4.6 Conoids 329 9.4.7 Example: Plücker's Conoid 330 9.4.8 Tangent Planes of Ruled Surfaces 331 9.5 Helical Surfaces 333 9.5.1 Mathematical Description 334 9.5.2 Special Helical Surfaces 335 9.5.3 Example: Intersection of a Helicoid and a Special Cylinder 336 9.6 Pipe Surfaces 339 10. Offsets 341 10.1 Offsets 343 10.2 Offset Curves 345 10.2.1 Offsets of Smooth Planar Curves 345 10.2.2 Math: Computing Offset Curves 345 10.2.3 Example: Offsets of a Circle and an Ellipse 346 10.2.4 Curve, Offset, and Evolute 347 10.2.5 Example: Cusps on an Ellipse Offset 347 10.2.6 Example: Silhouette of a Torus under Normal Projection 348 10.2.7 Offsets of Planar Polygons 349 10.3 Offset Surfaces 351 10.3.1 Math: Computing an Offset Surface 351 10.3.2 Example: Offsets of Spheres and Rotational Surfaces 352 10.3.3 Example: Offsets of Cylinder Surfaces 354 10.3.4 Example: Offsets of Cone Surfaces 355 10.3.5 Example: Offsets of Hyperbolic Paraboloids 356 10.3.6 Offsets of Polyhedral Surfaces 356 10.4 Trimming of Offsets 357 10.4.1 Trimming Offsets in Three Dimensions 358 10.4.2 Discrete Offsets of Planar Polygons 359 10.4.3 Discrete Offsets of Polyhedral Surfaces 360 10.5 Application of Offsets 361 10.5.1 Rolling Ball Blends 361 10.5.2 Geometric Roof Design 362 10.5.3 Designing Roofs of Constant Slope Using Offsets 364 10.5.4 Designing Roofs of Varying Slope 367 11. Freeform Surfaces 369 11.1 Freeform Surfaces 371 11.1.1 History of Freeform Surfaces in Architecture 372 11.1.2 History of Freeform Surfaces in CAGD 374 11.2 Bézier Surfaces 375 11.2.1 Translational Bézier Surfaces 375 11.2.2 General Bézier Surfaces 378 11.2.3 Properties of Bézier Surfaces 379 11.2.4 Bézier Surfaces of Degree (1,1) 379 11.2.5 Bézier Surfaces That are also Ruled Surfaces 380 11.2.6 Example: Surfaces from Parabolas with Vertical Axes 382 11.2.7 Bézier Surfaces Joined Smoothly 384 11.2.8 Example: Smooth Bézier Junction between a Parabolic Cylinder and a Plane 386 11.3 B-Spline Surfaces and NURBS Surfaces 387 11.3.1 Open and Closed Mode 389 11.3.2 Interpolating Spline Surfaces 390 11.4 Meshes 391 11.4.1 History of Meshes in Architecture 392 11.4.2 Geometry and Connectivity 394 11.4.3 Quadrilateral Meshes 397 11.4.4 Triangle Meshes -1 11.4.5 Hexagonal Meshes 399 11.4.6 Mesh Refinement 400 11.4.7 Mesh Decimation 403 11.4.8 Bad Meshes 403 11.4.9 Aesthetics of Meshes and Relaxation 404 11.5 Subdivision Surfaces 407 11.5.1 Motivation 407 11.5.2 History of Subdivision Surfaces 408 11.5.3 Quadratic B-Spline Surfaces via Subdivision 409 11.5.4 Doo-Sabin Subdivision Scheme 411 11.5.5 From Cubic B-Splines to Catmull-Clark Subdivision 412 11.5.6 Skew Quad Warning 415 11.5.7 Triangle-Based Subdivision 416 11.5.8 Multi-Resolution Modeling 418 12. Motions, Sweeping, and Shape Evolution 421 12.1 Motions, Sweeping, and Shape Evolution 423 12.2 Motions in the Plane 425 12.2.1 Example: Four-Bar Linkage 426 12.2.2 Example: Cardan Motion 429 12.2.3 Example: Cusps and Loops 430 12.2.4 Swept Areas and Envelopes 431 12.2.5 Example: Motion of the Frenet Frame of a Planar Curve 432 12.3 Spatial Motions 433 12.3.1 Extending Planar Motions to Three Dimensions 433 12.3.2 Example: A Ruled Surface with Ellipses as Cross Sections 434 12.3.3 Sweeping along a Planar Path and Moulding Surfaces 435 12.3.4 Developable Surfaces of Constant Slope 438 12.3.5 General Spatial Motions 439 12.3.6 Example: Superposing Uniform Rotations 440 12.4 Sweeping and Skinning 441 12.4.1 Sweeping a Profile along a Curved Path 441 12.4.2 Rotation Minimizing Frame and Quadrilateral Meshes with Planar Faces 442 12.4.3 Sweeping with Several Paths and Sources 443 12.4.4 Skinning Surfaces 444 12.4.5 Example: Skinning Surfaces in Art and Architecture 445 12.5 Curve Evolution 447 12.5.1 Constant Speed 447 12.5.2 Curvature Flow 448 12.5.3 A Simple Polygon Evolution 448 12.5.4 Example: Surface Design via a Curve Evolution 450 12.6 Metaballs and Modeling with Implicit Surfaces 451 12.6.1 Curves 451 12.6.2 Example: Isolines for Scientific Visualization 452 12.6.3 Example: Cassini Curves 453 12.6.4 Implicit Representation of Surfaces 454 12.6.5 Example: Rotational Surfaces with Cassini Profiles 454 12.6.6 Distance-Based Functions 455 12.6.7 Meta-Balls 456 13. Deformations 459 13.1 Deformations 461 13.2 Three-Dimensional Transformations 463 13.2.1 Math: Description of Transformations in Three Dimensions 463 13.2.2 Slice-Based Three-Dimensional Transformations 464 13.3 Twisting 465 13.4 Tapering 469 13.4.1 Math 469 13.4.2 Bulge 472 13.5 Shear Deformations 473 13.5.1 Math 474 13.6 Bending 477 13.7 Freeform Deformations 479 13.7.1 Planar Bézier Deformations of Degree (1,1) 479 13.7.2 Bézier Deformations of Degree (1,1,1) 481 13.7.3 Math 482 13.8 Inversions 485 13.8.1 Remarks on Envelopes of Spheres 488 13.9 Three-Dimensional Textures 489 13.9.1 Remark 490 14. Visualization and Analysis of Shapes 493 14.1 Visualization and Analysis of Shapes 495 14.2 Curvature of Surfaces 497 14.2.1 Curves Revisited 497 14.2.2 Example: Curvatures of a Sine Curve 498 14.2.3 The Osculating Paraboloid 499 14.2.4 Normal Curvatures 500 14.2.5 Classification of Surface Points 501 14.2.6 Example: Surfaces of Revolution 503 14.2.7 Example: Freeform Surfaces 504 14.2.8 Gaussian Curvature 505 14.2.9 Isometric Mappings and Cartography 506 14.2.10 Developable Surfaces 508 14.2.11 Mean Curvature 508 14.2.12 Visualization of the Curvature Behavior 509 14.2.13 Principal Curvature Lines 510 14.2.14 Remark (Umbilics) 511 14.3 Optical Lines for Quality Control 513 14.3.1 Generation of a Reflection Line 513 14.3.2 Math 514 14.3.3 Isophotes 515 14.3.4 Contour Generators 515 14.3.5 Example: Contour Generator of Simple Objects 516 14.3.6 Optical Lines for Shape Analysis 517 14.3.7 Example: Subdivision Surface 517 14.3.8 Implications on Aesthetic Design 518 14.4 Texture Mapping 519 14.4.1 Example: Texturing a Cone 522 14.5 Digital Elevation Models 523 14.5.1 Data Acquisition and Main Representations 524 14.5.2 Topographic Surfaces 524 14.6 Geometric Topology and Knots 527 14.6.1 Open versus Closed Surfaces 527 14.6.2 Deformations That Do Not Change Topology 528 14.6.3 Orientable versus Non-Orientable Surfaces 530 14.6.4 Euler's Formula 531 14.6.5 Euler's Formula for other Topological Types 532 14.6.6 Example: Torus 532 14.6.7 Example: Möbius Band 533 14.6.8 Example: Planar Domain with Holes 533 14.6.9 Classification of Closed Orientable Surfaces 534 14.6.10 Other Topological Invariants 535 14.6.11 Knots from the Mathematical Perspective 536 14.6.12 Example: Torus Knots 539 15. Developable Surfaces and Unfoldings 541 15.1 Developable Surfaces and Unfoldings 543 15.2 Surfaces That Can Be Built from Paper 545 15.2.1 Cylinders 546 15.2.2 Example: Cylinder of Revolution and Helixes 547 15.2.3 Example: Development of an Oblique Circular Cylinder 548 15.2.4 Cones 549 15.2.5 Strips Formed by Planar Quadrilaterals 551 15.2.6 Refinement of a PQ Strip with Perturbed Subdivision 552 15.2.7 Tangent Surfaces of Space Curves 554 15.2.8 These are All Developable Surfaces 556 15.2.9 Offsets are also Developable 556 15.2.10 Surfaces of Constant Slope 557 15.2.11 Example: Construction of a Roof 559 15.2.12 Example: Helical Developable Surface 560 15.2.13 Developable Surfaces through Curves 561 15.2.14 Example: D-Forms 562 15.2.15 The Development and the Inverse Operation 562 15.2.16 Developable Surfaces Related to Principal Curvature Lines 565 15.2.17 Strip Models of Doubly Curved Surfaces 566 15.2.18 Strips of Paper 568 15.2.19 Shortest Paths on Polyhedra and on Smooth Surfaces 569 15.3 Unfolding a Polyhedron 571 References 575 16. Digital Prototyping and Fabrication 577 16.1 Model Making and Architecture 579 16.1.1 History 580 16.1.2 Representation Models 583 16.1.3 Rapid Prototyping 585 16.1.4 Digital Fabrication and Assembly 587 16.2 Fabrication Techniques 589 16.2.1 Overview 589 16.3 Cutting-Based Processes 591 16.3.1 Sheet-Based Cutting Techniques 591 16.3.2 Laser Cutters and Plasma Cutters 592 16.3.3 Waterjet Cutter 592 16.3.4 Sheet Cutter 592 16.4 Additive Processes: Layered Fabrication 593 16.4.1 Fuse Deposition Modeling (FDM) 594 16.4.2 Powder-Based Processes 595 16.4.3 Stereolithography 596 16.5 Subtractive Techniques 597 16.5.1 Machining Models 597 16.5.2 Mills and Routers 598 16.5.3 Foam Cutters 599 16.5.4 Robotic Machining 599 16.6 Geometric Challenges Related to Machining and Rapid Prototyping 601 16.6.1 Aesthetics of Fabrication and Geometric Implications 602 16.6.2 Choosing Materials Based on Geometric Properties 603 16.6.3 Implications for Standardization 604 16.7 Assembly 605 16.7.1 Fastener-Based Assemblies 605 16.7.2 Geometry-Based Assemblies 606 16.7.3 Robotic Assembly 607 17. Geometry for Digital Reconstruction 609 17.1 Geometry for Digital Reconstruction 611 17.1.1 An Overview of the Digital Reconstruction Pipeline 613 17.2 Data Acquisition and Registration 617 17.2.1 Digitizer Arms 617 17.2.2 Optical Scanning Devices 617 17.2.3 Outlier Removal and Noise Reduction 621 17.2.4 Registration 622 17.2.5 Fully Automatic Registration and Surface Matching 624 17.3 The Polygon Phase 625 17.3.1 Triangulation 625 17.3.2 Voronoi Diagrams in the Plane 625 17.3.3 Two-Dimensional Delaunay Triangulation 627 17.3.4 Voronoi Diagrams in Three Dimensions 629 17.3.5 Surface Triangulations 630 17.3.6 Interactive Improvements 631 17.3.7 Relaxation and Smoothing 632 17.3.8 Mesh Decimation 632 17.4 Segmentation 635 17.5 Surface Fitting 639 17.5.1 Symmetries 640 17.5.2 Freeform Surface Fitting 642 17.6 The Surfaces Need to be Built 643 17.6.1 Approximation with Ruled Surfaces 644 17.6.2 Developable Surfaces 644 17.6.3 Translational Surfaces and More General Sweeping Surfaces 646 References 647 18. Shape Optimization Problems 649 18.1 Shape Optimization Problems 651 18.2 Remarks on Mathematical Optimization 653 18.2.1 Example (Minimization of a Quadratic Function) 654 18.2.2 Example (Roof Design via Least Squares Approximation) 655 18.2.3 Gradient Descent 656 18.3 Geometric Optimization 657 18.3.1 Minimal Surfaces 657 18.3.2 Example: Enneper Surface 660 18.3.3 Example: Helicoid 661 18.3.4 Example: Rotational and Helical Minimal Surface 662 18.3.5 Example: Scherk's Minimal Surface 664 18.3.6 Numerical Solution of the Plateau Problem 665 18.3.7 Surfaces with Constant Mean Curvature 666 18.3.8 Willmore Energy 668 18.3.9 Fair Curves and Polygons 669 18.3.10 Fair Curves on Surfaces 671 18.3.11 Fair Webs and Mesh Beautification 672 18.3.12 Geometric Constraints 673 18.4 Functional Optimization 675 18.4.1 Hanging Models 675 18.4.2 Membranes 676 References 677 19. Discrete Freeform Structures 679 19.1 Discrete Freeform Structures 681 19.2 Triangle Meshes 685 19.3 Quadrilateral Meshes with Planar Faces 687 19.3.1 Planar Quad Meshes of Simple Geometry 687 19.3.2 Example: Rotational PQ Mesh 688 19.3.3 Conjugate Curve Networks 690 19.3.4 Example: Hyperbolic Paraboloid 691 19.3.5 Negatively Curved Areas Cause Problems 691 19.3.6 Principal Curvature Lines 692 19.3.7 Planar Quad Meshes are Discrete Versions of Conjugate Curve Networks 692 19.3.8 A Planarization Algorithm 693 19.3.9 Limitations on Meshing 694 19.3.10 Combination of Subdivision and Planarization as a Design Tool 694 19.4 Parallel Meshes, Offsets, and Supporting Beam Layout 697 19.4.1 Parallel Meshes and Multilayer Constructions 697 19.4.2 Beams and Nodes 698 19.4.3 Triangle Meshes 700 19.5 Offset Meshes 701 19.5.1 Offset Surfaces Revisited 701 19.5.2 PQ Meshes with Exact Offsets 701 19.5.3 Discrete Gaussian Image and Characterization of Meshes with Precise Offsets 702 19.5.4 Vertex Offsets: Circular Meshes 704 19.5.5 Face Offsets: Conical Meshes 704 19.5.6 Meshes with Edge Offsets 708 19.5.7 Approximate Offsets: Computing a Support Structure for a PQ Mesh 709 19.6 Optimal Discrete Surfaces 711 19.6.1 Discrete Minimal Surfaces 711 19.6.2 The Christoffel Dual: Static Equilibrium of Diagonal Meshes 712 19.6.3 Beyond Quad Meshes 714 19.7 Future Research 717 References 719 Appendix A: Geometry Primer 721 A.1 Nomenclature 721 A.2 Basic Operations on Vectors 722 A.3 Complex Numbers 722 A.4 Collinear 723 A.5 Coordinates of a Vector 723 A.6 Co-Planar 723 A.7 Cross Product of Vectors 723 A.8 Direction Vector 723 A.9 Dot Product of Vectors 724 A.10 Interpolation 724 A.11 Length (Norm) of a Vector 724 A.12 Linear Interpolation 725 A.13 Mutual Positions of Lines 725 A.14 Normal to a Plane 725 A.15 Parallelogram 726 A.16 Planes 726 A.17 Polygon and Polyline 726 A.18 Position Vector 727 A.19 Positive and Negative Rotation 727 A.20 Radian Measure 727 A.21 Ratio 727 A.22 Regular Polygon 728 A.23 Straight Lines 728 A.24 Theorem of Pythagoras 729 A.25 Theorem of Thales 729 A.26 Vector 729 A.27 Zero Vector 729 List of Symbols 730 Index 731 A 731 B 731 C 732 D 734 E 735 F 735 G 736 H 736 I 737 K 737 L 737 M 738 N 739 O 739 P 740 Q 741 R 741 S 742 T 744 U 745 V 745 W 746 Photo Credits 747