Geometric Programming for Computer-Aided Design

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1 Geometric Programming for Computer-Aided Design Alberto Paoluzzi Dip. Informatica e Automazione, Università Roma Tre, Rome Italy with contributions from Valerio Pascucci Center for Applied Scientific Computing, L. Livermore National Laboratory, California Michele Vicentino WIND Telecomunicazioni Spa, Fraud Management System, Rome Claudio Baldazzi Saritel Spa, Internet Platforms for Service Providers, Pomezia, Rome and Simone Portuesi Dip. Informatica e Automazione, Università Roma Tre, Rome JOHN WILEY & SONS Chichester. New York. Brisbane. Toronto. Singapore

2 Contents Preface Book roadmap Acknowledgements xvii xviii xix I Programming and Geometry xxi 1 Introduction to FL and PLaSM Introduction to symbolic design programming Computational model Getting started with PLaSM Installing the language Using the language Programming at Function Level Elements of FL syntax Combining forms and functions Basics of PLaSM programming Expressions User-defined functions Built-in functions Geometric operators Pre-defined geometric operators Examples First programs Further examples Virtual Manhattan Virtual skyscraper Roof of S. Stefano Rotondo Annotated references 47 2 Geometric programming Basic programming Some operations on numbers Set operations Vector and matrix operations 61

3 viii CONTENTS String and character operations Other examples Pipeline paradigm Basic geometric programming Primitive shapes Non-primitive shapes (simplicial maps) Assembling shapes Primitive alignments Relative arrangements Examples Parametric nut stack Hierarchical temple 89 3 Elements of linear algebra Vector spaces Bases and coordinates PLaSM representation of vectors Affine spaces Operations on vectors and points Positive, affine and convex combinations Linear, affine and convex independence Convex coordinates Euclidean spaces Linear transformations and tensors Tensor operations Coordinate representation PLaSM matrix representation Determinant and inverse Orthogonal tensors PLaSM representation of tensors Elements of polyhedral geometry Basic concepts Convex sets Positive, affine and convex hulls Support, separation, extreme points Duality Boundary structure Polyhedral sets Extremal points Double description Polytopes Examples of polytopes Faces of polytopes Projection Simplicial complexes Definitions 144

4 CONTENTS ix Linear d-polyhedra Fractal d-simplex Polyhedral complexes Schlegel diagrams Nef polyhedra Locally adjoined pyramids Faces of Nef polyhedra Linear programming Geometry of linear programming Simplex method Some polyhedral algorithms Examples Platonic solids References 168 5Elements of differential geometry Curves Variable-free notation Reparametrization Orientation Differentiation Real-valued maps of a real variable Vector-valued maps of a real variable Real-valued maps of several real variables Vector-valued maps of several real variables Generalized implementation Fields and differential operators Gradient Divergence and Laplacian Curl Gradient, divergence and curl of a 3D field Differentiable manifolds Charts and atlases Differentiable manifolds Tangent spaces and maps Derivatives of a curve Examples Intrinsic properties of a surface First fundamental form Second fundamental form Gauss curvature Examples Color map of the Gauss curvature field 212

5 x CONTENTS II Graphics Affine transformations Preliminaries Transformations Points and vectors Orientations and rotations Homogeneous coordinates D Transformations Translation Scaling Reflection Rotation Shearing Generic transformation Tensor properties Fixed point transformations D Transformations Elementary transformations Rotations Rotations about affine axes Algebraic properties of rotations PLaSM implementation Pre-defined affine tensors User-defined affine tensors Examples Modeling applied 3D vectors and reference frames Graphic primitives Some background GKS PHIGS Open GL Open INVENTOR VRML Java 3D Basic primitives Points Polyline Polymarker Text Triangle strip Triangle fan Quadrilateral mesh Quotient set of polylines Basic definitions Interpolation of shapes 291

6 CONTENTS xi Shape compatibility Shape resemblance Examples Quadrilateral array with holes Annotated references Hierarchical structures Hierarchical graphs Local coordinates and modeling transformation Hierarchical structures Hierarchical structures in PLaSM Assembly using global coordinates Assembly with local coordinates Traversal Implementations Structure network in PHIGS Hierarchical scene graph in VRML Examples Living roommodeling Body model Umbrella modeling (1): structure Tree diagrams Array of aligned graphics symbols Graphic pipelines D pipeline Coordinate systems Normalization and device transformations Window-viewport mapping D pipeline View model Coordinate systems Transformations of coordinates View orientation View mapping Perspective transformation Workstation transformation Other implementations Scalable vector graphics (SVG) Open Inventor camera model Java 3D viewing model Examples PHIGS! pipeline display VRML camera implementation Viewing and rendering View-model 367

7 xii CONTENTS View parameters View volume Taxonomy of projections Perspective Parallel Hidden-surface removal Introduction Pre-processing Scene coherence Back-to-front and front-to-back Binary Space Partition HSR algorithms in image-space Illumination models Diffuse light Specular reflection Color models Shading models VRML rendering Illumination Color Shading Textures PLaSM lighting PLaSM texturing Examples Orthogonal projection on any viewplane Model of example house Cell extraction Exploded views Standard view models Annotated references 438 III Modeling Parametric curves Curve representations Polynomial parametric curves Linear curves Quadratic curves Cubic curves Higher-order Bézier curves Polynomial splines Cubic cardinal splines Cubic uniformb-splines Non-uniformpolynomial B-splines Multiresolution Bézier and B-splines 470

8 CONTENTS xiii 11.4 Rational curves and splines Rational Bézier curves NURBS Examples Shape design with a NURB spline Umbrella modeling (2): curved rods Splines implementation Cubic uniformsplines Non-uniformB-splines Non-uniformrational B-splines Parametric surfaces and solids Introduction Parametric representation Notable surface classes Profile product surfaces Ruled surfaces Cylinders and cones Tensor product surfaces Bilinear and biquadratic surfaces Bicubic surfaces Fixed-degrees tensor product surfaces NURB surfaces Higher-order tensor products Multivariate Bézier manifolds Transfinite methods Rationale of the approach Univariate case Multivariate case Transfinite Bézier Transfinite Hermite Connection surfaces Coons surfaces Thin solids generated by surfaces Examples Umbrella modeling (3): tissue canvas Helicoidal spiral volume Roof design for a sports building Constrained connection volume Basic solid modeling Representation scheme Definition and properties Taxonomy of representation schemes Enumerative schemes Flat enumerative schemes Hierarchical enumerative schemes 562

9 xiv CONTENTS 13.3 Decompositive schemes Simplicial schemes Convex-cell partitioning and covering Constructive Solid Geometry Boundary schemes Adjacency and incidence relations Euler equation Edge-based schemes Facet-based schemes Mass and inertia properties Examples Umbrella modeling (4): solid parts Dimension-independent PLaSM operators Inside Hierarchical Polyhedral Complexes Boolean operations Algorithm Optimizing by pruning and unpruning Interface to PLaSM operators Linear Programming implementation Boundary-to-interior operator Dimension-independent integration Tensor of inertia Generalized product of polyhedra Definitions Product of cell-decomposed polyhedra Intersection of extrusions Skeleton extraction Algorithm Extrusion, sweep, offset and Minkowski sum Background Unified approach Algorithms Annotated references Motion modeling Degrees of freedom Configuration space Animation with PLaSM Some definitions Generating flash animations Generating VRML animations Motion coordination Non-linear animation Network programming Modeling and animation cycle Extended Configuration Space 680

10 CONTENTS xv Introduction Rationale of the method ECS algorithm Encoding degrees of freedom Computation of FP Examples Umbrella modeling (5): animation Non-holonomic planar motion Anthropomorphic robot with 29 degrees of freedom Solving a 2D labyrinth 696 Appendix A Definition of MyFont 701 Appendix B PLaSM libraries 705 B.1 Standard 705 B.2 animation Library 719 B.3 colors Library 720 B.4 curves Library 723 B.5 derivatives Library 724 B.6 drawtree Library 725 B.7 flash Library 726 B.8 general Library 726 B.9 myfont Library 729 B.10 operations Library 730 B.11 primitives Library 732 B.12 shapes Library 735 B.13 splines Library 737 B.14 strings Library 739 B.15 surfaces Library 739 B.16 text Library 740 B.17 transfinite Library 741 B.18 vectors Library 741 B.19 viewmodels Library 744 Appendix C 747 REFERENCES 747 Index 759

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