Handbook of Computer Aided Geometrie Design

Transcription

1 Handbook of Computer Aided Geometrie Design Edited by Gerald Farin Josef Hoschek t Myung-Soo Kim 2002 ELSEVIER Amsterdam-Boston-London-New York-Oxford-Paris- San Diego-San Francisco-Singapore-Sydney-Tokyo

2 Contents Preface Contributors xxiii 1 A History of Curves and Surfaces in CAGD (G. Farin) INTRODUCTION EARLY DEVELOPMENTS DE CASTELJAU AND BtZIER PARAMETRIC CURVES RECTANGULAR SURFACES B-SPLINE CURVES AND NURBS TRIANGULAR PATCHES SUBDIVISION SURFACES SCIENTIFIC APPLICATIONS SHAPE INFLUENCES AND APPLICATIONS 14 2 Geometric Fundamentals (W. Boehm and H. Prautzsch) AFFINE FUNDAMENTALS Points and vectors Affine systems Barycentric coordinates Affine subspaces and parallelism Affine maps and axonometric images Affine combinations and A-frame CONIC SECTIONS AND QUADRICS Quadrics in affine space Tangents and polar planes Pascal's and Brianchon's theorems THE EUCLIDEAN SPACE Cartesian coordinates Gram-Schmidt orthogonalization Euclidean motions and orthogonal projections Quadrics in Euclidean space PROJECTIVE FUNDAMENTALS 32 vii

3 vii' Homogeneous coordinates Projective coordinates Projective maps The procedure of inhomogeneizing Repeated projective combinations Quadrics in projective space Parametrizing a quadric and its equation 2.5 DUALITY 2.6 OSCULATING CURVES AND SURFACES Curve and surface Curve and curve Surface and surface Contur lines, reflection lines and isophotes 2.7 DIFFERENTIAL FUNDAMENTALS Arc length and osculating plane Curvature and torsion The Frenet frame Curves an surfaces Meusnier's sphere and Dupin's indicatrix The curvatures of a surface Geometries for CAGD (H. Pottmann and S. Leopoldseder) 3.1 CURVES AND SURFACES IN PROJECTIVE GEOMETRY Bezier curves and surfaces as images of normal curves and surfaces NURBS curves and surfaces in projective geometry Duality and dual representation Developable surfaces as dual curves 3.2 SPHERE GEOMETRIES Models of Laguerre geometry Möbius geometry Applications of the cyclographic image of a curve in 3-space The medial axis transform in a sphere geometric approach Canal surfaces (in Laguerre and Möbius geometry) Rational curves and surfaces with rational offsets 3.3 LINE GEOMETRY Basics of line geometry Linear complexes in kinematics and reverse engineering Ruled surfaces Other applications of line geometry in geometric computing 3.4 APPROXIMATION IN SPACES OF GEOMETRIC OBJECTS Approximation in the space of spheres Approximation in the space of planes Approximation in line space 3.5 NON-EUCLIDEAN GEOMETRIES Hyperbolic geometry and geometric topology

4 ix Elliptic geometry and kinematics Isotropic geometry and analysis of functions and images 66 4 Bdzier Techniques (D. Hansford) WHY BEZIER TECHNIQUES? BEZIER CURVES Parametric curves Properties of Bdzier curves The de Casteljau algorithm for B6zier curves Bernstein polynomials Derivatives of Bdzier curves Degree elevation of Bdzier curves Interrogation techniques for Bdzier curves Basis conversion Piecewise Bdzier curves RECTANGULAR BEZIER PATCHES Bilinear patches Bdzier patches Properties of Bdzier patches Evaluation of Bdzier patches Derivatives of Bdzier patches Working with Bdzier patches Cl Bdzier patches TRIANGULAR BEZIER PATCHES Bdzier triangles introduced Properties of Bdzier triangles The de Casteljau algorithm for B6zier triangles Bivariate Bernstein polynomials Derivatives of Bdzier triangles Working with Bdzier triangles C 1 B6zier triangles Rational Techniques (H.J. Wolters) INTRODUCTION RATIONAL BEZIER CURVES Basic definitions Derivatives Fundamental algorithms Conics RATIONAL B-SPLINE CURVES Basic definitions Derivatives Fundamental algorithms GEOMETRIC CONTINUITY FOR RATIONAL CURVES RATIONAL CURVE APPROXIMATION AND INTERPOLATION. 123

5 5.5.1 Rational curve interpolation Rational curve approximation RATIONAL BEZIER SURFACES Basic definitions Derivatives Algorithms RATIONAL B-SPLINE SURFACES Basic definitions Derivatives Algorithms GEOMETRIC CONTINUITY FOR RATIONAL PATCHES INTERPOLATION AND APPROXIMATION ALGORITHMS RATIONAL SURFACE CONSTRUCTIONS Surfaces of revolution Canal and pipe surfaces CONCLUDING REMARKS Spline Basics (C. de Boor) PIECEWISE POLYNOMIALS B-SPLINES DEFINED SUPPORT AND POSITIVITY SPLIINE SPACES DEFINED SPECIFIC KNOT SEQUENCES THE POLYNOMIALS IN THE SPLINE SPACE: MARSDEN'S IDENTITY THE PIECEWISE POLYNOMIALS IN THE SPLINE SPACE DUAL FUNCTIONALS AND BLOSSOMS GOOD CONDITION CONVEX HULL DIFFERENTIATION AND INTEGRATION EVALUATION SPLINE FUNCTIONS VS SPLINE CURVES KNOT INSERTION VARIATION DIMINUTION AND SHAPE PRESERVATION: SCHOEN- BERG'S OPERATOR ZEROS OF A SPLINE, COUNTING MULTIPLICITY SPLINE INTERPOLATION: SCHOENBERG-WHITNEY SMOOTHING SPLINE LEAST-SQUARES SPLINE APPROXIMATION BACKGROUND Curve and Surface Constructions (D. Hansford and G. Farin) INTRODUCTION POLYNOMIAL CURVE METHODS Point Data Interpolation Point Data Approximation 170

6 xi Point and Tangent Data Interpolation C2 CUBIC SPLINE INTERPOLATION End Conditions Defining a Knot Sequence The Minimum Property POLYNOMIAL SURFACE METHODS Discrete Coons Patches Tensor Product Interpolation Approximation with Tensor Product Patches Bicubic Hermite Patches C2 BICUBIC SPLINE INTERPOLATION Finding Knot Sequences VOLUME DEFORMATIONS Geometric Continuity (J. Peters) MOTIVATING EXAMPLES Differentiation and evaluation GEOMETRIC CONTINUITY OF PARAMETRIC CURVES/SURFACES Joining parametric curve pieces Geometric continuity of edge-adjacent patches Geometrie continuity at a vertex Free-form surface splines EQUIVALENT AND ALTERNATIVE DEFINITIONS Matching intrinsic curve properties C' manifolds Tangent and normal continuity Global and regional reparametrization Implicit representation Generalized subdivision CONSTRUCTIONS Free-form surface splines of low degree ADDITIONAL LITERATURE Splines an Surfaces (M. Neamtu) INTRODUCTION SCALAR SPLINES ON SMOOTH SURFACES ALTERNATIVE METHODS FOR FUNCTIONS ON SURFACES Discrete surfaces Radial basis functions Variational methods Distance-weighting methods Transfinite methods Implicit methods Other types of splines Multiresolution methods 248

7 xii Visualization of surfaces an surfaces Box Splines (H. Prautzsch and W. Boehm) BOX SPLINES Inductive definition Geometrie definition Further definitions of Box splines Basic properties of Box splines Derivatives BOX SPLINE SURFACES 'Translates of Box splines Derivatives and polynomial properties Convexity Subdivision General subdivision Convergence under subdivision Bezier representation Bdzier representation of Symmetrie Box splines Generalized Box spline surfaces HALF-BOX SPLINES Inductive definition Basic properties Derivatives and polynomial structure HALF-BOX SPLINE SURFACES Translates of Half-Box splines Derivatives and polynomial properties Subdivision Bdzier representation Generalized Half-Box spline surfaces Finite Element Approximation with Splines (K. Höllig) INTRODUCTION SPLINES ON UNIFORM GRIDS Uniform B-splines Splines an bounded domains Hierarchical bases FINITE ELEMENT BASES Mesh-based elements WEB-basis R-functions Stability APPROXIMATION OF BOUNDARY VALUE PROBLEMS Essential boundary conditions Natural boundary conditions Mixed and higher order boundary conditions 299

8 Error estimates Implementation SUMMARY Subdivision Surfaces (M. Sabin) SUBDIVISION SURFACE DEFINITIONS INTRODUCTION SUBDIVISION CURVES The Chaikin construction Higher degree splines The 4-point interpolatory scheme BOX-SPLINES GENERALIZATIONS TO ARBITRARY TOPOLOGY SOME SPECIFIC SCHEMES The Doo-Sabin quadratic scheme The Catmull-Clark cubic scheme The Loop triangular mesh scheme The Butterfly interpolatory scheme ANALYSIS OF CONTINUITY AT THE SINGULARITIES Support Regular regions Neighbourhoods of singularities Limit point Natural configuration and characteristic map Curvatures Tuning subdivisions for better behaviour Jordan blocks Discontinuities of curvature Higher derivatives Precision set FIRST STEP ARTIFACTS CURRENT RESEARCH DIRECTIONS Square root of 3 scheme Subdivision over semiregular lattices Dual schemes and contact elements Unequal intervals Artifact analysis CONCLUSIONS Interrogation of Subdivision Surfaces (M. Sabin) SUBDIVISION SURFACE INTERROGATIONS HISTORICAL BACKGROUND THE CONVEX HULL PROPERTY Other hulls Hulls of non-positive bases Normal hulls 331

9 xiv Offset subdivision surfaces AN API FOR SUBDIVISION SURFACES EXAMPLE INTERROGATIONS Z-buffer imaging Raycast Plane section Surface-surface intersection Silhouette PERFORMANCE ISSUES CONCLUSIONS Multiresolution Techniques (L. P. Kobbelt) INTRODUCTION MULTIRESOLUTION REPRESENTATIONS FOR CURVES LIFTING GEOMETRIC SETTING MULTIRESOLUTION REPRESENTATIONS FOR SURFACES Coarse-to-fine hierarchies Fine-to-coarse hierarchies APPLICATIONS Multiresolution editing Geometry compression Algebraic Methods for Computer Aided Geometrie Design (T. W. Sederberg and J. Zheng) INTRODUCTION POLYNOMIALS, IDEALS, AND VARIETIES Notation and terminology Ideals and varieties Gröbner bases RESULTANTS Sylvester's resultant Bezout's resultant Dixon's resultant CURVE IMPLICITIZATION AND INVERSION Resultant-based method Gröbner basis technique Moving curve technique CURVE PARAMETRIZATION Planar algebraic curves Genus and rationality Parametrizing curves INTERSECTION COMPUTATIONS Parametric curve and implicit curve Implicit curve and implicit curve 378

10 XV Parametric curve and parametric curve SURFACES Implicit degree of a rational parametric surface Surface intersection curves Implicitization Parametrizaion OTHER ISSUES Scattered Data Interpolation: Radial Basis and Other Methods (S.K. Lodha and R. Franke) INTRODUCTION RADIAL INTERPOLATION Existence and uniqueness Computation of the interpolant Evaluation Applications OTHER LOCAL METHODS CONCLUSIONS Pythagorean-Hodograph Curves (R. T. Farouki) PREAMBLE POLYNOMIAL PH CURVES Planar PH curves Complex representation PH space curves CONSTRUCTION ALGORITHMS PH quintic Hermite interpolants Shape properties of PH quintics C2 PH quintic splines Spatial PH quintic Hermite interpolants Geometric Hermite interpolants Further constructions REAL TIME CNC INTERPOLATORS RATIONAL CURVES WITH RATIONAL OFFSETS Rational PH curves Improper parameterizations MINKOWSKI PH CURVES Minkowski metric of special relativity Minkowski metric defined by convex indicatrix CLOSURE Voronoi Diagrams (K. Sugihara) ORDINARY VORONOI DIAGRAM DELAUNAY DIAGRAM BASIC PROPERTIES OF THE VORONOI AND DELAUNAY DIAGRAMS432

11 xvi 18.4 ALGORITHMS APPLICATIONS Site retrieval Medial axis Offset curves and surfaces Interpolation EXTENSIONS Voronoi diagrams for general distances Additively weighted Voronoi diagram Multiplicatively weighted Voronoi diagram Power diagram Voronoi diagram based on L p distance Voronoi diagram based on elliptic distance Obstacle-avoidance Voronoi diagram Voronoi diagram in a river Crystal Voronoi diagram Voronoi diagram for lines and polygons Voronoi diagram for general figures CONCLUSION The Medial Axis Transform (H.I. Choi and C. Y. Han) INTRODUCTION MATHEMATICAL THEORY OF THE MEDIAL AXIS TRANSFORM Assumptions on the domain Medial axis transform Finiteness results Graph structure of medial axis transform Domain decomposition lemma ALGORITHMS Piecewise linear and circular arc boundary Domains with free-form boundaries Global decomposition algorithm CONCLUDING REMARKS Solid Modeling (V. Shapiro) INTRODUCTION A premise of informational completeness Outline MATHEMATICAL MODELS First postulates Continuum point set model of solidity Combinatorial model of solidity Generalizations COMPUTER REPRESENTATIONS Implicit and constructive 481

12 xvii Enumerative and combinatorial Boundary representation: a compromise Unification of representation schemes ALGORITHMS Fundamental computations Enabling algorithms APPLICATIONS Geometric design Analysis and simulation Dynamic analysis and lumped-parameter systems Planning and generation Manufacturing SYSTEMS Classical systems Parametric interaction Standards and interfaces CONCLUSIONS Unsolved problems and promising directions Summary Parametric Modeling (C.M. Hoffmann and R. Joan-Arinyo) INTRODUCTION PARAMETRIC MODELS VARIANT MODELING CONSTRAINT-BASED MODELING Constraints Modeling with constraints Solving geometric and equational constraints Degrees of freedom analysis FEATURE-BASED MODELING Features and the feature model A brief feature taxonomy Feature model construction Feature representation Features and constraints TRENDS Feature libraries Multiple views Semantic features Persistent naming OPEN PROBLEMS Constraint solving Features Semantics of parametric design Assembly-centric design 536

13 xviii 22 Sculptured Surface NC Machining (B.K. Choi, B.H. Kim, and R.B. Jerard) INTRODUCTION Overview of the sculptured surface machining process Information processing issues UNIT MACHINING OPERATIONS Tool path topology and milling-strategy options Ball-endmill UMOs Flat-endmill UMOs INTERFERENCE HANDLING CL-point interference CL-line interference Collisions TOOL PATH GENERATION METHODS The conventional approach and the C-space approach Geometric issues in conventional approach Geometric issues in the C-space approach GEOMETRIC ALGORITHMS CL-surface construction D PS-curve offsetting Area scan algorithm Point-sequence curve fairing Collision detection algorithms CONCLUSION Cyclides (W. Degen) INTRODUCTION THE GEOMETRY OF DUPIN CYCLIDES Dupin cyclides in classical differential geometry The three main types of Dupin cyclides and their parameter representations Implicit equations SUPERCYCLIDES Curves and surfaces in the pfojective space Basic properties of supercyclides CYCLIDES IN CAGD B6zier representation of cyclides Using cyclides as blendings Blending with supercyclides APPENDIX: STUDYING DUPIN CYCLIDES WITH LIE GEOMETRY Geometry Processing (T.A. Grandine) INTRODUCTION ROOT FINDING INTEGRATION 613

14 xix 24.4 COMPUTING MASS PROPERTIES Intersection Problems (N.M. Patrikalakis and T. Maekawa) INTRODUCTION CLASSIFICATION OF INTERSECTION PROBLEMS Classification by dimension Classification by type of geometric specification Classification by number system OVERVIEW OF NONLINEAR SOLVERS Brief review of local and global methods IPP algorithm CURVE/SURFACE INTERSECTION RPP curve/ia surface intersection RPP curve/rpp surface intersection IA curve/ia surface intersection IA curve/rpp surface intersection SURFACE/SURFACE INTERSECTIONS RPP/IA surface intersection RPP/RPP surface intersection IA/IA surface intersection CONCLUSION Reverse Engineering (T. Varady and R. Martin) INTRODUCTION THE BASIC PHASES OF REVERSE ENGINEERING DATA CAPTURE Laser scanners Multiple view registration TRIANGULATION AND DECIMATION Triangulation overview K6s's method RECONSTRUCTING FREE-FORM OBJECTS Segmentation strategies Fitting free-form surfaces Fitting feature surfaces RECONSTRUCTING CONVENTIONAL ENGINEERING OBJECTS Segmentation Fitting analytic surfaces Fitting extruded and rotational surfaces Constrained fitting for multiple curves and surfaces Reconstructing blend surfaces Building solid models Beautifying solid models CONCLUSION 676

15 xx 27 Vector and Tensor Field Visualization (G. Scheuermann and H. Hagen) INTRODUCTION VISUALIZATION PROCESS DATA SET TYPES AND INTERPOLATION METHODS DIRECT MAPPINGS TO GEOMETRIC PRIMITIVES Point-based methods Line-based methods Surface and volume-based methods ATTRIBUTE MAPPINGS STRUCTURE AND FEATURE BASED MAPPINGS Vector field topology Tensor field topology Feature detection algorithms Splines over Triangulations (F. Zeilfelder and H.-P. Seidel) INTRODUCTION BERNSTEIN-BZIER TECHNIQUES DIMENSION FINITE AND MACRO ELEMENTS INTERPOLATION TRIANGULAR B-SPLINES Kinematics and Animation (B. Jüttler and Wagner) INTRODUCTION THE KINEMATICAL MAPPING Coordinates Motions of a rigid body Euler parameters The kinematical mapping QUATERNIONS Fundamentals Homogeneous quaternions and the kinematical mapping Summary: homogeneous quaternion coordinates for 3D rotations MOTION DESIGN USING CURVES ON S Slerping Problems of slerping Other approaches Motion design desired features SPHERICAL RATIONAL MOTIONS SPATIAL RATIONAL MOTIONS Construction Special cases Affine control structure Some properties Interpolation schemes and applications 741

16 xxi Rational frames and sweeping surfaces CLOSURE Direct Rendering of Freeform Surfaces (G. Elber) INTRODUCTION SCAN-CONVERSION OF CURVES Forward differencing Adaptive forward differencing SURFACE COVERAGE AND RENDERING USING CURVES Coverage based on adaptive isoparametric curves Rendering using adaptive isoparametric curves RAY-TRACING Bezier clipping Ruled tracing EXTENSIONS Isometric texture mapping Machining using adaptive isoparametric curves Line-art rendering CONCLUSION Modeling and Processing with Quadric Surfaces (W. Wang) DEFINITION AND CLASSIFICATIONS Definition Euclidean classification Affine classification Projective classification PARAMETRIC REPRESENTATION Global rational parameterization Generalized stereographic projection Surface patches on quadrics FITTING, BLENDING, AND OFFSETTING Fitting Blending Offsetting INTERSECTION AND INTERFERENCE Computation of intersection curves Detecting interference ACKNOWLEDGMENTS 792 Index 797