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Handbook of Computer Aided Geometric Design

Book Description

This book provides a comprehensive coverage of the fields Geometric Modeling, Computer-Aided Design, and Scientific Visualization, or Computer-Aided Geometric Design. Leading international experts have contributed, thus creating a one-of-a-kind collection of authoritative articles. There are chapters outlining basic theory in tutorial style, as well as application-oriented articles. Aspects which are covered include:


Historical outline
Curve and surface methods
Scientific Visualization
Implicit methods
Reverse engineering.



This book is meant to be a reference text for researchers in the field as well as an introduction to graduate students wishing to get some exposure
to this subject.

Table of Contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Preface
  6. Contributors
  7. Chapter 1: A History of Curves and Surfaces in CAGD
    1. 1.1 INTRODUCTION
    2. 1.2 EARLY DEVELOPMENTS
    3. 1.3 DE CASTELJAU AND BÉZIER
    4. 1.4 PARAMETRIC CURVES
    5. 1.5 RECTANGULAR SURFACES
    6. 1.6 B-SPLINE CURVES AND NURBS
    7. 1.7 TRIANGULAR PATCHES
    8. 1.8 SUBDIVISION SURFACES
    9. 1.9 SCIENTIFIC APPLICATIONS
    10. 1.10 SHAPE
    11. 1.11 INFLUENCES AND APPLICATIONS
  8. Chapter 2: Geometric Fundamentals
    1. 2.1 AFFINE FUNDAMENTALS
    2. 2.2 CONIC SECTIONS AND QUADRICS
    3. 2.3 THE EUCLIDEAN SPACE
    4. 2.4 PROJECTIVE FUNDAMENTALS
    5. 2.5 DUALITY
    6. 2.6 OSCULATING CURVES AND SURFACES
    7. 2.7 DIFFERENTIAL FUNDAMENTALS
  9. Chapter 3: Geometries for CAGD
    1. 3.1 CURVES AND SURFACES IN PROJECTIVE GEOMETRY
    2. 3.2 SPHERE GEOMETRIES
    3. 3.3 LINE GEOMETRY
    4. 3.4 APPROXIMATION IN SPACES OF GEOMETRIC OBJECTS
    5. 3.5 NON-EUCLIDEAN GEOMETRIES
  10. Chapter 4: Bézier Techniques
    1. 4.1 WHY B éZIER TECHNIQUES?
    2. 4.2 BÉZIER CURVES
    3. 4.3 RECTANGULAR BÉZIER PATCHES
    4. 4.4 TRIANGULAR B éZIER PATCHES
  11. Chapter 5: Rational Techniques
    1. 5.1 INTRODUCTION
    2. 5.2 RATIONAL B éZIER CURVES
    3. 5.3 RATIONAL B-SPLINE CURVES
    4. 5.4 GEOMETRIC CONTINUITY FOR RATIONAL CURVES
    5. 5.5 RATIONAL CURVE APPROXIMATION AND INTERPOLATION
    6. 5.6 RATIONAL B éZIER SURFACES
    7. 5.7 RATIONAL B-SPLINE SURFACES
    8. 5.8 GEOMETRIC CONTINUITY FOR RATIONAL PATCHES
    9. 5.9 INTERPOLATION AND APPROXIMATION ALGORITHMS
    10. 5.10 RATIONAL SURFACE CONSTRUCTIONS
    11. 5.11 CONCLUDING REMARKS
  12. Chapter 6: Spline Basics
    1. 6.1 PIECEWISE POLYNOMIALS
    2. 6.2 B-SPLINES DEFINED
    3. 6.3 SUPPORT AND POSITIVITY
    4. 6.4 SPLINE SPACES DEFINED
    5. 6.5 SPECIFIC KNOT SEQUENCES
    6. 6.6 THE POLYNOMIALS IN THE SPLINE SPACE: MARSDEN ’S IDENTITY
    7. 6.7 THE PIECEWISE POLYNOMIALS IN THE SPLINE SPACE
    8. 6.8 DUAL FUNCTIONALS AND BLOSSOMS
    9. 6.9 GOOD CONDITION
    10. 6.10 CONVEX HULL
    11. 6.11 DIFFERENTIATION AND INTEGRATION
    12. 6.12 EVALUATION
    13. 6.13 SPLINE FUNCTIONS VS SPLINE CURVES
    14. 6.14 KNOT INSERTION
    15. 6.15 VARIATION DIMINUTION AND SHAPE PRESERVATION: SCHOENBERG’S OPERATOR
    16. 6.16 ZEROS OF A SPLINE, COUNTING MULTIPLICITY
    17. 6.17 SPLINE INTERPOLATION: SCHOENBERG-WHITNEY
    18. 6.18 SMOOTHING SPLINE
    19. 6.19 LEAST-SQUARES SPLINE APPROXIMATION
    20. BACKGROUND 6.20.
  13. Chapter 7: Curve and Surface Constructions
    1. 7.1 INTRODUCTION
    2. 7.2 POLYNOMIAL CURVE METHODS
    3. 7.3 C2 CUBIC SPLINE INTERPOLATION
    4. 7.4 POLYNOMIAL SURFACE METHODS
    5. 7.5 C2 BICUBIC SPLINE INTERPOLATION
    6. 7.6 VOLUME DEFORMATIONS
  14. Chapter 8: Geometric Continuity
    1. 8.1 MOTIVATING EXAMPLES
    2. 8.2 GEOMETRIC CONTINUITY OF PARAMETRIC CURVES AND SURFACES
    3. 8.3 EQUIVALENT AND ALTERNATIVE DEFINITIONS
    4. 8.4 CONSTRUCTIONS
    5. 8.5 ADDITIONAL LITERATURE
  15. Chapter 9: Splines on Surfaces
    1. 9.1 INTRODUCTION
    2. 9.2 SCALAR SPLINES ON SMOOTH SURFACES
    3. 9.3 ALTERNATIVE METHODS FOR FUNCTIONS ON SURFACES
  16. Chapter 10: Box Splines
    1. 10.1 BOX SPLINES
    2. 10.2 BOX SPLINE SURFACES
    3. 10.3 HALF-BOX SPLINES
    4. 10.4 HALF-BOX SPLINE SURFACES
  17. Chapter 11: Finite Element Approximation with Splines
    1. 11.1 INTRODUCTION
    2. 11.2 SPLINES ON UNIFORM GRIDS
    3. 11.3 FINITE ELEMENT BASES
    4. 11.4 APPROXIMATION OF BOUNDARY VALUE PROBLEMS
    5. 11.5 SUMMARY
    6. Acknowledgement.
  18. Chapter 12: Subdivision Surfaces
    1. 12.1 SUBDIVISION SURFACE DEFINITIONS
    2. 12.2 INTRODUCTION - SUBDIVISION CURVES
    3. 12.3 BOX-SPLINES
    4. 12.4 GENERALIZATIONS TO ARBITRARY TOPOLOGY
    5. 12.5 SOME SPECIFIC SCHEMES
    6. 12.6 ANALYSIS OF CONTINUITY AT THE SINGULARITIES
    7. 12.7 FIRST STEP ARTIFACTS
    8. 12.8 CURRENT RESEARCH DIRECTIONS
    9. 12.9 CONCLUSIONS
  19. Chapter 13: Interrogation of Subdivision Surfaces
    1. 13.1 SUBDIVISION SURFACE INTERROGATIONS
    2. 13.2 HISTORICAL BACKGROUND
    3. 13.3 THE CONVEX HULL PROPERTY
    4. 13.4 AN API FOR SUBDIVISION SURFACES
    5. 13.5 EXAMPLE INTERROGATIONS
    6. 13.6 PERFORMANCE ISSUES
    7. 13.7 CONCLUSIONS
  20. Chapter 14: Multiresolution Techniques
    1. 14.1 INTRODUCTION
    2. 14.2 MULTIRESOLUTION REPRESENTATIONS FOR CURVES
    3. 14.3 LIFTING
    4. 14.4 GEOMETRIC SETTING
    5. 14.5 MULTIRESOLUTION REPRESENTATIONS FOR SURFACES
    6. 14.6 APPLICATIONS
  21. Chapter 15: Algebraic Methods for Computer Aided Geometric Design
    1. 15.1 INTRODUCTION
    2. 15.2 POLYNOMIALS, IDEALS, AND VARIETIES
    3. 15.3 RESULTANTS
    4. 15.4 CURVE IMPLICITIZATION AND INVERSION
    5. 15.5 CURVE PARAMETRIZATION
    6. 15.6 INTERSECTION COMPUTATIONS
    7. 15.7 SURFACES
    8. 15.8 OTHER ISSUES
  22. Chapter 16: Scattered Data Interpolation: Radial Basis and Other Methods
    1. 16.1 INTRODUCTION
    2. 16.2 RADIAL INTERPOLATION
    3. 16.3 OTHER LOCAL METHODS
    4. 16.4 CONCLUSIONS
  23. Chapter 17: Pythagorean-Hodograph Curves
    1. 17.1 PREAMBLE
    2. 17.2 POLYNOMIAL PH CURVES
    3. 17.3 CONSTRUCTION ALGORITHMS
    4. 17.4 REAL-TIME CNC INTERPOLATORS
    5. 17.5 RATIONAL CURVES WITH RATIONAL OFFSETS
    6. 17.6 MINKOWSKI PH CURVES
    7. 17.7 CLOSURE
  24. Chapter 18: Voronoi Diagrams
    1. 18.1 ORDINARY VORONOI DIAGRAM
    2. 18.2 DELAUNAY DIAGRAM
    3. 18.3 BASIC PROPERTIES OF THE VORONOI AND DELAUNAY DIAGRAMS
    4. 18.4 ALGORITHMS
    5. 18.5 APPLICATIONS
    6. 18.6 EXTENSIONS
    7. 18.7 CONCLUSION
  25. Chapter 19: The Medial Axis Transform
    1. 19.1 INTRODUCTION
    2. 19.2 MATHEMATICAL THEORY OF THE MEDIAL AXIS TRANSFORM
    3. 19.3 ALGORITHMS
    4. 19.4 CONCLUDING REMARKS
  26. Chapter 20: Solid Modeling
    1. 20.1 INTRODUCTION
    2. 20.2 MATHEMATICAL MODELS
    3. 20.3 COMPUTER REPRESENTATIONS
    4. 20.4 ALGORITHMS
    5. 20.5 APPLICATIONS
    6. 20.6 SYSTEMS
    7. 20.7 CONCLUSIONS
  27. Chapter 21: Parametric Modeling
    1. 21.1 INTRODUCTION
    2. 21.2 PARAMETRIC MODELS
    3. 21.3 VARIANT MODELING
    4. 21.4 CONSTRAINT-BASED MODELING
    5. 21.5 FEATURE-BASED MODELING
    6. 21.6 TRENDS
    7. 21.7 OPEN PROBLEMS
  28. Chapter 22: Sculptured Surface NC Machining
    1. 22.1 INTRODUCTION
    2. 22.2 UNIT MACHINING OPERATIONS
    3. 22.3 INTERFERENCE HANDLING
    4. 22.4 TOOL PATH GENERATION METHODS AND CONSEQUENT GEOMETRIC ISSUES
    5. 22.5 GEOMETRIC ALGORITHMS
    6. 22.6 CONCLUSION
  29. Chapter 23: Cyclides
    1. 23.1 INTRODUCTION
    2. 23.2 THE GEOMETRY OF DUPIN CYCLIDES
    3. 23.3 SUPERCYCLIDES
    4. 23.4 CYCLIDES IN CAGD
    5. 23.5 APPENDIX: STUDYING DUPIN CYCLIDES WITH LIE GEOMETRY
  30. Chapter 24: Geometry Processing
    1. 24.1 INTRODUCTION
    2. 24.2 ROOT FINDING
    3. 24.3 INTEGRATION
    4. 24.4 COMPUTING MASS PROPERTIES
  31. Chapter 25: Intersection Problems
    1. 25.1 INTRODUCTION
    2. 25.2 CLASSIFICATION OF INTERSECTION PROBLEMS
    3. 25.3 OVERVIEW OF NONLINEAR SOLVERS
    4. 25.4 CURVE/SURFACE INTERSECTION
    5. 25.5 SURFACE/SURFACE INTERSECTIONS
    6. 25.6 CONCLUSION
  32. Chapter 26: Reverse Engineering
    1. 26.1 INTRODUCTION
    2. 26.2 THE BASIC PHASES OF REVERSE ENGINEERING
    3. 26.3 DATA CAPTURE
    4. 26.4 TRIANGULATION AND DECIMATION
    5. 26.5 RECONSTRUCTING FREE-FORM OBJECTS
    6. 26.6 RECONSTRUCTING CONVENTIONAL ENGINEERING OBJECTS
    7. 26.7 CONCLUSION
  33. Chapter 27: Vector and Tensor Field Visualization
    1. 27.1 INTRODUCTION
    2. 27.2 VISUALIZATION PROCESS
    3. 27.3 DATA SET TYPES AND INTERPOLATION METHODS
    4. 27.4 DIRECT MAPPINGS TO GEOMETRIC PRIMITIVES
    5. 27.5 ATTRIBUTE MAPPINGS
    6. 27.6 STRUCTURE AND FEATURE BASED MAPPINGS
  34. Chapter 28: Splines over Triangulations
    1. 28.1 INTRODUCTION
    2. 28.2 BERNSTEIN-B éZIER TECHNIQUES
    3. 28.3 DIMENSION OF SPLINE SPACES
    4. 28.4 FINITE ELEMENT AND MACRO ELEMENT METHODS
    5. 28.5 INTERPOLATION BY SPLINE SPACES
    6. 28.6 TRIANGULAR B-SPLINES
    7. Acknowledgment.
  35. Chapter 29: Kinematics and Animation
    1. 29.1 INTRODUCTION
    2. 29.2 THE KINEMATICAL MAPPING
    3. 29.6 SPATIAL RATIONAL MOTIONS
    4. 29.7 CLOSURE
  36. Chapter 30: Direct Rendering of Freeform Surfaces
    1. 30.1 INTRODUCTION
    2. 30.2 SCAN-CONVERSION OF CURVES
    3. 30.3 SURFACE COVERAGE AND RENDERING USING CURVES
    4. 30.4 RAY-TRACING
    5. 30.5 EXTENSIONS
    6. 30.6 CONCLUSION
  37. Chapter 31: Modeling and Processing with Quadric Surfaces
    1. 31.1 DEFINITION AND CLASSIFICATIONS
    2. 31.2 PARAMETRIC REPRESENTATION
    3. 31.3 FITTING, BLENDING, AND OFFSETTING
    4. 31.4 INTERSECTION AND INTERFERENCE
    5. 31.5 ACKNOWLEDGMENTS
  38. Index