Table of Contents

Cover image

Title page

Copyright

Preface

Contributors

Chapter 1: A History of Curves and Surfaces in CAGD

1.1 INTRODUCTION

1.2 EARLY DEVELOPMENTS

1.3 DE CASTELJAU AND BÉZIER

1.4 PARAMETRIC CURVES

1.5 RECTANGULAR SURFACES

1.6 B-SPLINE CURVES AND NURBS

1.7 TRIANGULAR PATCHES

1.8 SUBDIVISION SURFACES

1.9 SCIENTIFIC APPLICATIONS

1.10 SHAPE

1.11 INFLUENCES AND APPLICATIONS

Chapter 2: Geometric Fundamentals

2.1 AFFINE FUNDAMENTALS

2.2 CONIC SECTIONS AND QUADRICS

2.3 THE EUCLIDEAN SPACE

2.4 PROJECTIVE FUNDAMENTALS

2.5 DUALITY

2.6 OSCULATING CURVES AND SURFACES

2.7 DIFFERENTIAL FUNDAMENTALS

Chapter 3: Geometries for CAGD

3.1 CURVES AND SURFACES IN PROJECTIVE GEOMETRY

3.2 SPHERE GEOMETRIES

3.3 LINE GEOMETRY

3.4 APPROXIMATION IN SPACES OF GEOMETRIC OBJECTS

3.5 NON-EUCLIDEAN GEOMETRIES

Chapter 4: Bézier Techniques

4.1 WHY B éZIER TECHNIQUES?

4.2 BÉZIER CURVES

4.3 RECTANGULAR BÉZIER PATCHES

4.4 TRIANGULAR B éZIER PATCHES

Chapter 5: Rational Techniques

5.1 INTRODUCTION

5.2 RATIONAL B éZIER CURVES

5.3 RATIONAL B-SPLINE CURVES

5.4 GEOMETRIC CONTINUITY FOR RATIONAL CURVES

5.5 RATIONAL CURVE APPROXIMATION AND INTERPOLATION

5.6 RATIONAL B éZIER SURFACES

5.7 RATIONAL B-SPLINE SURFACES

5.8 GEOMETRIC CONTINUITY FOR RATIONAL PATCHES

5.9 INTERPOLATION AND APPROXIMATION ALGORITHMS

5.10 RATIONAL SURFACE CONSTRUCTIONS

5.11 CONCLUDING REMARKS

Chapter 6: Spline Basics

6.1 PIECEWISE POLYNOMIALS

6.2 B-SPLINES DEFINED

6.3 SUPPORT AND POSITIVITY

6.4 SPLINE SPACES DEFINED

6.5 SPECIFIC KNOT SEQUENCES

6.6 THE POLYNOMIALS IN THE SPLINE SPACE: MARSDEN ’S IDENTITY

6.7 THE PIECEWISE POLYNOMIALS IN THE SPLINE SPACE

6.8 DUAL FUNCTIONALS AND BLOSSOMS

6.9 GOOD CONDITION

6.10 CONVEX HULL

6.11 DIFFERENTIATION AND INTEGRATION

6.12 EVALUATION

6.13 SPLINE FUNCTIONS VS SPLINE CURVES

6.14 KNOT INSERTION

6.15 VARIATION DIMINUTION AND SHAPE PRESERVATION: SCHOENBERG’S OPERATOR

6.16 ZEROS OF A SPLINE, COUNTING MULTIPLICITY

6.17 SPLINE INTERPOLATION: SCHOENBERG-WHITNEY

6.18 SMOOTHING SPLINE

6.19 LEAST-SQUARES SPLINE APPROXIMATION

BACKGROUND 6.20.

Chapter 7: Curve and Surface Constructions

7.1 INTRODUCTION

7.2 POLYNOMIAL CURVE METHODS

7.3 C2 CUBIC SPLINE INTERPOLATION

7.4 POLYNOMIAL SURFACE METHODS

7.5 C2 BICUBIC SPLINE INTERPOLATION

7.6 VOLUME DEFORMATIONS

Chapter 8: Geometric Continuity

8.1 MOTIVATING EXAMPLES

8.2 GEOMETRIC CONTINUITY OF PARAMETRIC CURVES AND SURFACES

8.3 EQUIVALENT AND ALTERNATIVE DEFINITIONS

8.4 CONSTRUCTIONS

8.5 ADDITIONAL LITERATURE

Chapter 9: Splines on Surfaces

9.1 INTRODUCTION

9.2 SCALAR SPLINES ON SMOOTH SURFACES

9.3 ALTERNATIVE METHODS FOR FUNCTIONS ON SURFACES

Chapter 10: Box Splines

10.1 BOX SPLINES

10.2 BOX SPLINE SURFACES

10.3 HALF-BOX SPLINES

10.4 HALF-BOX SPLINE SURFACES

Chapter 11: Finite Element Approximation with Splines

11.1 INTRODUCTION

11.2 SPLINES ON UNIFORM GRIDS

11.3 FINITE ELEMENT BASES

11.4 APPROXIMATION OF BOUNDARY VALUE PROBLEMS

11.5 SUMMARY

Acknowledgement.

Chapter 12: Subdivision Surfaces

12.1 SUBDIVISION SURFACE DEFINITIONS

12.2 INTRODUCTION - SUBDIVISION CURVES

12.3 BOX-SPLINES

12.4 GENERALIZATIONS TO ARBITRARY TOPOLOGY

12.5 SOME SPECIFIC SCHEMES

12.6 ANALYSIS OF CONTINUITY AT THE SINGULARITIES

12.7 FIRST STEP ARTIFACTS

12.8 CURRENT RESEARCH DIRECTIONS

12.9 CONCLUSIONS

Chapter 13: Interrogation of Subdivision Surfaces

13.1 SUBDIVISION SURFACE INTERROGATIONS

13.2 HISTORICAL BACKGROUND

13.3 THE CONVEX HULL PROPERTY

13.4 AN API FOR SUBDIVISION SURFACES

13.5 EXAMPLE INTERROGATIONS

13.6 PERFORMANCE ISSUES

13.7 CONCLUSIONS

Chapter 14: Multiresolution Techniques

14.1 INTRODUCTION

14.2 MULTIRESOLUTION REPRESENTATIONS FOR CURVES

14.3 LIFTING

14.4 GEOMETRIC SETTING

14.5 MULTIRESOLUTION REPRESENTATIONS FOR SURFACES

14.6 APPLICATIONS

Chapter 15: Algebraic Methods for Computer Aided Geometric Design

15.1 INTRODUCTION

15.2 POLYNOMIALS, IDEALS, AND VARIETIES

15.3 RESULTANTS

15.4 CURVE IMPLICITIZATION AND INVERSION

15.5 CURVE PARAMETRIZATION

15.6 INTERSECTION COMPUTATIONS

15.7 SURFACES

15.8 OTHER ISSUES

Chapter 16: Scattered Data Interpolation: Radial Basis and Other Methods

16.1 INTRODUCTION

16.2 RADIAL INTERPOLATION

16.3 OTHER LOCAL METHODS

16.4 CONCLUSIONS

Chapter 17: Pythagorean-Hodograph Curves

17.1 PREAMBLE

17.2 POLYNOMIAL PH CURVES

17.3 CONSTRUCTION ALGORITHMS

17.4 REAL-TIME CNC INTERPOLATORS

17.5 RATIONAL CURVES WITH RATIONAL OFFSETS

17.6 MINKOWSKI PH CURVES

17.7 CLOSURE

Chapter 18: Voronoi Diagrams

18.1 ORDINARY VORONOI DIAGRAM

18.2 DELAUNAY DIAGRAM

18.3 BASIC PROPERTIES OF THE VORONOI AND DELAUNAY DIAGRAMS

18.4 ALGORITHMS

18.5 APPLICATIONS

18.6 EXTENSIONS

18.7 CONCLUSION

Chapter 19: The Medial Axis Transform

19.1 INTRODUCTION

19.2 MATHEMATICAL THEORY OF THE MEDIAL AXIS TRANSFORM

19.3 ALGORITHMS

19.4 CONCLUDING REMARKS

Chapter 20: Solid Modeling

20.1 INTRODUCTION

20.2 MATHEMATICAL MODELS

20.3 COMPUTER REPRESENTATIONS

20.4 ALGORITHMS

20.5 APPLICATIONS

20.6 SYSTEMS

20.7 CONCLUSIONS

Chapter 21: Parametric Modeling

21.1 INTRODUCTION

21.2 PARAMETRIC MODELS

21.3 VARIANT MODELING

21.4 CONSTRAINT-BASED MODELING

21.5 FEATURE-BASED MODELING

21.6 TRENDS

21.7 OPEN PROBLEMS

Chapter 22: Sculptured Surface NC Machining

22.1 INTRODUCTION

22.2 UNIT MACHINING OPERATIONS

22.3 INTERFERENCE HANDLING

22.4 TOOL PATH GENERATION METHODS AND CONSEQUENT GEOMETRIC ISSUES

22.5 GEOMETRIC ALGORITHMS

22.6 CONCLUSION

Chapter 23: Cyclides

23.1 INTRODUCTION

23.2 THE GEOMETRY OF DUPIN CYCLIDES

23.3 SUPERCYCLIDES

23.4 CYCLIDES IN CAGD

23.5 APPENDIX: STUDYING DUPIN CYCLIDES WITH LIE GEOMETRY

Chapter 24: Geometry Processing

24.1 INTRODUCTION

24.2 ROOT FINDING

24.3 INTEGRATION

24.4 COMPUTING MASS PROPERTIES

Chapter 25: Intersection Problems

25.1 INTRODUCTION

25.2 CLASSIFICATION OF INTERSECTION PROBLEMS

25.3 OVERVIEW OF NONLINEAR SOLVERS

25.4 CURVE/SURFACE INTERSECTION

25.5 SURFACE/SURFACE INTERSECTIONS

25.6 CONCLUSION

Chapter 26: Reverse Engineering

26.1 INTRODUCTION

26.2 THE BASIC PHASES OF REVERSE ENGINEERING

26.3 DATA CAPTURE

26.4 TRIANGULATION AND DECIMATION

26.5 RECONSTRUCTING FREE-FORM OBJECTS

26.6 RECONSTRUCTING CONVENTIONAL ENGINEERING OBJECTS

26.7 CONCLUSION

Chapter 27: Vector and Tensor Field Visualization

27.1 INTRODUCTION

27.2 VISUALIZATION PROCESS

27.3 DATA SET TYPES AND INTERPOLATION METHODS

27.4 DIRECT MAPPINGS TO GEOMETRIC PRIMITIVES

27.5 ATTRIBUTE MAPPINGS

27.6 STRUCTURE AND FEATURE BASED MAPPINGS

Chapter 28: Splines over Triangulations

28.1 INTRODUCTION

28.2 BERNSTEIN-B éZIER TECHNIQUES

28.3 DIMENSION OF SPLINE SPACES

28.4 FINITE ELEMENT AND MACRO ELEMENT METHODS

28.5 INTERPOLATION BY SPLINE SPACES

28.6 TRIANGULAR B-SPLINES

Acknowledgment.

Chapter 29: Kinematics and Animation

29.1 INTRODUCTION

29.2 THE KINEMATICAL MAPPING

29.6 SPATIAL RATIONAL MOTIONS

29.7 CLOSURE

Chapter 30: Direct Rendering of Freeform Surfaces

30.1 INTRODUCTION

30.2 SCAN-CONVERSION OF CURVES

30.3 SURFACE COVERAGE AND RENDERING USING CURVES

30.4 RAY-TRACING

30.5 EXTENSIONS

30.6 CONCLUSION

Chapter 31: Modeling and Processing with Quadric Surfaces

31.1 DEFINITION AND CLASSIFICATIONS

31.2 PARAMETRIC REPRESENTATION

31.3 FITTING, BLENDING, AND OFFSETTING

31.4 INTERSECTION AND INTERFERENCE

31.5 ACKNOWLEDGMENTS

Index