Front Cover -- Handbook of Computer Aided Geometric Design -- Copyright Page -- Preface -- Contents -- 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 BÉZIER TECHNIQUES? -- 4.2. BÉZIER CURVES -- 4.3. RECTANGULAR BÉZIER PATCHES -- 4.4. TRIANGULAR BÉZIER PATCHES -- Chapter 5. Rational Techniques -- 5.1. INTRODUCTION -- 5.2. RATIONAL BÉZIER CURVES -- 5.3. RATIONAL B-SPLINE CURVES -- 5.4. GEOMETRIC CONTINUITY FOR RATIONAL CURVES -- 5.5. RATIONAL CURVE APPROXIMATION AND INTERPOLATION -- 5.6. RATIONAL BÉZIER 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 -- 6.20. BACKGROUND -- 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/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 -- 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 -- 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-BÉZIER TECHNIQUES -- 28.3. DIMENSION -- 28.4. FINITE ELEMENT AND MACRO ELEMENTS -- 28.5. INTERPOLATION -- 28.6. TRIANGULAR B-SPLINES -- Chapter 29. Kinematics and Animation -- 29.1. INTRODUCTION -- 29.2. THE KINEMATICAL MAPPING -- 29.3. QUATERNIONS.

29.4. MOTION DESIGN USING CURVES ON S3 -- 29.5. SPHERICAL RATIONAL MOTIONS -- 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.