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by M.-S. Kim, J. Hoschek, G. Farin
Handbook of Computer Aided Geometric Design
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Title page
Table of Contents
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
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