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by Gerald Farin
Curves and Surfaces for CAGD, 5th Edition
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Title page
Table of Contents
The Morgan Kaufmann Series in Computer Graphics and Geometric Modeling
Copyright
Dedication
Preface
Chapter 1: P. Bézier: How a Simple System Was Born
Chapter 2: Introductory Material
2.1 Points and Vectors
2.2 Affine Maps
2.3 Constructing Affine Maps
2.4 Function Spaces
2.5 Problems
Chapter 3: Linear Interpolation
3.1 Linear Interpolation
3.2 Piecewise Linear Interpolation
3.3 Menelaos’ Theorem
3.4 Blossoms
3.5 Barycentric Coordinates in the Plane
3.6 Tessellations
3.7 Triangulations
3.8 Problems
Chapter 4: The de Casteljau Algorithm
4.1 Parabolas
4.2 The de Casteljau Algorithm
4.3 Some Properties of Bézier Curves
4.4 The Blossom
4.5 Implementation
4.6 Problems
Chapter 5: The Bernstein Form of a Bézier Curve
5.1 Bernstein Polynomials
5.2 Properties of Bézier Curves
5.3 The Derivatives of a Bézier Curve
5.4 Domain Changes and Subdivision
5.5 Composite Bézier Curves
5.6 Blossom and Polar
5.7 The Matrix Form of a Bézier Curve
5.8 Implementation
5.9 Problems
Chapter 6: Bézier Curve Topics
6.1 Degree Elevation
6.2 Repeated Degree Elevation
6.3 The Variation Diminishing Property
6.4 Degree Reduction
6.5 Nonparametric Curves
6.6 Cross Plots
6.7 Integrals
6.8 The Bézier Form of a Bézier Curve
6.9 The Weierstrass Approximation Theorem
6.10 Formulas for Bernstein Polynomials
6.11 Implementation
6.12 Problems
Chapter 7: Polynomial Curve Constructions
7.1 Aitken’s Algorithm
7.2 Lagrange Polynomials
7.3 The Vandermonde Approach
7.4 Limits of Lagrange Interpolation
7.5 Cubic Hermite Interpolation
7.6 Quintic Hermite Interpolation
7.7 Point-Normal Interpolation
7.8 Least Squares Approximation
7.9 Smoothing Equations
7.10 Designing with Bézier Curves
7.11 The Newton Form and Forward Differencing
7.12 Implementation
7.13 Problems
Chapter 8: B-Spline Curves
8.1 Motivation
8.2 B-Spline Segments
8.3 B-Spline Curves
8.4 Knot Insertion
8.5 Degree Elevation
8.6 Greville Abscissae
8.7 Smoothness
8.8 B-Splines
8.9 B-Spline Basics
8.10 Implementation
8.11 Problems
Chapter 9: Constructing Spline Curves
9.1 Greville Interpolation
9.2 Least Squares Approximation
9.3 Modifying B-Spline Curves
9.4 C2 Cubic Spline Interpolation
9.5 More End Conditions
9.6 Finding a Knot Sequence
9.7 The Minimum Property
9.8 C1 Piecewise Cubic Interpolation
9.9 Implementation
9.10 Problems
Chapter 10: W. Boehm: Differential Geometry I
10.1 Parametric Curves and Arc Length
10.2 The Frenet Frame
10.3 Moving the Frame
10.4 The Osculating Circle
10.5 Nonparametric Curves
10.6 Composite Curves
Chapter 11: Geometric Continuity
11.1 Motivation
11.2 The Direct Formulation
11.3 The γ, v, and β Formulations
11.4 G2 Cubic Splines
11.5 Interpolating G2 Cubic Splines
11.6 Higher-Order Geometric Continuity
11.7 Implementation
11.8 Problems
Chapter 12: Conic Sections
12.1 Projective Maps of the Real Line
12.2 Conics as Rational Quadratics
12.3 A de Casteljau Algorithm
12.4 Derivatives
12.5 The Implicit Form
12.6 Two Classic Problems
12.7 Classification
12.8 Control Vectors
12.9 Implementation
12.10 Problems
Chapter 13: Rational Bézier and B-Spline Curves
13.1 Rational Bézier Curves
13.2 The de Casteljau Algorithm
13.3 Derivatives
13.4 Osculatory Interpolation
13.5 Reparametrization and Degree Elevation
13.6 Control Vectors
13.7 Rational Cubic B-Spline Curves
13.8 Interpolation with Rational Cubics
13.9 Rational B-Splines of Arbitrary Degree
13.10 Implementation
13.11 Problems
Chapter 14: Tensor Product Patches
14.1 Bilinear Interpolation
14.2 The Direct de Casteljau Algorithm
14.3 The Tensor Product Approach
14.4 Properties
14.5 Degree Elevation
14.6 Derivatives
14.7 Blossoms
14.8 Curves on a Surface
14.9 Normal Vectors
14.10 Twists
14.11 The Matrix Form of a Bézier Patch
14.12 Nonparametric Patches
14.13 Problems
Chapter 15: Constructing Polynomial Patches
15.1 Ruled Surfaces
15.2 Coons Patches
15.3 Translational Surfaces
15.4 Tensor Product Interpolation
15.5 Bicubic Hermite Patches
15.6 Least Squares
15.7 Finding Parameter Values
15.8 Shape Equations
15.9 A Problem with Unstructured Data
15.10 Implementation
15.11 Problems
Chapter 16: Composite Surfaces
16.1 Smoothness and Subdivision
16.2 Tensor Product B-Spline Surfaces
16.3 Twist Estimation
16.4 Bicubic Spline Interpolation
16.5 Finding Knot Sequences
16.6 Rational Bézier and B-Spline Surfaces
16.7 Surfaces of Revolution
16.8 Volume Deformations
16.9 CONS and Trimmed Surfaces
16.10 Implementation
16.11 Problems
Chapter 17: Bézier Triangles
17.1 The de Casteljau Algorithm
17.2 Triangular Blossoms
17.3 Bernstein Polynomials
17.4 Derivatives
17.5 Subdivision
17.6 Differentiability
17.7 Degree Elevation
17.8 Nonparametric Patches
17.9 The Multivariate Case
17.10 S-Patches
17.11 Implementation
17.12 Problems
Chapter 18: Practical Aspects of Bézier Triangles
18.1 Rational Bézier Triangles
18.2 Quadrics
18.3 Interpolation
18.4 Cubic and Quintic Interpolants
18.5 The Clough–Tocher Interpolant
18.6 The Powell–Sabin Interpolant
18.7 Least Squares
18.8 Problems
Chapter 19: W. Boehm: Differential Geometry II
19.1 Parametric Surfaces and Arc Element
19.2 The Local Frame
19.3 The Curvature of a Surface Curve
19.4 Meusnier’s Theorem
19.5 Lines of Curvature
19.6 Gaussian and Mean Curvature
19.7 Euler’s Theorem
19.8 Dupin’s Indicatrix
19.9 Asymptotic Lines and Conjugate Directions
19.10 Ruled Surfaces and Developables
19.11 Nonparametric Surfaces
19.12 Composite Surfaces
Chapter 20: Geometric Continuity for Surfaces
20.1 Introduction
20.2 Triangle-Triangle
20.3 Rectangle-Rectangle
20.4 Rectangle-Triangle
20.5 “Filling in” Rectangular Patches
20.6 “Filling in” Triangular Patches
20.7 Theoretical Aspects
20.8 Problems
Chapter 21: Surfaces with Arbitrary Topology
21.1 Recursive Subdivision Curves
21.2 Doo–Sabin Surfaces
21.3 Catmull–Clark Subdivision
21.4 Midpoint Subdivision
21.5 Loop Subdivision
21.6 Subdivision
21.7 Interpolating Subdivision Surfaces
21.8 Surface Splines
21.9 Triangular Meshes
21.10 Decimation
21.11 Problems
Chapter 22: Coons Patches
22.1 Coons Patches: Bilinearly Blended
22.2 Coons Patches: Partially Bicubically Blended
22.3 Coons Patches: Bicubically Blended
22.4 Piecewise Coons Surfaces
22.5 Two Properties
22.6 Compatibility
22.7 Gordon Surfaces
22.8 Boolean Sums
22.9 Triangular Coons Patches
22.10 Problems
Chapter 23: Shape
23.1 Use of Curvature Plots
23.2 Curve and Surface Smoothing
23.3 Surface Interrogation
23.4 Implementation
23.5 Problems
Quick Reference of Curve and Surface Terms
List of Programs
Notation
References
Index
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