Curves and Surfaces
for CAGD
A Practical Guide
5th edition
Published by
Morgan-Kaufmann
Published 2002
499 pages
ISBN 1-55860-737-4
Table
of Contents
1 P. Bezier:
How a Simple System Was Born
2 Introductory
Material
2.1 Points and
Vectors 2.2 Affine Maps 2.3 Linear Interpolation 2.4 Piecewise Linear
Interpolation 2.5 Menelaos' Theorem 2.6 Barycentric Coordinates in
the Plane 2.7 Tessellations and Triangulations 2.8 Function Spaces
2.9 Problems
3 The de Casteljau
Algorithm
3.1 Parabolas 3.2
The de Casteljau Algorithm 3.3 Some Properties of Bezier Curves 3.4
The Blossom 3.5 Implementation 3.6 Problems
4 The Bernstein
Form of a Bezier Curve
4.1 Bernstein Polynomials
4.2 Properties of Bezier Curves 4.3 The Derivative of a Bezier Curve
4.4 Higher Order Derivatives 4.5 Derivatives and the de Casteljau
Algorithm 4.6 Subdivision 4.7 Blossom and Polar 4.8 The Matrix Form
of a Bezier Curve 4.9 Implementation 4.10 Problems
5 Bezier Curve
Topics
5.1 Degree Elevation
5.2 Repeated Degree Elevation 5.3 The Variation Diminishing Property
5.4 Degree Reduction 5.5 Nonparametric Curves 5.6 Cross Plots 5.7
Integrals 5.8 The Bezier Form of a Bezier Curve 5.9 The Barycentric
Form of a Bezier Curve 5.10 The Weierstrass Approximation Theorem
5.11 Formulas for Bernstein Polynomials 5.12 Implementation 5.13 Problems
6 Polynomial
Interpolation
6.1 Aitken's Algorithm
6.2 Lagrange Polynomials 6.3 The Vandermonde Approach 6.4 Limits of
Lagrange Interpolation 6.5 Cubic Hermite Interpolation 6.6 Quintic
Hermite Interpolation 6.7 The Newton Form and Forward Differencing
6.8 Implementation 6.9 Problems
7 Spline Curves
in Bezier Form
7.1 Global and
Local Parameters 7.2 Smoothness Conditions 7.3 C1 and C2 Continuity
7.4 Finding a C1 Parametrization 7.5 C1 Quadratic B-spline Curves
7.6 C2 Cubic B-spline Curves 7.7 Finding a Knot Sequence 7.8 Design
and Inverse Design 7.9 Implementation 7.10 Problems
8 Piecewise
Cubic Interpolation
8.1 C1 Piecewise
Cubic Hermite Interpolation 8.2 C1 Piecewise Cubic Interpolation I
8.3 C1 Piecewise Cubic Interpolation II 8.4 Point-Normal Interpolation
8.5 Font Design 8.6 Problems
9 Cubic Spline
Interpolation
9.1 The B-spline
Form 9.2 The Hermite Form 9.3 End Conditions 9.4 Finding a knot sequence
9.5 The Minimum Property 9.6 Implementation 9.7 Problems
10 B-splines
10.1 Motivation
10.2 Knot Insertion 10.3 The de Boor Algorithm 10.4 Smoothness of
B-spline Curves 10.5 The B-spline Basis 10.6 Two Recursion Formulas
10.7 Repeated Knot Insertion 10.8 B-spline Properties 10.9 B-spline
Blossoms 10.10 Approximation 10.11 B-spline Basics 10.12 Implementation
10.13 Problems
11 W. Boehm:
Differential Geometry I
11.1 Parametric
Curves and Arc Length 11.2 The Frenet Frame 11.3 Moving the Frame
11.4 The Osculating Circle 11.5 Nonparametric Curves 11.6 Composite
Curves
12 Geometric
Continuity
12.1 Motivation
12.2 The Direct Formulation 12.3 The gamma-formulation 12.4 The nu-
and beta- formulation 12.5 Comparison 12.6 G2 Cubic Splines 12.7 Interpolating
G2 cubic splines 12.8 Local Basis Functions for G2 Splines 12.9 Higher
Order Geometric Continuity 12.10 Implementation 12.11 Problems
14 Conic Sections
14.1 Projective
Maps of the Real Line 14.2 Conics as Rational Quadratics 14.3 A de
Casteljau Algorithm 14.4 Derivatives 14.5 The Implicit Form 14.6 Two
Classic Problems 14.7 Classification 14.8 Control Vectors 14.9 Implementation
14.10 Problems
15 Rational
Bezier and B-spline Curves
15.1 Rational Bezier
Curves 15.2 The de Casteljau Algorithm 15.3 Derivatives 15.4 Osculatory
Interpolation 15.5 Reparametrization and Degree Elevation 15.6 Control
Vectors 15.7 Rational Cubic B-spline Curves 15.8 Interpolation with
Rational Cubics 15.9 Rational B-splines of Arbitrary Degree 15.10
Implementation 15.11 Problems
16 Tensor
Product Patches
16.1 Bilinear Interpolation
16.2 The Direct de Casteljau Algorithm 16.3 The Tensor Product Approach
16.4 Properties 16.5 Degree Elevation 16.6 Derivatives 16.7 Blossoms
16.8 Normal Vectors 16.9 Twists 16.10 The Matrix Form of a Bezier
Patch 16.11 Nonparametric Patches 16.12 Tensor Product Interpolation
16.13 Bicubic Hermite Patches 16.14 Implementation 16.15 Problems
17 Composite
Surfaces and Spline Interpolation
17.1 Smoothness
and Subdivision 17.2 Tensor Product B-spline Surfaces 17.3 Twist Estimation
17.4 Bicubic Spline Interpolation 17.5 Finding Knot Sequences 17.6
Rational Bezier and B-spline Surfaces 17.7 Surfaces of Revolution
17.8 Volume Deformations 17.9 CONS and Trimmed Surfaces 17.10 Implementation
17.11 Problems
18 Bezier
Triangles
18.1 The de Casteljau
Algorithm 18.2 Triangular Blossoms 18.3 Bernstein Polynomials 18.4
Derivatives 18.5 Subdivision 18.6 Differentiability 18.7 Degree Elevation
18.8 Nonparametric Patches 18.9 Rational Bezier Triangles 18.10 Quadrics
18.11 Interpolation 18.11.1 Cubic and quintic interpolants 18.11.2
The Clough-Tocher interpolant 18.11.3 The Powell-Sabin interpolant
18.12 Implementation 18.13 Problems
19 Geometric
Continuity for Surfaces
19.1 Introduction
19.2 Triangle-Triangle 19.3 Rectangle-Rectangle 19.4 Rectangle-Triangle
19.5 ``Filling in'' Rectangular Patches 19.6 ``Filling in'' Triangular
Patches 19.7 Theoretical Aspects 19.8 Problems
20 Surfaces
with Arbitrary Topology
20.1 Doo-Sabin
Surfaces 20.2 Interpolation 20.3 S-Patches 20.4 Surface Splines 20.5
Problems
20 Coons Patches
20.1 Ruled Surfaces
20.2 Coons Patches: Bilinearly Blended 20.3 Coons Patches: Partially
Bicubically Blended 20.4 Coons Patches: Bicubically Blended 20.5 Piecewise
Coons Surfaces 20.6 Problems
21 Coons Patches:
Additional Material
21.1 Compatibility
21.2 Control Nets from Coons Patches 21.3 Translational Surfaces 21.4
Gordon Surfaces 21.5 Boolean Sums 21.6 Triangular Coons Patches 21.7
Implementation 21.8 Problems
22 W. Boehm:
Geometry II
22.1 Parametric
Surfaces and Arc Element 22.2 The Local Frame 22.3 The Curvature of
a Surface Curve 22.4 Meusnier's Theorem 22.5 Lines of Curvature 22.6
Gaussian and Mean Curvature 22.7 Euler's Theorem 22.8 Dupin's Indicatrix
22.9 Asymptotic Lines and Conjugate Directions 22.10 Ruled Surfaces
and Developables 22.11 Nonparametric Surfaces 22.12 Composite Surfaces
23 Interrogation
and Smoothing
23.1 Use of Curvature
Plots 23.2 Curve and Surface Smoothing 23.3 Surface Interrogation
23.4 Implementation 23.5 Problems
24 Evaluation
of Some Methods
24.1 Bezier Curves
or B-spline Curves? 24.2 Spline Curves or B-spline Curves? 24.3 The
Monomial or the Bezier Form? 24.4 The B-spline or the Hermite Form?
24.5 Triangular or Rectangular Patches?
25 Quick Reference
of Curve and Surface Terms
