An Introduction to NURBS: With Historical Perspective

The latest from a computer graphics pioneer, An Introduction to NURBS is the ideal resource for anyone seeking a theoretical and practical understanding of these very important curves and surfaces. Beginning with Bezier curves, the book develops a lucid explanation of NURBS curves, then does the same for surfaces, consistently stressing important shape design properties and the capabilities of each curve and surface type. Throughout, it relies heavily on illustrations and fully worked examples that will help you grasp key NURBS concepts and deftly apply them in your work. Supplementing the lucid, point-by-point instructions are illuminating accounts of the history of NURBS, written by some of its most prominent figures.

Whether you write your own code or simply want deeper insight into how your computer graphics application works, An Introduction to NURBS will enhance and extend your knowledge to a degree unmatched by any other resource.

Features:

• Presents vital information with applications in many different areas: CAD, scientific visualization, animation, computer games, and more.
• Facilitates accessiblity to anyone with a knowledge of first-year undergraduate mathematics.
• Details specific NURBS-based techniques, including making cusps with B-spline curves and conic sections with rational B-spline curves.
• Presents all important algorithms in easy-to-read pseudocode-useful for both implementing them and understanding how they work.
• Includes complete references to additional NURBS resources.

Contents

Preface

Chapter 1 - Curve and Surface Representation

1.1 Introduction

1.2 Parametric Curves

Extension to Three Dimensions

Parametric Line

1.3 Parametric Surfaces

1.4 Piecewise Surfaces

1.5 Continuity

Geometric Continuity

Parametric Continuity

Historical Perspective - Bezier Curves: A.R. Forrest

Chapter 2 - Bezier Curves

2.1 Bezier Curve Deffnition

Bezier Curve Algorithm

2.2 Matrix Representation of Bezier Curves

2.3 Bezier Curve Derivatives

2.4 Continuity Between Bezier Curves

2.5 Increasing the Flexibility of Bezier Curves

Degree Raising

Subdivision

Historical Perspective - B-splines: Richard F. Riesenfeld

Chapter 3 - B-spline Curves

3.1 B-spline Curve Deffnition

Properties of B-spline Curves

3.2 Convex Hull Properties of B-spline Curves

3.3 Knot Vectors

3.4 B-spline Basis Functions

B-spline Curve Controls

3.5 Open B-spline Curves

3.6 Nonuniform B-spline Curves

3.7 Periodic B-spline Curves

3.8 Matrix Formulation of B-spline Curves

3.9 End Conditions For Periodic B-spline Curves

Start and End Points

Start and End Point Derivatives

Controlling Start and End Points

Multiple Coincident Vertices

Pseudovertices

3.10 B-spline Curve Derivatives

3.11 B-spline Curve Fitting

3.12 Degree Elevation

Algorithms

3.13 Degree Reduction

Bezier Curve Degree Reduction

3.14 Knot Insertion and B-spline Curve Subdivision

3.15 Knot Removal

Pseudocode

3.16 Reparameterization

Historical Perspective - Subdivision: Tom Lyche, Elaine Cohen and Richard F. Riesenfeld

Chapter 4 - Rational B-spline Curves

4.1 Rational B-spline Curves (NURBS Curves)

Characteristics of NURBS

4.2 Rational B-spline Basis Functions and Curves

Open Rational B-spline Basis Functions and Curves

Periodic Rational B-spline Basis Functions and Curves

4.3 Calculating Rational B-spline Curves

4.4 Derivatives of NURBS Curves

4.5 Conic Sections

Historical Perspective - Rational B-splines: Lewis C. Knapp

Chapter 5 - Bezier Surfaces

5.1 Mapping Parametric Surfaces

5.2 Bezier Surfaces

Matrix Representation

5.3 Bezier Surface Derivatives

5.4 Transforming Between Surface Descriptions

Historical Perspective - Nonuniform Rational B-splines: Kenneth J. Versprille

Chapter 6 - B-spline Surfaces

6.1 B-spline Surfaces

6.2 Convex Hull Properties

6.3 Local Control

6.4 Calculating Open B-spline Surfaces

6.5 Periodic B-spline Surfaces

6.6 Matrix Formulation of B-spline Surfaces

6.7 B-spline Surface Derivatives

6.8 B-spline Surface Fitting

6.9 B-spline Surface Subdivision

6.10 Gaussian Curvature and Surface Fairness

Historical Perspective - Implementation: David F. Rogers

Chapter 7 - Rational B-spline Surfaces

7.1 Rational B-spline Surfaces (NURBS)

7.2 Characteristics of Rational B-spline Surfaces

Effects of positive homogeneous weighting factors on a single vertex

Effects of negative homogeneous weighting factors

Effects of internally nonuniform knot vector

Reparameterization

7.3 A Simple Rational B-spline Surface Algorithm

7.4 Derivatives of Rational B-spline Surfaces

7.5 Bilinear Surfaces

7.6 Sweep Surfaces

7.7 Ruled Rational B-spline Surfaces

Developable Surfaces

7.8 Surfaces of Revolution

7.9 Blending Surfaces

7.10 A Fast Rational B-spline Surface Algorithm

Naive Algorithms

A More Effcient Algorithm

Incremental Surface Calculation

Measure of Computational Effort

Appendices

A B-spline Surface File Format

B Problems and Projects

C Algorithms

Index