An Introduction to NURBS - 1st Edition - ISBN: 9781558606692, 9780080509204

An Introduction to NURBS

1st Edition

With Historical Perspective

Authors: David Rogers
eBook ISBN: 9780080509204
Hardcover ISBN: 9781558606692
Imprint: Morgan Kaufmann
Published Date: 21st July 2000
Page Count: 344
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Table of Contents


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 - Bézier Curves: A.R. Forrest

Chapter 2 - Bézier Curves

2.1 Bézier Curve Deffnition

Bézier Curve Algorithm

2.2 Matrix Representation of Bézier Curves

2.3 Bézier Curve Derivatives

2.4 Continuity Between Bézier Curves

2.5 Increasing the Flexibility of Bézier Curves

Degree Raising


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


3.10 B-spline Curve Derivatives

3.11 B-spline Curve Fitting

3.12 Degree Elevation


3.13 Degree Reduction

Bézier Curve Degree Reduction

3.14 Knot Insertion and B-spline Curve Subdivision

3.15 Knot Removal


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 - Bézier Surfaces

5.1 Mapping Parametric Surfaces

5.2 Bézier Surfaces

Matrix Representation

5.3 Bézier 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


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


A B-spline Surface File Format

B Problems and Projects

C Algorithms




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 Bézier 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.

Key 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.


Computer graphics professionals and CAD designers of all kinds, including: engineering designers, architectural engineers, professionals in engineering, scientific visualization, animation, and game development.


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© Morgan Kaufmann 2001
Morgan Kaufmann
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About the Authors

David Rogers Author

David F. Rogers, Ph.D., is the author of two computer graphics classics, Mathematical Elements for Computer Graphics and Procedural Elements for Computer Graphics, as well as works on fluid dynamics. His early research on the use of B-splines and NURBS for dynamic manipulation of ship hull surfaces led to significant commercial and scientific advances in a number of fields. Founder and former director of the Computer Aided Design/Interactive Graphics Group at the U.S. Naval Academy, Dr. Rogers was an original member of the USNA's Aerospace Engineering Department. He sits on the editorial boards of The Visual Computer and Computer Aided Design and serves on committees for SIGGRAPH, Computer Graphics International, and other conferences.

Affiliations and Expertise

The United States Naval Academy, Annapolis, Maryland, U.S.A.