An Introduction to NURBS

An Introduction to NURBS

With Historical Perspective

1st Edition - July 21, 2000
This is the Latest Edition
  • Author: David Rogers
  • eBook ISBN: 9780080509204
  • Hardcover ISBN: 9781558606692

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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.

Table of 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 - 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



Product details

  • No. of pages: 344
  • Language: English
  • Copyright: © Morgan Kaufmann 2000
  • Published: July 21, 2000
  • Imprint: Morgan Kaufmann
  • eBook ISBN: 9780080509204
  • Hardcover ISBN: 9781558606692
  • About the Author

    David Rogers

    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.

    Latest reviews

    (Total rating for all reviews)

    • WolfgangGeiss Tue Sep 03 2019

      Excellent book about NURBS. Easy

      Excellent book about NURBS. Easy written and understandable. Sources are available to learn how to implement NURBS.