An Introduction to NURBS

by

David F. Rogers


Table of Contents

Chapter 1   Curve and Surface Representation
Chapter 2   Bezier Curves
Chapter 3   B-spline Curves
Chapter 4   Rational B-spline Curves
Chapter 5   Bezier Surfaces
Chapter 6   B-spline Surfaces
Chapter 7   Rational B-spline Surfaces
Appendices
 
Preface
 
Chapter 1   Curve and Surface Representation
 
1.1Introduction
1.2Parametric Curves
   Extension to Three Dimensions
   Parametric Line
1.3Parametric Surfaces
1.4Piecewise Surfaces
1.5Continuity
   Geometric Continuity
   Parametric Continuity
 
Historical Perspective
  Bezier Curves: Robin Forrest    Top of page
 
Chapter 2   Bezier Curves
 
2.1Bezier Curve Definition
   Bezier Curve Algorithm
2.2Matrix Representation of Bezier Curves
2.3Bezier Curve Derivatives
2.4Continuity Between Bezier Curves
2.5Increasing the Flexibility of Bezier Curves
   Degree Elevation
   Subdivision
 
Historical Perspective
  Biography: Pierre Bezier
  B-splines: Rich Riesenfeld    Top of page
 
Chapter 3   B-spline Curves
 
3.1B-spline Curve Definition
   Properties of B-spline Curves
3.2Convex Hull Properties of B-spline Curves
3.3Knot Vectors
3.4B-spline Basis Functions
   B-spline Curve Controls
3.5Open B-spline Curves
3.6Nonuniform B-spline Curves
3.7Periodic B-spline Curves
3.8Matrix Formulation of B-spline Curves
3.9End Conditions for Periodic B-spline Curves
   Start and End Points
   Start and End Point Derivatives
   Controlling Start and End Points
    Multiple Coincident Vertices
    Pseudovertices
3.10B-spline Curve Derivatives
3.11B-spline Curve Fitting
3.12Degree Elevation
   Algorithms
3.13Degree Reduction
   Bezier Curve Degree Reduction
3.14Knot Insertion and B-spline Curve Subdivision
3.15Knot Removal
   Pseudocode
3.16Reparameterization
 
Historical Perspective
  Subdivision: Elaine Cohen, Tom Lyche and Rich Riesenfeld    Top of page
 
Chapter 4   Rational B-spline Curves
 
4.1Rational B-spline Curves (NURBS)
   Characteristics of NURBS
4.2Rational B-spline Basis Functions and Curves
   Open Rational B-spline Basis Functions and Curves
   Periodic Rational B-spline Basis Functions and Curves
4.3Calculating Rational B-spline Curves
4.4Derivatives of NURBS Curves
4.5Conic Sections
 
  Historical Perspective
    Rational B-splines: Lewis Knapp    Top of page
 
Chapter 5   Bezier Surfaces
 
5.1Mapping Parametric Surfaces
5.2Bezier Surface Definition and Characteristic
   Matrix Representation
5.3Bezier Surface Derivatives
5.4Transforming Between Surface Descriptions
 
  Historical Perspective
    Nonuniform Rational B-splines: Ken Versprill    Top of page
 
Chapter 6   B-spline Surfaces
 
6.1B-spline Surfaces
6.2Convex Hull Properties
6.3Local Control
6.4Calculating Open B-spline Surfaces
6.5Periodic B-spline Surfaces
6.6Matrix Formulation of B-spline Surfaces
6.7B-spline Surface Derivatives
6.8B-spline Surface Fitting
6.9B-spline Surface Subdivision
6.10Gaussian Curvature and Surface Fairness
 
Historical Perspective
  Implementation: Al Adams and Dave Rogers    Top of page
 
Chapter 7   Rational B-spline Surfaces
 
7.1Rational B-spline Surfaces (NURBS)
7.2Characteristics of Rational B-spline Surfaces
   Positive Homogeneous Weighting Factors
   Negative Homogeneous Weighting Factors
   Internally Nonuniform Knot Vector
   Reparameterization
7.3A Simple Rational B-spline Surface Algorithm
7.4Derivatives of Rational B-spline Surfaces
7.5Bilinear Surfaces
7.6Sweep Surfaces
7.7Ruled Rational B-spline Surfaces
   Developable Surfaces
7.8Surfaces of Revolution
7.9Blending Surfaces
7.10A Fast Rational B-spline Surface Algorithm
   Naive Algorithms
   A More Efficient Algorithm
   Incremental Surface Calculation
   Measure of Computational Effort    Top of page
 
  Appendices
 
A B-spline Surface File Format
BProblems
CAlgorithms    Top of page
 
  References
 
  Index
 
  About the Author

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Copyright © 2000 David F. Rogers. All rights reserved.