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An Introduction to NURBS: With Historical Perspective

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An Introduction to NURBS: With Historical Perspective

David Rogers

ISBN 1558606696
Pages 344


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.

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 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 Bézier 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 - 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 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 References Index