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## With Historical Perspective

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### David F. Rogers

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1st Edition - July 21, 2000

Author: David F. Rogers

Hardback ISBN:

9 7 8 - 1 - 5 5 8 6 0 - 6 6 9 - 2

eBook ISBN:

9 7 8 - 0 - 0 8 - 0 5 0 9 2 0 - 4

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… Read more

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Immediately download your ebook while waiting for your print delivery. No promo code is needed.

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.

- 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.

Preface Chapter 1 - Curve and Surface Representation1.1 Introduction1.2 Parametric CurvesExtension to Three Dimensions Parametric Line1.3 Parametric Surfaces1.4 Piecewise Surfaces 1.5 Continuity Geometric Continuity Parametric ContinuityHistorical Perspective - Bézier Curves: A.R. ForrestChapter 2 - Bézier Curves2.1 Bézier Curve DeffnitionBézier Curve Algorithm2.2 Matrix Representation of Bézier Curves 2.3 Bézier Curve Derivatives 2.4 Continuity Between Bézier Curves 2.5 Increasing the Flexibility of Bézier CurvesDegree Raising SubdivisionHistorical Perspective - B-splines: Richard F. RiesenfeldChapter 3 - B-spline Curves3.1 B-spline Curve DeffnitionProperties of B-spline Curves3.2 Convex Hull Properties of B-spline Curves 3.3 Knot Vectors 3.4 B-spline Basis FunctionsB-spline Curve Controls3.5 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 CurvesStart and End Points Start and End Point Derivatives Controlling Start and End Points Multiple Coincident Vertices Pseudovertices3.10 B-spline Curve Derivatives 3.11 B-spline Curve Fitting 3.12 Degree Elevation Algorithms3.13 Degree ReductionBézier Curve Degree Reduction3.14 Knot Insertion and B-spline Curve Subdivision 3.15 Knot RemovalPseudocode3.16 Reparameterization Historical Perspective - Subdivision: Tom Lyche, Elaine Cohen and Richard F. RiesenfeldChapter 4 - Rational B-spline Curves4.1 Rational B-spline Curves (NURBS Curves)Characteristics of NURBS4.2 Rational B-spline Basis Functions and CurvesOpen Rational B-spline Basis Functions and Curves Periodic Rational B-spline Basis Functions and Curves4.3 Calculating Rational B-spline Curves 4.4 Derivatives of NURBS Curves 4.5 Conic Sections Historical Perspective - Rational B-splines: Lewis C. KnappChapter 5 - Bézier Surfaces5.1 Mapping Parametric Surfaces5.2 Bézier Surfaces Matrix Representation5.3 Bézier Surface Derivatives 5.4 Transforming Between Surface Descriptions Historical Perspective - Nonuniform Rational B-splines: Kenneth J. VersprilleChapter 6 - B-spline Surfaces6.1 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. RogersChapter 7 - Rational B-spline Surfaces7.1 Rational B-spline Surfaces (NURBS)7.2 Characteristics of Rational B-spline SurfacesEffects of positive homogeneous weighting factors on a single vertex Effects of negative homogeneous weighting factors Effects of internally nonuniform knot vector Reparameterization7.3 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 SurfacesDevelopable Surfaces7.8 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 EffortAppendicesA B-spline Surface File FormatB Problems and Projects C AlgorithmsReferencesIndex

- No. of pages: 344
- Language: English
- Published: July 21, 2000
- Imprint: Morgan Kaufmann
- Hardback ISBN: 9781558606692
- eBook ISBN: 9780080509204

DR

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.