## The NURBS BookThe NURBS book covers all aspects of non-uniform rational B-splines necessary to design geometry in a computer aided environment. Basic B-spline features, curve and surface algorithms, and state-of-the-art geometry tools are all discussed. Detailed code for design algorithms and computational tricks are covered too, in a lucid, easy-to-understand style, with a minimum of mathematics and using numerous worked examples. The book is a must for students, researchers, and implementors whose work involves the use of splines. "Some years ago a few researchers joked about NURBS, saying that the acronym really stands for Nobody Understands Rational B-Splines write the authors in their foreword; they formulate the aim of changing NURBS to EURBS, that is, Everybody ... . There is no doubt that they have achieved this goal. ... ... I highly recommend the book to anyone who is interested in a detailed description of NURBS. It is extremely helpful for students, teachers and designers of geometric modelling systems. ..." - H. Pottmann, Computer Aided Design, 1996 |

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#### Review: The Nurbs Book

User Review - Greg - GoodreadsThis taught me everything I needed to be able to understand and use non-uniform rational bsplines for my work. I didn't need quite all the detail in the book - I'm not designing a nurbs library, just ... Read full review

### Contents

Curve and Surface Basics | 1 |

12 Power Basis Form of a Curve | 5 |

13 Bezier Curves | 9 |

14 Rational Bezier Curves | 25 |

15 Tensor Product Surfaces | 34 |

EXERCISES | 43 |

BSpline Basis Functions | 47 |

22 Definition and Properties of Bspline Basis Functions | 50 |

92 Global Interpolation | 363 |

922 GLOBAL CURVE INTERPOLATION WITH END DERIVATIVES SPECIFIED | 369 |

923 CUBIC SPLINE CURVE INTERPOLATION | 370 |

924 GLOBAL CURVE INTERPOLATION WITH FIRST DERIVATIVES SPECIFIED | 372 |

925 GLOBAL SURFACE INTERPOLATION | 375 |

93 Local Interpolation | 381 |

932 LOCAL PARABOLIC CURVE INTERPOLATION | 387 |

933 LOCAL RATIONAL QUADRATIC CURVE INTERPOLATION | 391 |

23 Derivatives of Bspline Basis Functions The derivative of a basis function is given by | 59 |

24 Further Properties of the Basis Functions | 63 |

25 Computational Algorithms | 67 |

EXERCISES | 78 |

Bspline Curves and Surfaces | 81 |

33 The Derivatives of a Bspline Curve | 91 |

34 Definition and Properties of Bspline Surfaces | 100 |

35 Derivatives of a Bspline Surface | 110 |

EXERCISES | 116 |

Rational Bspline Curves and Surfaces | 117 |

43 Derivatives of a NURBS Curve | 125 |

44 Definition and Properties of NURBS Surfaces | 128 |

45 Derivatives of a NURBS Surface | 136 |

EXERCISES | 138 |

Fundamental Geometric Algorithms | 141 |

53 Knot Refinement | 161 |

54 Knot Removal | 178 |

55 Degree Elevation | 187 |

56 Degree Reduction | 211 |

EXERCISES | 226 |

Advanced Geometric Algorithms | 228 |

62 Surface Tangent Vector Inversion | 234 |

63 Transformations and Projections of Curves and Surfaces | 235 |

64 Reparameterization of NURBS Curves and Surfaces | 240 |

65 Curve and Surface Reversal | 262 |

66 Conversion Between Bspline and Piecewise Power Basis Forms | 264 |

EXERCISES | 278 |

73 The Quadratic Rational Bezier Arc | 290 |

74 Infinite Control Points | 294 |

75 Construction of Circles | 297 |

76 Construction of Conies | 309 |

77 Conic Type Classification and Form Conversion | 319 |

78 Higher Order Circles | 325 |

EXERCISES | 329 |

83 The General Cylinder | 333 |

84 The Ruled Surface | 336 |

85 The Surface of Revolution | 339 |

86 Nonuniform Scaling of Surfaces | 347 |

87 A Threesided Spherical Surface | 350 |

Curve and Surface Fitting | 360 |

934 LOCAL CUBIC CURVE INTERPOLATION | 394 |

935 LOCAL BICUBIC SURFACE INTERPOLATION | 398 |

94 Global Approximation | 404 |

941 LEAST SQUARES CURVE APPROXIMATION | 406 |

942 WEIGHTED AND CONSTRAINED LEAST SQUARES CURVE FITTING | 412 |

943 LEAST SQUARES SURFACE APPROXIMATION | 418 |

944 APPROXIMATION TO WITHIN A SPECIFIED ACCURACY | 423 |

95 Local Approximation | 436 |

951 LOCAL RATIONAL QUADRATIC CURVE APPROXIMATION | 437 |

952 LOCAL NONRATIONAL CUBIC CURVE APPROXIMATION | 440 |

EXERCISES | 451 |

103 Skinned Surfaces | 456 |

104 Swept Surfaces | 471 |

105 Interpolation of a Bidirectional Curve Network | 484 |

106 Coons Surfaces | 495 |

112 Control Point Repositioning | 510 |

113 Weight Modification | 517 |

1131 MODIFICATION OF ONE CURVE WEIGHT | 519 |

1132 MODIFICATION OF TWO NEIGHBORING CURVE WEIGHTS | 525 |

1133 MODIFICATION OF ONE SURFACE WEIGHT | 530 |

114 Shape Operators | 532 |

1142 FLATTENING | 541 |

1143 BENDING | 546 |

115 Constraintbased Curve and Surface Shaping | 554 |

1152 CONSTRAINTBASED SURFACE MODIFICATION | 561 |

123 NURBS Within the Standards | 579 |

l232 STEP | 582 |

1233 PHIGS | 584 |

124 Data Exchange to and from a NURBS System | 585 |

132 Data Types and Portability | 593 |

133 Data Structures | 595 |

134 Memory Allocation | 600 |

135 Error Control | 606 |

136 Utility Routines | 612 |

137 Arithmetic Routines | 615 |

138 Example Programs | 617 |

139 Additional Structures | 622 |

1310 System Structure | 625 |

REFERENCES | 628 |

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### Common terms and phrases

### References to this book

### References from web pages

Piegl la,, Tiller W. — The NURBS Book (Monographs in Visual **...**

Название: The NURBS Book (Monographs in Visual Communication) Авторы: Piegl la,, Tiller W. Аннотация:. The second, revised edition of this book covers all ...

lib.mexmat.ru/ books/ 4302

The NURBS book

The NURBS book. Purchase this Book. Source. Pages: 646. Year of Publication: 1995. ISBN:3-540-55069-0. Authors. Les Piegl, Univ. of South Florida, Tampa ...

portal.acm.org/ citation.cfm?id=208469&

The NUNBS Book

Welcome to the The NURBS Book, a publication of Springer-Verlag. The book covers all geometrical and algorithmic aspects of (Non-Uniform Rational B-splines) ...

www.cse.usf.edu/ ~lap/ nurbs.htm

NURBS Toolbox

For a detailed explanation of NURBS and how to manipulate them, I can strongly recommend the book by Les Piegl and Wayne Tiller called 'The NURBS Book' ISBN ...

www.aria.uklinux.net/ nurbs.php3

Infoscience: Record #25483: The Nurbs book

infoscience. subtitle. Français | English. EPFL > infoscience > Record #25483: The Nurbs book. My actions. Identification. Identification ...

infoscience.epfl.ch/ record/ 25483

DIVERSITY IN GRAPHICS RESEARCH AT USF Dr. Les A. Piegl

of the best selling text, THE NURBS BOOK, published by Springer-Verlag. Dr. Piegl has been Editor-in-Chief. of COMPUTER-AIDED DESIGN (CAD) in the last ...

www.cs.hku.hk/ seminars/ 2003/ 030226b.pdf

sciencedirect - Computer-Aided Design : Splines with Pictures and **...**

While The NURBS book by Piegl and Tiller [1], reviewed in 1996 in CAD, goes into greater detail by including pieces of code, and de Boor's B-form basics [2] ...

linkinghub.elsevier.com/ retrieve/ pii/ S0010448502001094

Notes: Code is based on the Piegl Nurbs book. The NURBS Book, Les **...**

The NURBS Book, Les Piegl and Wayne Tiller, Springer 1997 */ #define GNURBS_MAX_ORDER 16 // // find span - see the NURBS Book // page 68 // int findspan ...

developer.blaxxun.com/ developer/ contact/ 3d/ nurbs/ spec/ gnurbsutil.cpp

thenurbsbook

The NURBS Book by Les Piegl and Wayne Tiller. Springer-Verlag, 1997, 646 pages, 333 figures. ISBN 3-540-61545-8. Table of Contents. Curve and surface basics ...

www.piegl.com/ TheNURBSBook.html

B-SPLINES AND NURBS CURVES AND SURFACES

Piegl, L. and Tiller, W: The NURBS Book, 2nd. Edition, Springer Verlag Berlin, 1997. NURBS surfaces are included here as a particular case. However, some ...

personales.unican.es/ iglesias/ CD-CG/ Bsplines.pdf