# The NURBS Book

Springer Science & Business Media, 14 nov 1996 - 646 pagine
Until recently B-spline curves and surfaces (NURBS) were principally of interest to the computer aided design community, where they have become the standard for curve and surface description. Today we are seeing expanded use of NURBS in modeling objects for the visual arts, including the film and entertainment industries, art, and sculpture. NURBS are now also being used for modeling scenes for virtual reality applications. These applications are expected to increase. Consequently, it is quite appropriate for The.N'URBS Book to be part of the Monographs in Visual Communication Series. B-spline curves and surfaces have been an enduring element throughout my pro fessional life. The first edition of Mathematical Elements for Computer Graphics, published in 1972, was the first computer aided design/interactive computer graph ics textbook to contain material on B-splines. That material was obtained through the good graces of Bill Gordon and Louie Knapp while they were at Syracuse University. A paper of mine, presented during the Summer of 1977 at a Society of Naval Architects and Marine Engineers meeting on computer aided ship surface design, was arguably the first to examine the use of B-spline curves for ship design. For many, B-splines, rational B-splines, and NURBS have been a bit mysterious.

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### Sommario

 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
 923 CUBIC SPLINE CURVE INTERPOLATION 367 924 GLOBAL CURVE INTERPOLATION WITH FIRST DERIVATIVES SPECIFIED 369 925 GLOBAL SURFACE INTERPOLATION 372 93 Local Interpolation 378 932 LOCAL PARABOLIC CURVE INTERPOLATION 384 933 LOCAL RATIONAL QUADRATIC CURVE INTERPOLATION 388 934 LOCAL CUBIC CURVE INTERPOLATION 391 935 LOCAL BICUBIC SURFACE INTERPOLATION 395

 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 208 EXERCISES 223 Advanced Geometric Algorithms 225 62 Surface Tangent Vector Inversion 231 63 Transformations and Projections of Curves and Surfaces 232 64 Reparameterization of NURBS Curves and Surfaces 237 65 Curve and Surface Reversal 259 66 Conversion Between Bspline and Piecewise Power Basis Forms 261 EXERCISES 275 Conics and Circles 277 73 The Quadratic Rational Bezier Arc 287 74 Infinite Control Points 291 75 Construction of Circles 294 76 Construction of Conies 306 77 Conic Type Classification and Form Conversion 316 78 Higher Order Circles 322 EXERCISES 326 Construction of Common Surfaces 329 83 The General Cylinder 330 84 The Ruled Surface 333 85 The Surface of Revolution 336 86 Nonuniform Scaling of Surfaces 344 87 A Threesided Spherical Surface 347 Curve and Surface Fitting 357 92 Global Interpolation 360 922 GLOBAL CURVE INTERPOLATION WITH END DERIVATIVES SPECIFIED 366
 94 Global Approximation 401 941 LEAST SQUARES CURVE APPROXIMATION 403 942 WEIGHTED AND CONSTRAINED LEAST SQUARES CURVE FITTING 406 943 LEAST SQUARES SURFACE APPROXIMATION 412 944 APPROXIMATION TO WITHIN A SPECIFIED ACCURACY 417 95 Local Approximation 430 951 LOCAL RATIONAL QUADRATIC CURVE APPROXIMATION 431 952 LOCAL NONRATIONAL CUBIC CURVE APPROXIMATION 434 EXERCISES 445 Advanced Surface Construction Techniques 448 103 Skinned Surfaces 450 104 Swept Surfaces 465 105 Interpolation of a Bidirectional Curve Network 478 106 Coons Surfaces 489 Shape Modification Tools 502 112 Control Point Repositioning 504 113 Weight Modification 511 1131 MODIFICATION OF ONE CURVE WEIGHT 513 1132 MODIFICATION OF TWO NEIGHBORING CURVE WEIGHTS 519 1133 MODIFICATION OF ONE SURFACE WEIGHT 524 114 Shape Operators 526 1142 FLATTENING 535 1143 BENDING 540 115 Constraintbased Curve and Surface Shaping 548 1152 CONSTRAINTBASED SURFACE MODIFICATION 555 Standards and Data Exchange 564 123 NURBS Within the Standards 573 l232 STEP 576 1233 PHIGS 578 124 Data Exchange to and from a NURBS System 579 Bspline Programming Concepts 586 132 Data Types and Portability 587 133 Data Structures 589 134 Memory Allocation 594 135 Error Control 600 136 Utility Routines 606 137 Arithmetic Routines 609 138 Example Programs 611 139 Additional Structures 616 1310 System Structure 619 REFERENCES 622 INDEX 634 Copyright

### Riferimenti a questo libro

 Geometric Tools for Computer GraphicsAnteprima non disponibile - 2003
 3D Computer Graphics: A Mathematical Introduction with OpenGLAnteprima limitata - 2003
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