# The NURBS Book

Springer Science & Business Media, Jan 1, 1997 - Computers - 646 pages
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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### 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 Copyright