The NURBS Book

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