The NURBS Book

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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

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