The NURBS BookSpringer 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. |
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 |
Altre edizioni - Visualizza tutto
Parole e frasi comuni
algorithm approximation assume B-spline curve basis functions Bernstein polynomials Bézier curve Bézier form Bézier segment Bézier surface C¹ continuous circle circular arc compute conic constraints control polygon convex hull coordinate cubic curve curve interpolation curve or surface curves and surfaces defined degree elevation degree reduced denote derivatives direction equations error example geometric hence Input internal knots Kmax knot insertion knot refinement knot removal knot span knot vector linear matrix method multiplicity nonrational nonzero NURBS curve NURBS surface obtain Output P₁ P₂ parameterization patch Pi,j piecewise Piegl plane polynomial power basis form projection quadratic curve rational Bézier reparameterization representation routine ruled surface section curves shown in Figure shows spline surface interpolation surface of revolution tangent tensor product three-dimensional three-dimensional space Ui+1 Ui+p Ui+p+1 unclamped vspan w₁ weights yields
Riferimenti a questo libro
Geometric Tools for Computer Graphics Philip Schneider,David H. Eberly Anteprima non disponibile - 2003 |
3D Computer Graphics: A Mathematical Introduction with OpenGL Samuel R. Buss Anteprima limitata - 2003 |