Farin G. Curves and Surfaces for CAGD. A Practical Guide
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492 Index
Bezier, P., 64, 89, 195, 301, 304
Bezier patch, trivariate, 302
Bezier points, 45
Bezier polygon, 45
Bezier surface, 248
Bezier tetrahedron, 333
biaffine, 247
bias,
195
bicubic Hermite interpolant, 405
bicubic Hermite patch, 276
bihnear interpolant, 245, 271, 292,
400
bilinearly blended Coons patch, 291,
402,408
binomial coefficient, 57
binormal, 181
Birkhoff,
G., 414, 420
Blaschke, W., 45
blending function, 402, 412
blossom, 31, 51, 74, 313
of tensor product surface, 258
for triangular patches, 313
Bobrow, J., 305
Boehm, W., 30, 43, 50, 119, 189, 194,
195,
202, 314, 340, 364, 367,
376
Bol,G.,l 89
Bonneau, G., 305
Boolean sum, 412, 414
boundary edge, 394
boundary point, 39
boundary vertex, 394
brain imaging, 302
Brianchon, C., 206
Brown, J., 41, 291
Brunet, P., 293, 305, 367
Brunnett, G., 305
B-spline
polygon, 289, 419, 431
B-splines, 399,431
butterfly scheme, 391
canonical coordinates, 181
canonical form, 184
cardinal, 99, 103, 106
Casale, M., 305
Catmull-Clark surfaces, 383, 392
Catmull, E., 383
Catmull-Rom spline, 173
centroid, 15, 18
Ceva's theorem, 36
Chaikin, G., 132
Chang, G., 74
Chang, S., 116, 305
Charrot, P, 330, 367, 417
Chebychev polynomials, 399
Chiyokura, H., 410
chord length, 180
parametrization, 170
Chou,J.,238
Chrysler, 399
circle, 2, 5, 159, 221, 239, 299
osculating, 184
circulant matrix, 383, 386
clamped end condition, 155
Clark,].,
383
clay model, 6
closed mesh, 395
Clough-Tocher interpolant, 343
Cobb,
J., 339
Cohen, E., 68, 132, 163
Cohn-Vossen, S., 425
collinear, 25
compatibility, 408
complementary segment of a conic,
220
computational fluid dynamics, 399
computation count, 268
conic precision, 240
conic section, 44, 205, 209, 337, 359,
364
conjugate direction, 361
connection matrix, 202
CONS,
304
control net, 248
control points, 45
infinite, 224
control polygon, 45
of a conic, 211
control vectors, 238
conversion of
B-spline
surfaces, 288
convex, 41, 85
convex combination, 16, 273
convex hull, 16,37,41,48
Index 493
convex hull property, 48, 60, 98, 224,
335
of rational Bezier curves, 228
of Bezier patches, 254
of Bezier triangles, 312
convexity, 270
convexity of surfaces, 264
convex set, 16
Coons patch
discrete, 272
Coons patch, bilinearly blended, 291,
402
Coons, S., 2, 205, 227, 399
coordinate-free, 13
coordinate system, 13
corner, 170
corner-cutting, 132
corner of a curve, 162
corner tv^ist, 290
correction surface, 413
Coxeter, H., 15
Cox,M., 119, 142
Cramer's rule, 34
cross-boundary derivative, 257, 319,
403
cross-boundary normal derivative, 344
cross plot, 87
cross ratio, 206, 213, 219, 231
cross ratio theorem, 206
Crouch, P., 168
cubic Hermite interpolant, 416
cubic Hermite interpolation, 404, 413,
415
cubic Hermite polynomial, 104, 199,
403
cubic precision, 176, 240
curvature, 167, 183,419
Gaussian, 270
curvature,
derivative of, 188,421
of nonparametric curve, 187
of rational Bezier curve, 185
signed, 235, 243
curvature plot, 159, 163, 419
curve on surface, 304
cusp,
166
cylinder, 300
Dahmen. W., 68
Daniel, M, 432
data transfer, 83
Daubisse, J., 432
Davis,
P., 22, 57, 90
de Boor algorithm, 123
rational, 242
for rational B-splines, 241
de Boor, C, 22, 119, 123, 132, 142,
235,285,309
de Casteljau, P., 2, 30, 43, 50, 68, 119,
245,
309
de Casteljau algorithm, 45, 68, 247
degenerate patch, 261
Degen, W., 202
degree elevation, 238, 326
for B-splines, 144
for Bernstein polynomials, 92
for Bezier curves, 81, 132
for Bezier patches, 254
for rational Bezier curves, 236
degree reduction, 85
for B-spline curves, 133
for Bezier triangles, 333
Delaunay triangulation, 39
depth
of an
S-patch,
330
de Rham G., 132
derivative
of a B-spline, 144
of a conic, 215
of a rational Bezier curve, 233
DeRose, T, 17, 50, 202, 260, 301, 330,
367,
390, 434
design, 49
developable surface, 270, 358
diagonalization, 378
difference operator, G6, 256
Dill, J., 420, 425
dimension
of B-spline spaces, 140
directional derivative, 316
directrix, 9
Dirichlet tessellation, 37
discrete Coons patch, 272
discriminant, 351
divided differences, 119
494 Index
do Carmo, M., 202
domain, 68, 246
Doo,
D., 380
Doo-Sabin surfaces, 380, 392
duck, 167
Dupin's indicatrix, 359, 364, 376
Dyn, N., 41, 202, 379, 391
ellipse, 220, 359
elliptic point, 357
end condition
Bessel, 158, 306
clamped, 155
natural, 157
not-a-knot, 158
endpoint interpolation, 61
of Bezier curves, 49
Epstein, M., 162
euclidean map, 19
euclidean space, 14
Euler, L., 17, 179, 359
Euler-Poincare formula, 395
extraordinary vertex, 381, 385, 388
extrapolation, 68
fairness of a curve, 420
Farin, G., 84, 202, 309, 326, 327, 422
Farouki, R., 222, 305, 425, 432
Faux, I., 74
Ferguson, J., 2, 285, 290
fine
tune,
197
Fiorot, J., 227
first fundamental form, 350
flat point, 357
flat spot, 404
Floater, M., 233, 242
FMILL, 172
Foley, T, 162, 163
font design, 19, 71
Ford, 399
Ford, H., 12
Forrest, A. R., 64, 84, 205, 213, 227,
420,
425
Forsey, D., 377
forward difference, 64
forward differencing, 114
four-point scheme, 379
four tangent theorem, 213
fractals, 79
Frederickson, L., 309
Frenet frame, 181
Frenet frame continuity, 202
Frenet-Serret formulas, 183
functional curve, 86
Gallier,].,
119
Garland, M, 396
Gauss, C., 179, 357
Gaussian curvature, 270, 357, 425
Gauss-Seidel iteration, 157
Geise, G., 195, 202
generalized Vandermonde matrix, 101
General Motors, 399
generatrix, 300, 361
genus,
395
geodesic, 363
geometric continuity, 189, 191, 222
geometric programming, 17
geometry matrix, 265
Goldman, R., 13, 17, 50, 144, 189, 202,
314
Gonska, H., 57
Goodman, T., 202, 234
Gordon surface, 410
spline-blended, 412
Gordon, W., 2, 399,410,414
gradient, of implicit conic, 217
Grassmann, H., 41
Gregory, J., 202, 330, 367, 379, 391,
408,417
Gregory's square, 409
Greville abscissa, 135, 148
Hagen, H., 202, 293
Hahn, J., 367
Hansford, D., 40, 340
harmonic function, 8
Hartley, R, 163
hat function, 23
Heckbert, R, 396
Index 495
Hermite form, 435
Hermite interpolation, 102, 375, 413
Hermite patch, 276
Herron, G., 367, 376
hierarchical
B-spline
surface, 377
Hilbert, D., 425
Hinds, J., 305, 425
hodograph, 64
Hollig,K., 119,235
Holman, T., 434
homogeneous coordinates, 210
Horner's scheme, 75
Hoschek,].,
153, 367,424
hyperbola, 220, 247, 359
hyperbolic paraboloid, 246
hyperbolic point, 357
idempotent operator, 414
identity map, 18
IGES,
143, 217
image plane, 206
image registration, 22
implicit form
of conic, 216
of quadric, 340
of surface, 349
tolerances, 225
improper parametrization, 237
incenter, 345
infinite control point, 224, 238
inflection point, 429
integrals, 88
integrals of tensor product patches, 267
interference checking, 49
interior vertex, 394
interpolating
conic, 218
polynomial, 95
quadric, 340
intrinsic geometry, 357
invariance under affine parameter
transformations, 435
of Bezier curves, 48
isoparametric curve, 246, 252, 410
isophotes, 425
isotropic directions, 351
Jeannin, P., 227
Jensen, T., 367
JoUes, S., 74
Jones,
A., 367, 427
Judd, C, 163
junction points, 170
Kahmann, J., 367, 376
Kaufmann, E., 427
Kiciak, P., 367
Kimura, E, 410
Kjellander, J., 424
Klass,
R., 235, 425, 427
Klein bottle, 308
Klucewicz, I., 291
knot insertion for rational B-splines,
242
knot multiplicities, in IGES, 143
Kobbelt, L., 389
Korovkin, R, 57, 90, 91
Kronecker Delta, 61
Kruth, J., 240
Lacombe, C., 417
Lagrange interpolation, 99
Lagrange polynomial, 98, 411
Lane, J., 68
Laplacian smoothing, 281
Lasser, D., 305
least squares, 108, 149, 278, 346
tensor product, 278
Lee,
E., 50, 162, 220, 242
Leibniz formula, 33
Levin, D., 41, 391
Levin, S., 379
Levoy, M., 394
Lien, S., 116
Liming, R., 205, 219
linear independence of B-splines, 141
linear interpolation, 25
bivariate, 36
linear interpolation, repeated, 44
linearly independent, 22
linear operator, 23, 45
linear parameter transformation, 48
496 Index
linear precision, 62, 98, 171
of Bezier triangles, 327
of rational curves, 228
linear space, 14, 22
line of curvature, 356
of a surface of revolution, 356
Little, R, 40
Liu, D., 367, 420
Lobachevsky, N., 119
local control, 129
local coordinates, 48
local parameter, 28
local support, 140
lofted surface, 269, 411
loop,
170
Loop,
C, 330, 387
Lorentz, L., 57
Lounsbery, M., 390
Lucian, M., 242
Lyche, T, 132
main normal vector, 181
Manning,].,
195, 202
Mann, S., 260, 390
Mansfield, L., 119, 142
Marhsall,J.,417
mass point, 41
matrix circulant, 383, 386
matrix form of curves, 74
Ma, W., 240
maxmin criterion, 40
McConalogue, D., 163, 174
mean curvature, 357, 425
Meier, J., 57
Memke's theorem, 189, 194
Menelaos' theorem, 30
meridian, 299
Meusnier's theorem, 354
Micchelli,C., 132, 189,202
Michelangelo, 394
digital, 394
Miller, J., 305
minimal support of B-splines, 140
minmax box, 49, 69
minmax criterion, 40
Mitchell, A., 417
mixed partials, 257
Moebius, R, 17, 26, 34, 206
Monge, C, 179
monomial basis, 75, 432
monomial form, Gl
of Bezier patches, 265
monomials, 23, 99
Mortenson, M., 74
multiaffine, 31, 53
multiaffine function, 313
multiplicities, of end knots, 143
multiplicity
of a knot, 127
multiresolution, 397
Munchmeyer, R, 425
Musse, J., 367
Nachman, L., 403
NAPOLEON, 11
Nasri, A., 390
natural end condition, 157, 162
nested multiplication, IS
neutral set, 39
Neville, ^l
New^ton form, 114
New^ton iteration, 153
Nielson, G., 107, 163, 194, 199, 367,
414,416,417
nine parameter interpolant, 341
nonparametric curve, 187
nonparametric patch, 266
normal curvature, 354
normal equations, 86, 150, 347
normal section, 354
normal vector, 261
numerical control, 1
NURBS,
227
O'Dell,
C, 163
open mesh, 395
order of contact, 202
orthonormal matrix, 19
osculant, of tensor product surface, 259
osculating circle, 184
osculating plane, 73, 182, 203
Index 497
osculatory interpolation, 234
Oslo algorithm, 132
Overhauser spline, 173, 177
parabola, 2, 5, 43, 44, 173, 185, 209,
220,
227, 240, 247, 339
parabolic cylinder, 339
parabolic point, 357
paraboloid, 339, 359
parallel projection, 19, 20
parameter correction, 153
parametrization, 170, 179, 222, 339
centripetal, 150, 162
chord length, 162
equidistant, 161
for tensor product interpolation, 295
uniform, 161
partial derivatives, 256
partition, 23
of unity, 141
Pascal's theorem, 218
Patterson, R., 205, 236
Pegna, J., 364, 376
Penna, M., 205
Petersen, C, 327, 333
Peters, G., 285
Peters,].,
386, 392
piecev^ise Bezier curve, 137
piecewise bilinear interpolation, 255
piecewise linear interpolation, 82, 84
Piegl, L., 224, 227, 339
Piper, B., 107, 367, 375
pixel, 116
Poeschl, T, 425
point, 14
at infinity, 224
polar, 73
of tensor product surface, 259
polar form, 351
polygon, 29
polynomial space, 22
Pottmann, H., 202, 421
Powell, M., 345
pov^er basis, 432
Pratt, M., 74
Pratt, v., 116
Prautzsch, H., 85, 119, 132, 144
principal curvatures, 357
principal direction, 356
projective geometry, 205
projective invariance, 48
projective map, 205, 208
projector, 414
quadratic precision, 342
quadric, 337, 396
quintic Hermite Interpolation, 106, 177
radial line, 322
radius of curvature, 184
Rajan, V., 432
Ramshaw, L., 50, 68, 74, 119, 247, 336
range, 246
rank, 20
ratio,
24, 27, 206, 286
rational linear parameter transformation,
213
rational linear transformation, 208
rational quadratic form for conies, 211
rational quadratics, 337
recursive subdivision, 380
reflection lines, 425
regular parametrization, 180
Reif,
U., 377, 386
Renyi, A., 119
Renz, W., 424
reparametrization, 87, 180, 191, 201,
299,311,349,360
of rational Bezier curves, 235
of rational B-splines, 242
of surfaces, 376
Riesenfeld, R., 2, 68, 132
rigid body motion, 19
Rippa, S., 41
Ris,
G., 367
Rochetti, R., 116
rotation, 18
rough sketch, 197
roundoff,
435
ruled surface, 246, 269, 361
ruling, 270
Runge phenomenon, 102
498 Index
S. Parry, 301
Sabin, M., 235, 309, 345, 380, 389, 420
Sablonniere, P., 309, 327
Sakkalis, T, 222
Sapidis, N., 419
Sarraga, R., 367, 375
scaling, 18
scattered data interpolation, 341
Schelske, J., 264
Schneider, R, 240
Schoenberg, L, 84, 119
Schoenberg-Whitney theorem, 150
Schulze, G., 293
Schumaker, L., 68, 327
second fundamental form, 353
Sederberg, X, 49, 64, 237, 260, 301
Seidel,H.,50, 314
Selesnick, S., 293
self-conjugate, 361
semicircle, 224
Sequin, C, 367
Shantz,M., 116, 305
shape preservation, 101, 270
shear, 19, 24
Shirman, L., 367
shoulder point, 213, 219
shoulder tangent, 219
signed curvature, 187
simplest scheme, 386
slanted font, 19
smoothness
of piecewise Bezier curves, 137
soHd modeling, 338
sphere, 300
spline, 2, 167, 173, 240, 421, 424
spline interpolation, 412
iterative, 157
stability of the monomial form, 432
Staerk, E., 68
Stancu, D., 309
standard cubic, 55
standard form
for conies, 213
of rational Bezier curve, 236, 299
star, 396
Steiner, J., 213
stochastic matrix, 385
storage for
B-spline
curves, 435
straight line, 25
strain energy, 167
Su, B., 420
subdivision, 6S
formula, 68
of Bezier curves, 68
of Bezier patches, 259
of Bezier triangles, 321
of rational Bezier curves, 232
in terms of Bernstein polynomials, 92
subspace, 22
surface, 9
butterfly, 391
conical, 270
cylindrical, 270
developable, 270
lofted, 269
ruled, 269
translational, 272
surface of revolution, 299
Sw^immer, A., 41
symmetric function, 313
symmetry, 61
tangent length, 435
tangent plane of Bezier triangle, 318
tangent ribbon, 404
tangent vector, 181
Taylor series, 67
tension, 195
tensor product, 246, 412
interpolant, 405, 435
surfaces, 245
tetrahedron, 333
texture mapping, 425
theorema egregium, 357
Thiessen region, 37
three tangent theorem, 44
Tiller, W., 227, 339
topology of triangulations, 394
torsion, 183
continuity, 202
of nonparametric curve, 187
of rational Bezier curve, 185, 186,
234
Index 499
torus,
300
transfinite interpolation, 399, 410
translation, 14, 18
translational surface, 264, 272, 282, 290
transposant, 304
triangle
area of, 35
numbers, 310
triangulation, 39
data dependent, 41
trimmed surfaces, 304
trinomial coefficient, 315
trivariate B-splines, 302
trivial reject, 69
twist estimation, 406
twist incompatibility, 417
twist, variable, 409
twist vector, 257, 264
type check, 17
umbilical point, 355
uniform parametrization, 222
UNISURF, 1
unit interval, 28
unit sphere, 339
unit square, 246
variable twists, 409
variation diminishing property, 29, 41,
84,
98, 202
for Bezier patches, 254
of rational Bezier curves, 228
vectors, 14
Vernet, P., 89
Veron, M., 367
Versprille, K., 224, 238
vertex split, 397
Vigo,
M., 305
Vinacua, A., 367
volume distortion, 301
Voronoi diagram, 37
Wang, X., 64
Weierstrass approximation theorem, 90
weight point, 214, 219, 230, 231, 235,
336
weights, 211
of rational Bezier curves, 227
Wolter, R, 364, 376
Worsey, A., 235
Yamaguchi, R, 157
valency, 381
Vandermonde, 100, 275
vanWijk,].,
367
zero twist, 273, 290, 404
Zhou, J., 84
This Page Intentionally Left Blank
Plate I.
An automobile.
Figure courtesy of
Mercedes-Benz, FRG.
Plate II.
Color rendering of the
hood. Figure courtesy of
Mercedes-Benz, FRG.
Plate
111.
Wire frame rendering of
the hood. Figure courtesy
of Mercedes-Benz, FRG.