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| Ïîèñê ïî óêàçàòåëÿì |
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| Yamaguchi F. — Curves and surfaces in computer aided geometric design |
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| Ïðåäìåòíûé óêàçàòåëü |
Acceleration, normal 27
Acceleration, tangential 27
Adini’s method 118 119
Approximation, B-spline- 233—336
Approximation, Bernstein- 169—232
APT system 3
Area of a surface 42 43 47 48
B-spline curve segment 233—334
B-spline curve segment, derivatives 309—311
B-spline curve segment, interpolation of a sequence of points 247 to 250 325—327
B-spline curve segment, matrix representation 327—329
B-spline curve segment, properties 311 312
B-spline curve type (1) 281 282
B-spline curve type (2) 283
B-spline curve type (3) 294—296
B-spline, B-spline function, approximation 233—336
B-spline, B-spline function, definition 270—273
B-spline, B-spline function, normalized 273
B-spline, B-spline function, properties 270—274 291—294
B-spline, B-spline function, recursive calculation 285—291
B-spline, B-spline function, relation to Bernstein basis 293 294
B-spline, B-spline function, variation diminishing property 294
Bernoulli — Euler law 135
Bernstein approximation 169—232
Bernstein approximation, derivatives 190
Bernstein approximation, properties 189—193
Bernstein approximation, smoothing effect 193
Bernstein basis function 183 189—193
Bernstein basis vector 195
Bernstein polynomial 174
Bezier curve segment 169—232
Bezier curve segment, connection 213 214
Bezier curve segment, derivatives 198 199
Bezier curve segment, finite difference representation 194 195
Bezier curve segment, increase of degree 204-209
Bezier curve segment, partitioning 209—212
Bezier curve segment, various representation 193—198
Bezier net 217
Bezier polygon 173
Bezier surface patch 216—232
Bezier surface patch, connection 221—226
Bilinear transformation 36
Binomial expansion 182 183
Binormal vector, unit- 28
Blending function 77
Boundary condition matrix 111
C-spline 139
Cartesian product surface patch 132 133
Cauchy’s relation 19
Chaikin’s algorithm 320—325
Characteristic net 217
Characteristic polygon 173
Circular arc, circle, approximation 78—80 177 178 243—245
Circular arc, circle, approximation by Bezier curve segments 177 178
Circular arc, circle, approximation by cubic B-spline curve segments 243 to 245
Circular arc, circle, approximation by Ferguson curve segments 78—80
Circular arc, circle, approximation error 80 178
Circular arc, circle, approximation, passing through 3 points 354 355
Common perpendicular, length 352 353
Condition for determining a tangent vector to a curve on a surface 48
Conic section 337—346
Connection of Bezier curve segments 213 214
Connection of Bezier surface patches 221—226
Connection of Coons bi-cubic surface patches 96 97 107—109
Connection of curve segments 33—35
Connection of Ferguson surface patches 89—91
Connection of surface patches 57—59
Continuity, clss  - 20 44
Control point 16
Convex combination 171 241 274
Convex hull property 241 311
Crease 15
Cross partial derivative vector 98 112—122
Cubic spline curves 145—160
Cubic spline curves, using circular arc length 159 160
Curvature 24—27
Curvature of a surface 49—51
Curvature vector 24—27
Curvature, average(mean) 51
Curvature, Gauss- 51
Curvature, normal 50
Curvature, principal 51
Curvature, radius 25
Curvature, total 51
Curve defining polygon 173
Curve generation by geometric processing 320—325
Curve segment, B-spline 233—334
Curve segment, Bezier- 169—232
Curve segment, conic section 337—346
Curve segment, cubic B-spline 233—251
Curve segment, cubic B-spline, geometrical relations among derivatives 342 343
Curve segment, cubic Bezier 169—182
Curve segment, cubic/cubic rational polynomial 346 347
Curve segment, Ferguson- 73—80
Curve segment, Hermite- 72—86
Curve segment, increase of degree 85 86 204—209
Curve segment, Lagrange- 64—71
Curve segment, non-uniform B-spline- 296
Curve segment, rational polynomial 337—350
Curve segment, T-comc 347—350
Curve segment, uniform cubic B-spline 233—251
Cusp 15
Cusp by cubic B-spline curve segments 245
De Boor’s algorithm 312—315
Degeneration of a surface patch 44
Degeneration, formation of triangular surface patch 59 to 61
Derivatives of B-sphne curve segments 309—311
Derivatives of Bernstein polynomials 198 199
Derivatives of Bezier curve segments 198 199
Descartes’ sign rule 192
Determination of a point by linear operations 199—204
Determination of a point on a B-sphne curve segment by linear operations 312—315
Determination of a point on a cubic Bezier curve segment 178 179
Divided difference 67
Divided difference, interpolation formula with remainder (Newton’s formula) 68
ellipse 340
Equation of a normal plane 23
Equation of a plane passing through 3 points 353
Equation of a straight line segment 351
Equation of a tangent plane 46
Equation of an osculating plane 23
Error in approximating a circle 80 178
Finite difference, computation 32 33
Finite difference, determination of a point, on a curve 32 33
Finite difference, determination of a point, on a surface 51—55
Finite difference, matrix 33 52—55
Finite difference, operator 32
Finite difference, representation of a Bezier curve segment 194 195
First fundamental matrix of a surface 46—48
FMILL 89
Forrest’s method 117
Frenet frame 29
Frenet — Serret formula 31
Function, B-spline- 236 237 270—273
Function, Bernstein basis- 182 183 189—193
Function, blending- 77
Function, Coons blending- 77
Function, full spline basis- 291
Function, Hermite- 72—83
Function, truncated power- 138
Function, uniform B-spline- 296
Functional determinant 44
Gauss curvature 51
Gauss quadrature 42
Hermite interpolation 72—134
Hermite polynomial 72 —134
Hyperbola 340
Increase of degree of a Bezier curve segment 204—209
Increase of degree of a Ferguson curve segment 85 86
Independence of coordinate axes 12—14
| Interpolation, Hermite- 72—134
Interpolation, Lagrange- 64—71
Interpolation, of a sequence of points by a B-sphne curve 247—250 325—327
Interpolation, spline- 135—168
Intersection of 2 curves 43 44
Intersection of 3 planes 354
Intersection of a curve and a plane 43
Invariance of shape under coordinate transformation 18 19
Inverse transformation of a uniform cubic, B-sphne curve 247—250 325—327
Jacobian 44 60—62
Knot, additional 272
Knot, extended 272
Knot, insertion 316—320
Knot, interior 272
Knot, multiple 291
Knot, pseudo 292
Kronecker delta 92
Lagrange’s formula 65
Leibnitz’ formula 356
Length of a curve 42
Length of a curve, on a surface 47
Length of common perpendicular 352 353
Line segment by a Bezier curve segment 176 177
Line segment by a cubic B-spline curve segment 243
Line segment, perpendicular bisector 351
Local uniqueness 242
Marsden’s identity 358 359
Mathematical model 3 4
Matrix representation of a B-spline curve segment 327—329
Matrix, boundary condition- 111
Matrix, finite difference- 33 52—55
Matrix, first fundamental- (of a surface) 46—48
Matrix, second fundamental- (of a surface) 50
Minimal interpolation property 140—144
Model, mathematical 3 4
Model, physical 3 4
Moving frame 29
Net, surface defining- 217
Normal plane 23
Normal plane, equation 23
Normal vector, unit- 46
Normal, principal- 23
Normalized B-spline 273
Offset surfaces 63
Order of spline 136
Ordinary point 20
Osculating plane 23
parabola 342
Parametric cubic curve 31
Parametric representation of a curve 20
Parametric representation of a surface 44
Partitioning of Coons bi-cubic surface patches 122 123
Partitioning of cubic Bezier curve segments 179—182
Partitioning of curve segments 39
Partitioning of Ferguson curve segments 84 85
Pentagonal surface 9 10
Perpendicular bisector of a line segment 351 352
Physical model 3 4
Plane curve, condition to be 29
Plane, normal 23
Plane, osculating 23
Plane, passing through 3 points 353
Plane, rectifying 24
Plane, tangent 46
Plane, vector representation 353
Point, control- 16
Point, inflection- 27
Point, knot- 240
Point, ordinary 20
Point, regular 20
Point, singular 20
Polynomial, Bernstein- 189
Polynomial, Lagrange- 65
Power basis vector 195
Principal normal, vector, unit- 46
Principle of minimizing total bending energy 136 140
Product of differences 64
Property, minimal interpolation- 140—144
Property, of B-spline curve segments 311 312
Property, of Bernstein polynomials 189—193
Property, of Bezier curve segments 189—193
Property, variation diminishing- 192 294 312
Rectifying plane 24
Regular 20
Second fundamental matrix of a surface 50
Shape control, global 16 17
Shape control, local 16 17
Shape control, of a Coons bi-cubic surface patch 126—131
Shift operator 193 194
Singular point 20
SKETCHPAD 3 4
Space curve 14
Space curve, condition to be 29
Spatial uniqueness 11
SPLINE 135—168
Spline curve 145—160
Spline Function 136 137
Spline interpolation 135—168
Spline surface 163—168
Spline, basis- 160—163 270—273
Spline, bending rigidity 136
Spline, by Bezier curve segments 214—216
Spline, C-spline 139
Spline, end conditions of parametric- 151—159
Spline, fundamental 163
Spline, mathematical 136
Spline, minimal interpolation property 140—144
Spline, natural 138—144
Spline, periodic 159
Spline, physical 136
Spline, smoothing 144
Spline, stored bending energy 136
Spline, under tension 7
Subsphne basis 291
Surface of revolution 264
Surface of revolution by cubic B-spline curve segments 264—270
Surface patch, B-sphne- 334 335
Surface patch, Bezier- 216—226
Surface patch, bicubic Coons- 111 112
Surface patch, bilinear 100
Surface patch, Boolean sum type 133
Surface patch, Cartesian product 132
Surface patch, Coons-(1964) 91—101
Surface patch, Coons-(1967) 102—112
Surface patch, Ferguson- 87—91
Surface patch, general B-spline surface 334 335
Surface patch, loft 132
Surface patch, tensor product 132
Surface patch, triangular 228—231
Tangent plane 46
Tangent plane, equation 46
Tangent vector, unit- 21
Tangent, equation of a tangent line 21
Torsion 28
Torsion of a cubic curve 41
Transformation, bilinear 36
Transformation, homographic 36
Transformation, invariance of shape under coordinate- 18 19
Transformation, inverse 247—250 262—264 325—327
Transformation, inverse, curve 247—250 325—327
Transformation, inverse, surface 262—264
Transformation, parameter 35—39
Transformation, rational formula 36
Triangular area coordinate 229
Triangular surface patch 228—231
Triangular surface patch by degeneration 59 60 131 226—228
Truncated power function 138
Twist vector 91 98 104 112—116
Twist vector, method of determination 117 —122
Types of surfaces 132—133
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