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Micchelli C.A. Ч Mathematical Aspects of Geometric Modeling
Micchelli C.A. Ч Mathematical Aspects of Geometric Modeling

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Ќазвание: Mathematical Aspects of Geometric Modeling

јвтор: Micchelli C.A.


This monograph examines in detail certain concepts that are useful for the modeling of curves and surfaces and emphasizes the mathematical theory that underlies these ideas. The two principal themes of the text are the use of piecewise polynomial representation (this theme appears in one form or another in every chapter), and iterative refinement, also called subdivision. Here, simple iterative geometric algorithms produce, in the limit, curves with complex analytic structure. In the first three chapters, the de Casteljau subdivision for Bernstein-Bezier curves is used to introduce matrix subdivision, and the Lane-Riesenfield algorithm for computing cardinal splines is tied into stationary subdivision. This ultimately leads to the construction of prewavelets of compact support. The remainder of the book deals with concepts of "visual smoothness" of curves, along with the intriguing idea of generating smooth multivariate piecewise polynomials as volumes of "slices" of polyhedra.

язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1987

 оличество страниц: 256

ƒобавлена в каталог: 26.10.2010

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ѕредметный указатель
Affinely independent points      155
Affinely independent vectors      156 242
Affinely independent vectors,      243
B-patches      227Ч233
B-patches as B-splines      247Ч251
B-patches, pyramid schemes and      234Ч237
B-patches, subdivision by      242Ч247
B-splines as volumes      149Ч155
B-splines, $\textit{n}th$ degree forward      58
B-splines, B-patches as      247Ч251
B-splines, basis      123Ч131
B-splines, basis, dual      215Ч219
B-splines, basis, variation diminishing property of      143Ч146
B-splines, bivariate      154Ч155 156 200 201Ч202
B-splines, correspondence on historical development      200Ч204
B-splines, fine knot quadratic      61
B-splines, geometrically continuous      133
B-splines, multivariate, Bernstein Ч Bezier polynomials and      169Ч171
B-splines, multivariate, defined      155
B-splines, multivariate, degree-raising formula for      169 171
B-splines, multivariate, multiple points and explicit formula      169Ч176
B-splines, multivariate, smoothness and recursions      155Ч169
B-splines, multivariate, square of      171
B-splines, multivariate, truncated power functions and      189Ч191
B-splines, Polya frequency functions and      203
B-splines, quadratic      59 61
B-splines, SchoenbergТs geometric construction of      150Ч153
B-splines, series, blossoming, univariate case      215Ч219
B-splines, sum of products of      171
B-splines, univariate      153 200Ч201 203Ч204
Bernstein operator      8Ч9
Bernstein operator, $\beta$-splines      106Ч107
Bernstein Ч Bezier basis      2
Bernstein Ч Bezier curve, reparameterization      26 27
Bernstein Ч Bezier manifold, multivariate,      27
Bernstein Ч Bezier polynomials, blossoming, univariate case      211 215
Bernstein Ч Bezier polynomials, dual basis for      214
Bernstein Ч Bezier polynomials, multivariate of degree n      27
Bernstein Ч Bezier polynomials, multivariate, B-splines and      169Ч171
Bernstein Ч Bezier polynomials, multivariate, on affine space      221
Bernstein Ч Bezier polynomials, multivariate, recursion      208 210 229Ч230 232
Bias      106
Binary fractions      7
Binomial theorem      139 140 141
Biorthogonal vectors      183
Bivariate B-splines      154 155 156 200 201Ч202
Blending functions      115Ч116
Blossoming, B-spline series, univariate case      215Ч219
Blossoming, Bernstein Ч Bezier polynomials, univariate case      211Ч215
Blossoming, de CasteljauТs triangle for a cubic curve      213
Blossoming, multivariate      219Ч222
Blossoming, polynomials      211Ч212
Blossoming, up recurrence formula for quadratic Bernstein Ч Bezier polynomials      232
Brunn Ч Minkowski inequality      154
Budan Ч Fourier lemma      127Ч128
Cardinal spline functions      58 61
Cauchy Ч Binet formula      38
Cholesky factorization of matrices      113
Common supporting hyperplane      161
Connection matrices      116Ч121 123Ч131
Connects, defined      34
Continuity, Frenet frame      113Ч115 116Ч117
Continuity, geometric of order n      110
Continuity, geometric, B-splines      133
Continuity, geometric, reparameterization matrices and      105Ч110
Continuity, modulus of      14
Control point, control polygon paradigm      1Ч10
Convergence, de Casteljau algorithm      10
Convergence, de Casteljau subdivision to cubic curves      4
Convergence, matrix subdivision      10Ч20
Convergence, stationary subdivision      67Ч83
Convex polyhedrons      160
Corner cutting      3 34Ч38 40
Correspondence on historical development of B-splines      200Ч204
Cube splines      192Ч200
Cubic curves, blossoming de CasteljauТs triangle for      213
Cubic curves, convergence of de Casteljau subdivison to      4
Cubic curves, corner cutting      34 36
Cubic curves, de Casteljau evaluation of      208
Curvature vector for planar curves      106
Curvatures of vectors      112
Curvatures, de Casteljau recurrence formula      207Ч211
Curvatures, de Casteljau subdivision      1Ч10
Curvatures, de Casteljau tableau      3
Curvatures, de Rahm Ч Chaikin algorithm      55Ч58
Curvatures, Frenet equation and      110Ч115
Degree raising formula for multivariate B-splines      169Ч171
Dual basis for B-splines      215Ч219
Dual basis for Bernstein Ч Bezier polynomials      214
Dual functionals      135 143
Exceptional point      160
Faa di Bruno formula      109
Frenet equation      110Ч115
Frenet frame      111Ч117
Frenet matrices      110Ч115 117 121Ч122
General position, sets in      156
Geometric continuity B-splines      133
Geometric continuity B-splines of order n      110
Geometric continuity B-splines, reparameterization matrices and      105Ч110
Gram matrix of vectors      113
Hermite Ч Gennochi formula      201
Histogram of vectors      223Ч224
Hoelder continuous functions      17
HoelderТs inequality      16Ч17
Hurwitz matrices      86Ч87 89Ч90
Hurwitz polynomials      83Ч95
Jordan normal form of matrices      246
Knots, fine-knot quadratic B-splines      61
Knots, insertion and variation diminishing property of the B-spline basis      143Ч146
Knots, insertion for multivariate B-splines      163Ч166
Knots, insertion, identity for truncated power functions      179Ч181
Knots, knot regions for bivariate quadratic B-splines      157
Knots, lifting      151
Lagrange polynomials      215
Lane Ч Riesenfeld subdivision      61Ч67
Laurent polynomials      68 83 84 95 98 102
Lifting knots      151
Lifting of curves      115Ч122
Lineal polynomials      219Ч227
Linear cube spline      194
M-patches      237Ч242
Mask of stationary subdivision      68
Matrices, Cholesky factorization      113
Matrices, connection      116Ч121 123Ч131
Matrices, connects, defined      34
Matrices, Frenet      110Ч115 117 121Ч122
Matrices, Gram matrix of vectors      113
Matrices, Hurwitz      86Ч87 89Ч90
Matrices, Jordan normal form      246
Matrices, permanent of      28Ч29
Matrices, permutation      5
Matrices, reparameterization and geometric continuity      105Ч110
Matrices, reparameterization defined      109
Matrices, reparameterization examples      20Ч29
Matrices, signum of      30Ч31
Matrices, stochastic      29Ч34
Matrices, strictly totally positive      41Ч43
Matrices, subdivision      6Ч10
Matrices, totally positive, defined      37
Matrices, totally positive, product of two totally positive matrices      37Ч38
Matrices, totally positive, strictly totally positive      41Ч43
Matrices, totally positive, variation diminishing curves and      38Ч53
Matrix subdivision      (see also УStationary subdivisionФ)
Matrix subdivision, convergence criteria      10Ч20
Matrix subdivision, corner cutting      34Ч38
Matrix subdivision, de Casteljau subdivision      1 10
Matrix subdivision, example      33
Matrix subdivision, introduced      1
Matrix subdivision, reparameterization examples      20Ч29
Matrix subdivision, stochastic matrices      29Ч34
Matrix subdivision, total positiviti and variation diminishing curves      38Ч53
Modulus of continuity      14
Multiaffine functions      211Ч212
Multiple points of multivariate B-splines      169Ч176
Multivariate B-splines, Bernstein Ч Bezier polynomials and      169Ч171
Multivariate B-splines, defined      155
Multivariate B-splines, degree-raising formula for      169Ч171
Multivariate B-splines, multiple points and explicit formula      169Ч176
Multivariate B-splines, smoothness and recursions      155Ч169
Multivariate B-splines, square of      171
Multivariate B-splines, truncated power functions and      189Ч191
Multivariate Bernstein Ч Bezier manifold,      27
Multivariate Bernstein Ч Bezier, polynomials, of degree n      27
Multivariate blossoming      219 222
Multivariate partial fraction, decomposition      172 176
Multivariate truncated powers, identities      186Ч192
Multivariate truncated powers, smoothness and recursion      176Ч186
One periodic functions      22
Partial fraction decomposition, multivariate      172Ч176
Patches, B-patches      227Ч233
Patches, B-patches as B-splines      247Ч251
Patches, B-patches, pyramid schemes and      234Ч237
Patches, B-patches, subdivision by      242Ч247
Patches, M-patches      237Ч242
Permanent of matrices      28Ч29
Permutation matrix      5
Piecewise polynomial curves, B-spline basis      123Ч131
Piecewise polynomial curves, connection matrices and      123Ч131
Piecewise polynomial curves, curvatures and Frenet equation, Frenet matrices      110Ч115
Piecewise polynomial curves, diminishing property of the B-spline basis      143Ч146
Piecewise polynomial curves, dual functionals      135 143
Piecewise polynomial curves, introduced      105
Piecewise polynomial curves, projection and lifting of curves      115Ч122
Piecewise polynomial curves, reparameterization matrices      105Ч110
Piecewise polynomial surfaces correspondence on historical, cube splines      192Ч200
Piecewise polynomial surfaces correspondence on historical, development of B-splines      200Ч204
Piecewise polynomial surfaces correspondence on historical, geometric methods, B-splines as volumes      149Ч155
Piecewise polynomial surfaces correspondence on historical, geometric methods, introduced      149
Piecewise polynomial surfaces correspondence on historical, geometric methods, multiple points and explicit formula      169Ч176
Planar control polygon      2
Point zeros of multiplicity n      124
Polarization      (see УBlossomingФ)
Polya frequency functions and B-splines      203
Polyhedrons, convex      160
Polynomials      (see УPiecewiseФ УPolynomial УPiecewise УRecursionФ)
Polynomials, Bernstein Ч Bezier      (see УBernstein Ч Bezier polynomialsФ)
Polynomials, blossom of      211Ч212
Polynomials, Hurwitz      83Ч95
Polynomials, Lagrange      215
Polynomials, Laurent      68 83 84 95 98 102
Polynomials, lineal      219Ч227
Polynomials, Roth Ч Hurwitz      88
Polytopes      160
Pre-wavelets      100
Projection of curves      115 122 116
Proper triangle      158
pyramid schemes      234Ч242
Quadratic B-splines      58 61
Quadratic M-patches      238
Quadratic truncated power      180 187
Recursion, B-patches      227Ч230
Recursion, Bernstein Ч Bezier polynomials      208Ч210 229Ч230 232
Recursion, blossoming and      (see УBlossomingФ)
Recursion, de Casteljau      207Ч211
Recursion, introduced      207
Recursion, M-patches      238
Recursion, multivariate B-splines      155Ч169 166Ч169
Recursion, multivariate truncated powers      176 186
Recursion, pyramid schemes      234
Recursion, truncated power functions      181Ч182
Recursive triangles      209Ч210
Regular curves      108
Reparameterization matrices and geometric continuity      105Ч110
Reparameterization matrices, defined      109
Reparameterization matrices, examples      20Ч29
Reparameterization of the Bernstein Ч Bezier curve      26Ч27
Ripplets      96
RolleТs Theorem      128
Roth Ч Hurwitz criterion      87
Roth Ч Hurwitz polynomials      88
Schoenberg operator      64
SchoenbergТs geometric construction of B-splines      150Ч153
Sets in general position      156
Signum of matrices      30Ч31
Smoothness, multivariate B-splines      155 169
Smoothness, multivariate truncated powers      176Ч186
Splines, $\beta$-splines      106Ч107
Splines, B-splines      (see УB-splinesФ)
Splines, cardinal spline functions      58 61
Splines, cube splines      192 200
Standard d-simplex      27
Stationary subdivision      (see also УMatrix subdivisionФ)
Stationary subdivision, cardinal spline functions      58Ч61
Stationary subdivision, convergence      67 83
Stationary subdivision, de Rahm Ч Chaikin algorithm      55Ч58
Stationary subdivision, Hurwitz polynomials and      83Ч95
Stationary subdivision, introduced      55
Stationary subdivision, Lane Ч Riesenfeld subdivision      61Ч67
Stationary subdivision, mask      68
Stationary subdivision, symbol      69
Stationary subdivision, wavelet decomposition      96 103
Stochastic matrices      29Ч34
Strictly totally positive matrices      41 43
Subdivision by B-patches      242Ч247
Subdivision matrices      6Ч10
Subdivision operator      68
Subdivision, matrix      (see УMatrix subdivisionФ)
Subdivision, stationary      (see УStationary subdivisionФ)
SylvesterТs determinantal identity      43
Symbol of stationary subdivision      69
Tension      106
Totally positive matrices, defined      37
Totally positive matrices, product of two totally positive matrices      37Ч38
Totally positive matrices, strictly totally positive      41Ч43
Totally positive matrices, variation diminishing curves and      38Ч53
Truncated powers, multivariate, identities      186Ч192
Truncated powers, multivariate, smoothness and recursion      176Ч186
Univariate B-splines      153 200 201 203Ч204
Variation diminishing curves and total positivity      38Ч53
Variation diminishing property of the B-spline basis      143Ч146
Vectors, affinely independent      156 242 243
Vectors, biorthogonal      183
Vectors, curvature of      112
Vectors, curvature, for planar curves      106
Vectors, Gram matrix of      113
Vectors, histogram of      223Ч224
Wavelet decomposition      96Ч103
Zero counting convention      124Ч125
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