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Micchelli C.A. — Mathematical Aspects of Geometric Modeling

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Название: Mathematical Aspects of Geometric Modeling

Автор: Micchelli C.A.

Аннотация:

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.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1987

Количество страниц: 256

Добавлена в каталог: 26.10.2010

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Предметный указатель

Affinely independent points      155
Affinely independent vectors      156 242
Affinely independent vectors,      243
B-patches      227—233
B-patches as B-splines      247—251
B-patches, pyramid schemes and      234—237
B-patches, subdivision by      242—247
B-splines as volumes      149—155
B-splines, $\textit{n}th$ degree forward      58
B-splines, B-patches as      247—251
B-splines, basis      123—131
B-splines, basis, dual      215—219
B-splines, basis, variation diminishing property of      143—146
B-splines, bivariate      154—155 156 200 201—202
B-splines, correspondence on historical development      200—204
B-splines, fine knot quadratic      61
B-splines, geometrically continuous      133
B-splines, multivariate, Bernstein — Bezier polynomials and      169—171
B-splines, multivariate, defined      155
B-splines, multivariate, degree-raising formula for      169 171
B-splines, multivariate, multiple points and explicit formula      169—176
B-splines, multivariate, smoothness and recursions      155—169
B-splines, multivariate, square of      171
B-splines, multivariate, truncated power functions and      189—191
B-splines, Polya frequency functions and      203
B-splines, quadratic      59 61
B-splines, Schoenberg’s geometric construction of      150—153
B-splines, series, blossoming, univariate case      215—219
B-splines, sum of products of      171
B-splines, univariate      153 200—201 203—204
Bernstein operator      8—9
Bernstein operator, $\beta$ -splines      106—107
Bernstein — Bezier basis      2
Bernstein — Bezier curve, reparameterization      26 27
Bernstein — Bezier manifold, multivariate,      27
Bernstein — Bezier polynomials, blossoming, univariate case      211 215
Bernstein — Bezier polynomials, dual basis for      214
Bernstein — Bezier polynomials, multivariate of degree n      27
Bernstein — Bezier polynomials, multivariate, B-splines and      169—171
Bernstein — Bezier polynomials, multivariate, on affine space      221
Bernstein — Bezier polynomials, multivariate, recursion      208 210 229—230 232
Bias      106
Binary fractions      7
Binomial theorem      139 140 141
Biorthogonal vectors      183
Bivariate B-splines      154 155 156 200 201—202
Blending functions      115—116
Blossoming, B-spline series, univariate case      215—219
Blossoming, Bernstein — Bezier polynomials, univariate case      211—215
Blossoming, de Casteljau’s triangle for a cubic curve      213
Blossoming, multivariate      219—222
Blossoming, polynomials      211—212
Blossoming, up recurrence formula for quadratic Bernstein — Bezier polynomials      232
Brunn — Minkowski inequality      154
Budan — Fourier lemma      127—128
Cardinal spline functions      58 61
Cauchy — Binet formula      38
Cholesky factorization of matrices      113
Common supporting hyperplane      161
Connection matrices      116—121 123—131
Connects, defined      34
Continuity, Frenet frame      113—115 116—117
Continuity, geometric of order n      110
Continuity, geometric, B-splines      133
Continuity, geometric, reparameterization matrices and      105—110
Continuity, modulus of      14
Control point, control polygon paradigm      1—10
Convergence, de Casteljau algorithm      10
Convergence, de Casteljau subdivision to cubic curves      4
Convergence, matrix subdivision      10—20
Convergence, stationary subdivision      67—83
Convex polyhedrons      160
Corner cutting      3 34—38 40
Correspondence on historical development of B-splines      200—204
Cube splines      192—200
Cubic curves, blossoming de Casteljau’s 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      207—211
Curvatures, de Casteljau subdivision      1—10
Curvatures, de Casteljau tableau      3
Curvatures, de Rahm — Chaikin algorithm      55—58
Curvatures, Frenet equation and      110—115
Degree raising formula for multivariate B-splines      169—171
Dual basis for B-splines      215—219
Dual basis for Bernstein — Bezier polynomials      214
Dual functionals      135 143
Exceptional point      160
Faa di Bruno formula      109
Frenet equation      110—115
Frenet frame      111—117
Frenet matrices      110—115 117 121—122
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      105—110
Gram matrix of vectors      113
Hermite — Gennochi formula      201
Histogram of vectors      223—224
Hoelder continuous functions      17
Hoelder’s inequality      16—17
Hurwitz matrices      86—87 89—90
Hurwitz polynomials      83—95
Jordan normal form of matrices      246
Knots, fine-knot quadratic B-splines      61
Knots, insertion and variation diminishing property of the B-spline basis      143—146
Knots, insertion for multivariate B-splines      163—166
Knots, insertion, identity for truncated power functions      179—181
Knots, knot regions for bivariate quadratic B-splines      157
Knots, lifting      151
Lagrange polynomials      215
Lane — Riesenfeld subdivision      61—67
Laurent polynomials      68 83 84 95 98 102
Lifting knots      151
Lifting of curves      115—122
Lineal polynomials      219—227
Linear cube spline      194
M-patches      237—242
Mask of stationary subdivision      68
Matrices, Cholesky factorization      113
Matrices, connection      116—121 123—131
Matrices, connects, defined      34
Matrices, Frenet      110—115 117 121—122
Matrices, Gram matrix of vectors      113
Matrices, Hurwitz      86—87 89—90
Matrices, Jordan normal form      246
Matrices, permanent of      28—29
Matrices, permutation      5
Matrices, reparameterization and geometric continuity      105—110
Matrices, reparameterization defined      109
Matrices, reparameterization examples      20—29
Matrices, signum of      30—31
Matrices, stochastic      29—34
Matrices, strictly totally positive      41—43
Matrices, subdivision      6—10
Matrices, totally positive, defined      37

Matrices, totally positive, product of two totally positive matrices      37—38
Matrices, totally positive, strictly totally positive      41—43
Matrices, totally positive, variation diminishing curves and      38—53
Matrix subdivision      (see also “Stationary subdivision”)
Matrix subdivision, convergence criteria      10—20
Matrix subdivision, corner cutting      34—38
Matrix subdivision, de Casteljau subdivision      1 10
Matrix subdivision, example      33
Matrix subdivision, introduced      1
Matrix subdivision, reparameterization examples      20—29
Matrix subdivision, stochastic matrices      29—34
Matrix subdivision, total positiviti and variation diminishing curves      38—53
Modulus of continuity      14
Multiaffine functions      211—212
Multiple points of multivariate B-splines      169—176
Multivariate B-splines, Bernstein — Bezier polynomials and      169—171
Multivariate B-splines, defined      155
Multivariate B-splines, degree-raising formula for      169—171
Multivariate B-splines, multiple points and explicit formula      169—176
Multivariate B-splines, smoothness and recursions      155—169
Multivariate B-splines, square of      171
Multivariate B-splines, truncated power functions and      189—191
Multivariate Bernstein — Bezier manifold,      27
Multivariate Bernstein — Bezier, polynomials, of degree n      27
Multivariate blossoming      219 222
Multivariate partial fraction, decomposition      172 176
Multivariate truncated powers, identities      186—192
Multivariate truncated powers, smoothness and recursion      176—186
One periodic functions      22
Partial fraction decomposition, multivariate      172—176
Patches, B-patches      227—233
Patches, B-patches as B-splines      247—251
Patches, B-patches, pyramid schemes and      234—237
Patches, B-patches, subdivision by      242—247
Patches, M-patches      237—242
Permanent of matrices      28—29
Permutation matrix      5
Piecewise polynomial curves, B-spline basis      123—131
Piecewise polynomial curves, connection matrices and      123—131
Piecewise polynomial curves, curvatures and Frenet equation, Frenet matrices      110—115
Piecewise polynomial curves, diminishing property of the B-spline basis      143—146
Piecewise polynomial curves, dual functionals      135 143
Piecewise polynomial curves, introduced      105
Piecewise polynomial curves, projection and lifting of curves      115—122
Piecewise polynomial curves, reparameterization matrices      105—110
Piecewise polynomial surfaces correspondence on historical, cube splines      192—200
Piecewise polynomial surfaces correspondence on historical, development of B-splines      200—204
Piecewise polynomial surfaces correspondence on historical, geometric methods, B-splines as volumes      149—155
Piecewise polynomial surfaces correspondence on historical, geometric methods, introduced      149
Piecewise polynomial surfaces correspondence on historical, geometric methods, multiple points and explicit formula      169—176
Planar control polygon      2
Point zeros of multiplicity n      124
Polarization      (see “Blossoming”)
Polya frequency functions and B-splines      203
Polyhedrons, convex      160
Polynomials      (see “Piecewise” “Polynomial “Piecewise “Recursion”)
Polynomials, Bernstein — Bezier      (see “Bernstein — Bezier polynomials”)
Polynomials, blossom of      211—212
Polynomials, Hurwitz      83—95
Polynomials, Lagrange      215
Polynomials, Laurent      68 83 84 95 98 102
Polynomials, lineal      219—227
Polynomials, Roth — Hurwitz      88
Polytopes      160
Pre-wavelets      100
Projection of curves      115 122 116
Proper triangle      158
pyramid schemes      234—242
Quadratic B-splines      58 61
Quadratic M-patches      238
Quadratic truncated power      180 187
Recursion, B-patches      227—230
Recursion, Bernstein — Bezier polynomials      208—210 229—230 232
Recursion, blossoming and      (see “Blossoming”)
Recursion, de Casteljau      207—211
Recursion, introduced      207
Recursion, M-patches      238
Recursion, multivariate B-splines      155—169 166—169
Recursion, multivariate truncated powers      176 186
Recursion, pyramid schemes      234
Recursion, truncated power functions      181—182
Recursive triangles      209—210
Regular curves      108
Reparameterization matrices and geometric continuity      105—110
Reparameterization matrices, defined      109
Reparameterization matrices, examples      20—29
Reparameterization of the Bernstein — Bezier curve      26—27
Ripplets      96
Rolle’s Theorem      128
Roth — Hurwitz criterion      87
Roth — Hurwitz polynomials      88
Schoenberg operator      64
Schoenberg’s geometric construction of B-splines      150—153
Sets in general position      156
Signum of matrices      30—31
Smoothness, multivariate B-splines      155 169
Smoothness, multivariate truncated powers      176—186
Splines, $\beta$ -splines      106—107
Splines, B-splines      (see “B-splines”)
Splines, cardinal spline functions      58 61
Splines, cube splines      192 200
Standard d-simplex      27
Stationary subdivision      (see also “Matrix subdivision”)
Stationary subdivision, cardinal spline functions      58—61
Stationary subdivision, convergence      67 83
Stationary subdivision, de Rahm — Chaikin algorithm      55—58
Stationary subdivision, Hurwitz polynomials and      83—95
Stationary subdivision, introduced      55
Stationary subdivision, Lane — Riesenfeld subdivision      61—67
Stationary subdivision, mask      68
Stationary subdivision, symbol      69
Stationary subdivision, wavelet decomposition      96 103
Stochastic matrices      29—34
Strictly totally positive matrices      41 43
Subdivision by B-patches      242—247
Subdivision matrices      6—10
Subdivision operator      68
Subdivision, matrix      (see “Matrix subdivision”)
Subdivision, stationary      (see “Stationary subdivision”)
Sylvester’s determinantal identity      43
Symbol of stationary subdivision      69
Tension      106
Totally positive matrices, defined      37
Totally positive matrices, product of two totally positive matrices      37—38
Totally positive matrices, strictly totally positive      41—43
Totally positive matrices, variation diminishing curves and      38—53
Truncated powers, multivariate, identities      186—192
Truncated powers, multivariate, smoothness and recursion      176—186
Univariate B-splines      153 200 201 203—204
Variation diminishing curves and total positivity      38—53
Variation diminishing property of the B-spline basis      143—146
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      223—224
Wavelet decomposition      96—103
Zero counting convention      124—125

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