Curves and fractal dimension:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
New York u.a.
Springer
1995
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 323 S. Ill., graph. Darst. |
| ISBN: | 0387940952 3540940952 |
Staff View
MARC
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| 245 | 1 | 0 | |a Curves and fractal dimension |c Claude Tricot |
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Record in the Search Index
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| adam_text | Contents
Foreword, by Michel Mendes France v
Introduction vii
Part I. Sets of Null Measure on the Line
1. Perfect Sets and Their Measure 1
1.1 Duality set—measure 1
1.2 Closed sets and contiguous intervals 2
1.3 Perfect sets 4
1.4 Binary trees and the power of perfect sets 5
1.5 Symmetrical perfect sets 8
1.6 Tree representation of perfect sets 9
1.7 Bibliographical notes 12
2. Covers and Dimension 13
2.1 What is a null measure? 13
2.2 Hierarchy of sets of null measure 15
2.3 Cantor Minkowski measure 16
2.4 Space filling and the order of growth 19
2.5 Orders of growth and dimension 20
2.6 Equivalent definitions of the dimension 23
2.7 Examples of computing the dimension 25
2.8 Some properties of the dimension 26
2.9 Upper and lower dimensions 27
2.10 Bibliographical notes 29
3. Contiguous Intervals and Dimension 33
3.1 Borel s logarithmic rarefaction 33
3.2 Index of Besicovitch Taylor 34
3.3 Equivalent orders of growth 34
3.4 The contiguous intervals and the fractal dimension 36
3.5 Algorithms to compute the dimension 38
3.6 Bibliographical notes 40
Contents XI
Part II. Rectifiable Curves
4. What Is a Curve? 43
4.1 Some types of sets in the plane 43
4.2 Velocities, trajectories 44
4.3 The definition of a curve 45
4.4 Bibliographical notes 46
5. Polygonal Curves and Length 47
5.1 Rectifiability 47
5.2 Hausdorff distance 47
5.3 Polygonal approximations 50
5.4 The length of a curve 51
5.5 Two distinct notions 54
5.6 Measuring the length by compass 56
5.7 Bibliographical notes 57
6. Parameterized Curves, Support of a Measure 59
6.1 Parameterization by arc length 59
6.2 Image measure 60
6.3 Length by instantaneous velocity 60
6.4 The devil staircase 62
6.5 Length by the average of local velocity 67
6.6 Bibliographical notes 70
7. Local Geometry of Rectifiable Curves 71
7.1 Tangent, cone, convex hulls 71
7.2 Relations between local properties 73
7.3 Counterexamples 75
7.4 Tangent almost everywhere 78
7.5 Local length, almost everywhere 80
7.6 Rectifiability revisited 81
7.7 Bibliographical notes 82
8. Length, by Intersections with Straight Lines 85
8.1 Intersections, projections 85
8.2 The measure of families of straight lines 86
8.3 Family of lines intersecting a set 89
8.4 The case of convex sets 91
8.5 Length by secant lines 93
8.6 The length by projections 97
8.7 Application: practical computation of length 98
8.8 The length by random intersections 100
8.9 Buffon needle 101
XII Contents
8.10 Bibliographical notes 102
9. The Length by the Area of Centered Balls 105
9.1 Minkowski sausage 105
9.2 Length by the area of sausages 106
9.3 Convergence of the algorithm of the sausages 110
9.4 Reduction of balls to parallel segments 112
9.5 Bibliographical notes 114
Part III. Nonrectifiable Curves
10. Curves of Infinite Length 115
10.1 What is infinite length? 115
10.2 Two examples 116
10.3 Dimension 119
10.4 Some examples of dimensions of curves 120
10.5 Classical covers: balls and boxes 123
10.6 Covers by figures of any kind 128
10.7 Covering curves by crosses 130
10.8 Bibliographical notes 133
11. Fractal Curves 135
11.1 What is a fractal curve? 135
11.2 A fractal curve is nowhere rectifiable 137
11.3 Diameter, size 139
11.4 Characterization of a fractal curve 141
12. Graphs of Nondifferentiable Functions 143
12.1 Curves parameterized by the abscissa 143
12.3 Size of local arcs 144
12.3 Variation of a function 145
12.4 Fractal dimension of a graph 148
12.5 Holder exponent 150
12.6 Functions defined by series 152
12.7 Weierstrass function 154
12.8 Fractal dimension and the structure function 157
12.9 Functions constructed by diagonal affinities 160
12.10Invariance under change of scale 163
12.11 The Weierstrass Mandelbrot function 166
12.12 The spectrum of invariant functions 168
12.13 Computing the dimensions of the graphs 170
12.14 Bibliographical notes 174
13. Curves Constructed by Similarities 177
Contents XIII
13.1 Similarities 177
13.2 Self similar structure 179
13.3 Generator 180
13.4 Self similar structure on [0,1] 182
13.5 Parameterization of the generator 183
13.6 The limit curve T 185
13.7 Simplicity criterion 187
13.8 Similarity and dimension exponent 190
13.9 Examples 192
13.10 The natural parameterization 195
13.11 The algorithm of local sizes 199
13.12 Bibliographical notes 201
14. Deviation, and Expansive Curves 203
14.1 Introducing new notions 203
14.2 Deviation of a set 203
14.3 Constant deviation along a curve 206
14.4 Definition of an expansive curve 208
14.5 Expansivity criterion 209
14.6 Expansivity and self similarity 213
14.7 How to construct an expansive curve 214
14.8 Bibliographical notes 220
15. The Constant Deviation Variable Step Algorithm 221
15.1 A unified analysis of expansive curves 221
15.2 The covering index 222
15.3 Convex hulls and Minkowski sausages 223
15.4 A theorem on the dimension: the discrete form 225
15.5 Applications 227
15.6 Statistical self similarity 232
15.7 Curves of uniform deviation 234
15.8 Applications 236
15.9 The dimension of a curve 239
15.10 Bibliographical notes 242
16. Scanning a Curve with Straight Lines 243
16.1 Directional dimension 243
16.2 Comparing the dimensions 245
16.3 Examples and applications 246
16.4 Coordinate systems 247
16.5 Intersections by straight lines 251
16.6 Essential upper bound 253
16.7 Uniform intersections 255
16.8 Intersection with an average curve 256
16.9 Bibliographical notes 258
XIV Contents
17. Lateral Dimension of a Curve 259
17.1 Semisausages 259
17.2 Other expressions of the lateral dimensions 260
17.3 Possible values of the lateral dimension 263
17.4 Examples 264
17.5 The inverse Minkowski operation 268
17.6 Bibliographical notes 271
18. Dimensional Homogeneity 273
18.1 Local structures of some curves 273
18.2 Local dimension 274
18.3 The packing dimension 277
18.4 Possible values of the packing dimension 279
18.5 The ^ stabilization 282
18.6 Bibliographical notes 283
Part IV. Annexes, References and Index
A. Upper Limit and Lower Limit 285
A.I Convergence 285
A.2 Nonconvergent sequences 287
A.3 Nonconvergent functions 288
A.4 Limits of the ratio log f(e)/ log g(e) 289
A.5 Some applications 291
B. Two Covering Lemmas 293
B.I Vitali s lemma 293
B.2 Covers by homothetic convex sets 296
C. Convex Sets in the Plane 301
C.I Convexity 301
C.2 Size of a convex set 302
C.3 Breadth of a convex set 305
C.4 Area of a convex set 310
C.5 Convex hull 310
C.6 Perimeter of the convex hull 312
C.7 Area of the convex hull of a curve 313
References 315
Index 318
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T17:43:54Z |
| institution | BVB |
| isbn | 0387940952 3540940952 |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006598009 |
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| physical | XIV, 323 S. Ill., graph. Darst. |
| publishDate | 1995 |
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| spelling | Tricot, Claude Verfasser aut Courbes et dimension fractale Curves and fractal dimension Claude Tricot New York u.a. Springer 1995 XIV, 323 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Courbes planes Fractales Curves, Plane Fractals Ebene Kurve (DE-588)4150970-5 gnd rswk-swf Fraktal (DE-588)4123220-3 gnd rswk-swf Fraktal (DE-588)4123220-3 s DE-604 Ebene Kurve (DE-588)4150970-5 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006598009&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tricot, Claude Curves and fractal dimension Courbes planes Fractales Curves, Plane Fractals Ebene Kurve (DE-588)4150970-5 gnd Fraktal (DE-588)4123220-3 gnd |
| subject_GND | (DE-588)4150970-5 (DE-588)4123220-3 |
| title | Curves and fractal dimension |
| title_alt | Courbes et dimension fractale |
| title_auth | Curves and fractal dimension |
| title_exact_search | Curves and fractal dimension |
| title_full | Curves and fractal dimension Claude Tricot |
| title_fullStr | Curves and fractal dimension Claude Tricot |
| title_full_unstemmed | Curves and fractal dimension Claude Tricot |
| title_short | Curves and fractal dimension |
| title_sort | curves and fractal dimension |
| topic | Courbes planes Fractales Curves, Plane Fractals Ebene Kurve (DE-588)4150970-5 gnd Fraktal (DE-588)4123220-3 gnd |
| topic_facet | Courbes planes Fractales Curves, Plane Fractals Ebene Kurve Fraktal |
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