Space filling curves:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Springer-Verlag
1994
|
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xv, 193 Seiten Illustrationen, Diagramme |
| ISBN: | 3540942653 0387942653 9780387942650 |
Staff View
MARC
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface vii
Acknowledgments ix
Illustration Credits xi
Chapter 1. Introduction 1
1.1. A Brief History of Space Filling Curves 1
1.2. Notation 2
1.3. Definitions and Netto s Theorem 4
1.4. Problems 6
Chapter 2. Hubert s Space Filling Curve 9
2.1. Generation of Hilbert s Space Filling Curve 9
2.2. Nowhere Differentiability of the Hilbert Curve 12
2.3. A Complex Representation of the Hilbert
Curve 13
2.4. Arithmetization of the Hilbert Curve 18
2.5. An Analytic Proof of the Nowhere
Differentiability of the Hilbert Curve 19
2.6. Approximating Polygons for the Hilbert Curve 21
2.7. Moore s Version of the Hilbert Curve 24
2.8. A Three Dimensional Hilbert Curve 26
2.9. Problems 29
Chapter 3. Peano s Space Filling Curve 31
3.1. Definition of Peano s Space Filling Curve 31
3.2. Nowhere Differentiability of the Peano Curve 34
3.3. Geometric Generation of the Peano Curve 34
3.4. Proof that the Peano Curve and the Geometric
Peano Curve are the Same 36
3.5. Cesaro s Representation of the Peano Curve 40
3.6. Approximating Polygons for the Peano Curve 42
3.7. Wunderlich s Versions of the Peano Curve 43
xiv Contents
3.8. A Three Dimensional Peano Curve 45
3.9. Problems 46
Chapter 4. Sierpinski s Space Filling Curve 49
4.1. Sierpinski s Original Definition 49
4.2. Geometric Generation and Knopp s
Representation of the Sierpinski Curve 51
4.3. Representation of the Sierpiriski Knopp Curve
in Terms of Quaternaries 56
4.4. Nowhere Differentiability of the Sierpinski
Knopp Curve 58
4.5. Approximating Polygons for the Sierpinski
Knopp Curve 60
4.6. Polya s Generalization of the Sierpiriski Knopp
Curve 62
4.7. Problems 67
Chapter 5. Lebesgue s Space Filling Curve 69
5.1. The Cantor Set 69
5.2. Properties of the Cantor Set 71
5.3. The Cantor Function and the Devil s Staircase 74
5.4. Lebesgue s Definition of a Space Filling Curve 76
5.5. Approximating Polygons for the Lebesgue
Curve 79
5.6. Problems 82
Chapter 6. Continuous Images of a Line Segment 85
6.1. Preliminary Remarks and a Global
Characterization of Continuity 85
6.2. Compact Sets 91
6.3. Connected Sets 94
6.4. Proof of Netto s Theorem 97
6.5. Locally Connected Sets 98
6.6. A Theorem by Hausdorff 99
6.7. Pathwise Connectedness 101
6.8. The Hahn Mazurkiewicz Theorem 106
6.9. Generation of Space Filling Curves by
Stochastically Independent Functions 108
6.10. Representation of a Space Filling Curve by an
Analytic Function 112
6.11. Problems 115
Chapter 7. Schoenberg s Space Filling Curve 119
7.1. Definition and Basic Properties 119
7.2. The Nowhere Differentiability of the
Schoenberg Curve 121
Contents xv
7.3. Approximating Polygons 123
7.4. A Three Dimensional Schoenberg Curve 127
7.5. An No Dimensional Schoenberg Curve 128
7.6. Problems 129
Chapter 8. Jordan Curves of Positive Lebesgue Measure 131
8.1. Jordan Curves 131
8.2. Osgood s Jordan Curves of Positive Measure 132
8.3. The Osgood Curves of Sierpiriski and Knopp 136
8.4. Other Osgood Curves 140
8.5. Problems 142
Chapter 9. Fractals 145
9.1. Examples 145
9.2. The Space where Fractals are Made 149
9.3. The Invariant Attractor Set 154
9.4. Similarity Dimension 156
9.5. Cantor Curves 159
9.6. The Heighway Dragon 162
9.7. Problems 165
Appendix 169
A.I. Computer Programs 169
A.1.1. Computation of the Nodal Points of
the Hilbert Curve 169
A.1.2. Computation of the Nodal Points of
the Peano Curve 170
A.1.3. Computation of the Nodal Points of
the Sierpiriski Knopp Curve 171
A. 1.4. Plotting Program for the Approximating
Polygons of the Schoenberg Curve 172
A.2. Theorems from Analysis 173
A.2.1. Binary and Other Representations 173
A.2.2. Condition for Non Differentiability 174
A.2.3. Completeness of the Euclidean Space 174
A.2.4. Uniform Convergence 174
A.2.5. Measure of the Intersection of a
Decreasing Sequence of Sets 174
A.2.6. Cantor s Intersection Theorem 175
A.2.7. Infinite Products 175
References 177
Index 187
|
| any_adam_object | 1 |
| author | Sagan, Hans 1928-2000 |
| author_GND | (DE-588)1116307014 |
| author_facet | Sagan, Hans 1928-2000 |
| author_role | aut |
| author_sort | Sagan, Hans 1928-2000 |
| author_variant | h s hs |
| building | Verbundindex |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/62 |
| dewey-search | 516.3/62 |
| dewey-sort | 3516.3 262 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV009676036 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:39:03Z |
| institution | BVB |
| isbn | 3540942653 0387942653 9780387942650 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006399664 |
| oclc_num | 29791898 |
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| physical | xv, 193 Seiten Illustrationen, Diagramme |
| publishDate | 1994 |
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| series2 | Universitext |
| spelling | Sagan, Hans 1928-2000 (DE-588)1116307014 aut Space filling curves Hans Sagan Space-filling curves New York Springer-Verlag 1994 xv, 193 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Universitext Courbe Courbes algébriques Courbes sur les surfaces Courbes sur les surfaces ram Fractale Fractales Géométrie différentielle Topologie Topologie ram Curves on surfaces Topology Topologie (DE-588)4060425-1 gnd rswk-swf Kurve (DE-588)4033824-1 gnd rswk-swf Fläche (DE-588)4129864-0 gnd rswk-swf Raumfüllende Kurve (DE-588)4374972-0 gnd rswk-swf Abbildung Mathematik (DE-588)4000044-8 gnd rswk-swf Topologie (DE-588)4060425-1 s DE-604 Kurve (DE-588)4033824-1 s Abbildung Mathematik (DE-588)4000044-8 s Fläche (DE-588)4129864-0 s Raumfüllende Kurve (DE-588)4374972-0 s Erscheint auch als Online-Ausgabe 978-1-4612-0871-6 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006399664&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sagan, Hans 1928-2000 Space filling curves Courbe Courbes algébriques Courbes sur les surfaces Courbes sur les surfaces ram Fractale Fractales Géométrie différentielle Topologie Topologie ram Curves on surfaces Topology Topologie (DE-588)4060425-1 gnd Kurve (DE-588)4033824-1 gnd Fläche (DE-588)4129864-0 gnd Raumfüllende Kurve (DE-588)4374972-0 gnd Abbildung Mathematik (DE-588)4000044-8 gnd |
| subject_GND | (DE-588)4060425-1 (DE-588)4033824-1 (DE-588)4129864-0 (DE-588)4374972-0 (DE-588)4000044-8 |
| title | Space filling curves |
| title_alt | Space-filling curves |
| title_auth | Space filling curves |
| title_exact_search | Space filling curves |
| title_full | Space filling curves Hans Sagan |
| title_fullStr | Space filling curves Hans Sagan |
| title_full_unstemmed | Space filling curves Hans Sagan |
| title_short | Space filling curves |
| title_sort | space filling curves |
| topic | Courbe Courbes algébriques Courbes sur les surfaces Courbes sur les surfaces ram Fractale Fractales Géométrie différentielle Topologie Topologie ram Curves on surfaces Topology Topologie (DE-588)4060425-1 gnd Kurve (DE-588)4033824-1 gnd Fläche (DE-588)4129864-0 gnd Raumfüllende Kurve (DE-588)4374972-0 gnd Abbildung Mathematik (DE-588)4000044-8 gnd |
| topic_facet | Courbe Courbes algébriques Courbes sur les surfaces Fractale Fractales Géométrie différentielle Topologie Curves on surfaces Topology Kurve Fläche Raumfüllende Kurve Abbildung Mathematik |
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