Rational families of circles and bicircular quartics: = Rationale Kreisscharen und bizirkulare Quartiken
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Hochschulschrift/Dissertation Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
2012
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| Schlagwörter: | |
| Links: | https://open.fau.de/handle/openfau/2301 https://nbn-resolving.org/urn:nbn:de:bvb:29-opus-34320 http://d-nb.info/1024608662/34 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025261686&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | 139 S. |
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| 245 | 1 | 0 | |a Rational families of circles and bicircular quartics |b = Rationale Kreisscharen und bizirkulare Quartiken |c von Thomas Rainer Werner |
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Datensatz im Suchindex
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| adam_text | IMAGE 1
CONTENTS
1 INTRODUCTION 5
1.1 MAIN RESULTS 5
PROJECTIVE SPACE OF CIRCLES 5
SOLUTION OF THE GENERAL EQUATION OF DEGREE FOUR 5
GEOMETRIC AND COMPUTATIONAL RESULTS 6
1.2 MOTIVATION 6
QUADRATIC FAMILIES OF CIRCLES 6
INVERSIVE GEOMETRY 7
SYSTEMATIC ASPECTS 8
1.3 HISTORY AND SOURCES 9
ANCIENT MATHEMATICS 9
FROM THE RENAISSANCE TO THE INDUSTRIAL REVOLUTION 9
IRISH MATHEMATICS IN THE 19TH CENTURY 10
IMPORTANT SOURCES 10
2 ALGEBRAIC G E O M E T R Y 1 4
2.1 BASIC TERMS 14
AFFINE AND PROJECTIVE SPACE 14
ALGEBRAIC CURVES 16
2.2 GEOMETRIC TOOLS 18
POINTS ON AN ALGEBRAIC CURVE 18
DUAL CURVE AND PLIICKER EQUATIONS 19
3 CIRCLES 2 1
3.1 REAL AND COMPLEX CIRCLES 21
CLASSICAL DEFINITION 21
CIRCLES IN C = R 2 22
DESCRIPTION USING A MOBIUS TRANSFORMATION 23
COMPLIANCE WITH THE CLASSICAL DEFINITION 23
CIRCLES IN C 2 AND P 2(C) 24
3.2 THE SPACE OF CIRCLES 25
REAL AND IMAGINARY CIRCLES, NULLCIRCLES 25
THE SPACE OF CIRCLES 26
TANGENTIAL PIANOS OF FI 27
THE POLAR PLANE OF A REPRESENTANT C 28
3.3 ELEMENTARY GEOMETRY 29
POLARITY AND ORTHOGONALITY 29
1
HTTP://D-NB.INFO/1026129257
IMAGE 2
CONTENTS
ASSOCIATED CIRCLES 30
ANGLE OF INTERSECTION 31
4 TRANSFORMATIONS 3 3
4.1 TRANSLATIONS, ROTATIONS AND DILATIONS 33
TRANSLATIONS 33
ROTATIONS 34
DILATIONS 35
4.2 INVERSIONS 36
THE UNIT INVERSION E 36
GENERAL INVERSIONS 37
MODIFICATION OF THE CONSTRUCTION 38
INVERSION OF ALGEBRAIC CURVES 38
4.3 PROPERTIES OF THE INVERSION 40
IMAGES OF LINES AND CIRCLES UNDER INVERSION 40
INVARIANT CIRCLES 42
COMMUTING INVERSIONS 43
ANGLES 43
5 PROJECTIVE S P A C E O F CIRCLES 4 6
5.1 THE PROJECTIVE SPACE OF CIRCLES P(CIRC) 46
MOTIVATION 46
DEFINITION 47
CENTER AND RADIUS, INFINITELY LARGE CIRCLES 48
5.2 THE ACTION OF TRANSFORMATIONS ON P(CIRC) 49
TRANSLATIONS, ROTATIONS AND DILATIONS 49
THE UNIT INVERSION 51
GENERAL INVERSIONS 51
5.3 ELEMENTARY GEOMETRY IN P(IRC) 53
THE POLAR PLANE IN P(FFITC) 53
POLARITY AND ANGLES 54
5.4 THE INVERSIVE GROUP 55
REFLECTIONS AND INVERSIONS 55
CONNECTION TO MOBIUS TRANSFORMATIONS 56
THE TRANSFORMATION MATRIX M 57
6 LINEAR FAMILIES O F CIRCLES 5 9
6.1 LINES IN P(ITC) 59
LINEAR FAMILIES OF CIRCLES 59
BASE POINTS 61
NULLCIRCLES 62
6.2 THE CONJUGATED FAMILY 62
DEFINITION 62
REGULAR CONFIGURATIONS 64
SINGULAR CONFIGURATIONS 65
2
IMAGE 3
CONTENTS
7 INVERSION O F A LINEAR FAMILY 6 9
7.1 INVERSION ABOUT A CIRCLE OF THE FAMILY 69
STABILIZING INVERSIONS 69
REPRESENTATION OF THE INVERSION 70
FIXED POINTS - EIGENCIRCLES 71
7.2 INVERSION ABOUT A GENERAL CIRCLE 73
PROJECTION ONTO A LINEAR FAMILY 73
INDUCED ACTION ON A LINEAR FAMILY 74
EIGENCIRCLES OF THE INDUCED ACTION 75
OPEN QUESTIONS CONCERNING THE INDUCED ACTION 76
7.3 NORMALIZED INVERSION 77
DEFINITION AND BASIC PROPERTIES 77
INVERSIONS BETWEEN TWO GIVEN CIRCLES 78
8 BICIRCULAR QUARTICS 8 2
8.1 INVERSE AND PEDAL CURVE OF A CONIC SECTION 82
INVERSE OF A CONIC SECTION 82
PEDAL CURVE OF A CONIC SECTION 84
8.2 BICIRCULAR QUARTICS 85
DEFINITION 85
MULTICIRCULAR CURVES 87
C-Q-FORM OF A BICIRCULAR QUARTIC 87
8.3 ENVELOPE OF A RATIONAL FAMILY OF CIRCLES 89
PARAMETRIZATION OF A RATIONAL FAMILY OF CIRCLES 89
ENVELOPE OF A RATIONAL FAMILY OF CIRCLES 90
9 INVERSION O F BICIRCULAR QUARTICS 9 5
9.1 BICIRCULAR QUARTICS AS ENVELOPES 95
GEOMETRIC PROPERTIES OF THE C-Q-FORM 95
DEGENERATION OF THE QUADRIC Q T 98
DETERMINATION OF THE RATIONAL FAMILY OF CIRCLES 100
9.2 INVERSION OF BICIRCULAR QUARTICS 102
THE IMAGE UNDER THE UNIT INVERSION 102
CIRCULAR CUBICS 104
INVERSIONS AND THE THREE CIRCLES FORM 104
INVERSION OF MULTICIRCULAR CURVES 106
BICIRCULAR CURVES ARE ANALLAGMATIC CURVES 107
9.3 SINGULARITIES 110
REDUCIBILITY AND NUMBER OF SINGULARITIES 110
POSSIBLE TYPES OF SINGULARITIES 112
SINGULARITIES IN THE C-Q-FORM AND THE THREE CIRCLES FORM 113
1 0 CLASSICAL BICIRCULAR CURVES 1 1 5
10.1 MEANS FOR CLASSIFYING BICIRCULAR CURVES 115
ALGEBRAIC PROPERTIES 115
3
IMAGE 4
CONTENTS
* SINGULARITIES 116
GEOMETRY IN CITE AND P(ITC) 117
10.2 LIST OF CLASSICAL BICIRCULAR CURVES 118
TORIC SECTIONS 118
CENTRAL TORIC SECTIONS AND VILLARCEAU CIRCLES 120
SPIRIC SECTIONS, CASSINIAN CURVES, HIPPOPEDES 122
CARTESIAN OVALS - LIMAGONS 124
CIRCULAR CUBICS - CONCHOIDS OF DE SLUZE 125
11 APPLICATIONS O F BICIRCULAR CURVES 1 2 6
11.1 ELECTRICAL NETWORKS 126
CALCULATION OF IMPEDANCE 126
RC- AND RL-CIRCUIT 127
RC- AND RL-CASCADE 128
11.2 GENERALIZATION IN DIFFERENT DIMENSIONS 129
BICIRCULAR QUARTICS AS A GENERAL CONCEPT 129
CYCLIDES - BISPHERICAL SURFACES 130
POLYNOMIALS OF DEGREE FOUR 131
12 S U M M A R Y 1 3 4
4
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| indexdate | 2024-12-20T16:14:25Z |
| institution | BVB |
| language | English |
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| spellingShingle | Werner, Thomas Rainer Rational families of circles and bicircular quartics = Rationale Kreisscharen und bizirkulare Quartiken Kreis (DE-588)4032962-8 gnd Kreisschar (DE-588)4735676-5 gnd Inversion Mathematik (DE-588)4162235-2 gnd Analytische Geometrie (DE-588)4001867-2 gnd |
| subject_GND | (DE-588)4032962-8 (DE-588)4735676-5 (DE-588)4162235-2 (DE-588)4001867-2 (DE-588)4113937-9 |
| title | Rational families of circles and bicircular quartics = Rationale Kreisscharen und bizirkulare Quartiken |
| title_alt | Rationale Kreisscharen und bizirkulare Quartiken |
| title_auth | Rational families of circles and bicircular quartics = Rationale Kreisscharen und bizirkulare Quartiken |
| title_exact_search | Rational families of circles and bicircular quartics = Rationale Kreisscharen und bizirkulare Quartiken |
| title_full | Rational families of circles and bicircular quartics = Rationale Kreisscharen und bizirkulare Quartiken von Thomas Rainer Werner |
| title_fullStr | Rational families of circles and bicircular quartics = Rationale Kreisscharen und bizirkulare Quartiken von Thomas Rainer Werner |
| title_full_unstemmed | Rational families of circles and bicircular quartics = Rationale Kreisscharen und bizirkulare Quartiken von Thomas Rainer Werner |
| title_short | Rational families of circles and bicircular quartics |
| title_sort | rational families of circles and bicircular quartics rationale kreisscharen und bizirkulare quartiken |
| title_sub | = Rationale Kreisscharen und bizirkulare Quartiken |
| topic | Kreis (DE-588)4032962-8 gnd Kreisschar (DE-588)4735676-5 gnd Inversion Mathematik (DE-588)4162235-2 gnd Analytische Geometrie (DE-588)4001867-2 gnd |
| topic_facet | Kreis Kreisschar Inversion Mathematik Analytische Geometrie Hochschulschrift |
| url | https://open.fau.de/handle/openfau/2301 https://nbn-resolving.org/urn:nbn:de:bvb:29-opus-34320 http://d-nb.info/1024608662/34 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025261686&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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