Geometry of curves:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
Champan & Hall/CRC
2000
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| Series: | Chapman & Hall CRC mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 361 S. graph. Darst. |
| ISBN: | 1584881666 |
Staff View
MARC
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Record in the Search Index
| _version_ | 1804128171076878336 |
|---|---|
| adam_text | IMAGE 1
GEOMETRY
OF CURVES
J O HN W. R U T T ER
CHAPMAN & HALL/CRC BOCA RATON LONDON NEW YORK WASHINGTON, D.C.
IMAGE 2
CONTENTS
INTRODUCTION 0.1 0.2 0.3 0.4 0.5 0.6 0.7 CARTESIAN COORDINATES
POLAR COORDINATES THE ARGAND DIAGRAM POLAR EQUATIONS ANGLES
ORTHOGONAL AND PARALLEL VECTORS TRIGONOMETRY
1
1
2
4
4
5
6
7
1 LINES, CIRCLES, AND CONICS 9
1.1 LINES 9
1.2 THE CIRCLE 13
1.3 CONICS 15
1.4 THE ELLIPSE IN CANONICAL POSITION 16
1.5 THE HYPERBOLA IN CANONICAL POSITION 18
1.6 THE PARABOLA IN CANONICAL POSITION 21
1.7 CLASSICAL GEOMETRIC CONSTRUCTIONS OF CONICS 22
1.8 POLAR EQUATION OF A CONIC WITH A FOCUS AS POLE 23
1.9 HISTORY AND APPLICATIONS OF CONICS 26
2 CONICS: GENERAL POSITION 31
2.1 GEOMETRICAL METHOD FOR DIAGONALISATION 31
2.2 ALGEBRA 33
2.3 ALGEBRAIC METHOD FOR DIAGONALISATION 37
2.4 TRANSLATING TO CANONICAL FORM 38
2.5 CENTRAL CONICS REFERRED TO THEIR CENTRE 40
2.6 PRACTICAL PROCEDURES FOR DEALING WITH THE GENERAL CONIC 45
2.7 RATIONAL PARAMETRISATIONS OF CONICS 55
IMAGE 3
VIII CONTENTS
3 SOME HIGHER CURVES 59
3.1 THE SEMICUBICAL PARABOLA: A CUSPIDAL CUBIC 61
3.2 A CRUNODAL CUBIC 62
3.3 AN ACNODAL CUBIC 63
3.4 A CUBIC WITH TWO PARTS 64
3.5 HISTORY AND APPLICATIONS OF ALGEBRAIC CURVES 66
3.6 A TACHNODAL QUARTIC CURVE 66
3.7 LIMAGONS 68
3.8 EQUI-ANGULAR (LOGARITHMIC) SPIRAL 69
3.9 ARCHIMEDEAN SPIRAL 71
3.10 APPLICATION - WATT S CURVES 73
4 PARAMETERS, TANGENTS, NORMAL 77
4.1 PARAMETRIC CURVES 77
4.2 TANGENTS AND NORMALS AT REGULAER POINTS 85
4.3 NON-SINGULAR POINTS OF ALGEBRAIC CURVES 88
4.4 PARAMETRISATION OF ALGEBRAIC CURVES 89
4.5 TANGENTS AND NORMALS AT NON-SINGULAR POINTS 91
4.6 ARC-LENGTH PARAMETRISATION 93
4.7 SOME RESULTS IN ANALYSIS 96
5 CONTACT, INFLEXIONS, UNDULATIONS 103
5.1 CONTACT 103
5.2 INVARIANCE OF POINT-CONTACT ORDER 108
5.3 INFLEXIONS AND UNDULATIONS 112
5.4 GEOMETRICAL INTERPRETATION OF N-POINT CONTACT 120
5.5 AN ANALYTICAL INTERPRETATION OF CONTACT 121
6 CUSPS, NON-REGULAR POINTS 123
6.1 CUSPS 123
6.2 TANGENTS AT CUSPS 124
6.3 CONTACT BETWEEN A LINE AND A CURVE AT A CUSP 126
6.4 HIGHER SINGULARITIES 128
7 CURVATURE 133
7.1 CARTESIAN COORDINATES 133
7.2 CURVES GIVEN BY POLAR EQUATION 142
7.3 CURVES IN THE ARGAND DIAGRAM 146
7.4 AN ALTERNATIVE FORMULA 147
8 CURVATURE: APPLICATIONS 151
8.1 INFLEXIONS OF PARAMETRIC CURVES AT REGULAER POINTS 151
8.2 VERTICES AND UNDULATIONS AT REGULAER POINTS 153
8.3 CURVATURE OF ALGEBRAIC CURVES 157
IMAGE 4
10 LIMAGONS
10.1 10.2 10.3 10.4
10.5 10.6 10.7 10.8
THE EQUATION CURVATURE NON-REGULAR POINTS INFLEXIONS
VERTICES UNDULATIONS THE FIVE CLASSES OF LIMACONS AN ALTERNATIVE
EQUATION
CONTENTS IX
8.4 LIMITING CURVATURE OF ALGEBRAIC CURVES AT CUSPS 160
9 CIRCLE OF CURVATURE 167
9.1 CENTRE OF CURVATURE AND CIRCLE OF CURVATURE 167
9.2 CONTACT BETWEEN CURVES AND CIRCLES 173
177 177 179 179
180 180 181 181 182
11 EVOLUTES 183
11.1 DEFINITION AND SPECIAL POINTS 183
11.2 A MATRIX METHOD FOR CALCULATING EVOLUTES 186
11.3 EVOLUTES OF THE CYCLOID AND THE CARDIOID 187
12 PARALLELS, INVOLUTES 195
12.1 PARALLELS OF A CURVE 195
12.2 INVOLUTES 203
13 ROULETTES 209
13.1 GENERAL ROULETTES 209
13.2 PARAMETRISATION OF CIRCLES 214
13.3 CYCLOIDS: ROLLING A CIRCLE ON A LINE 215
13.4 TROCHOIDS: ROLLING A CIRCLE ON OR IN A CIRCLE 218
13.5 RIGID MOTIONS 234
13.6 NON-REGULAR POINTS AND INFLEXIONS OF ROULETTES 237
14 ENVELOPES 243
14.1 EVOLUTES AS A MODEL 244
14.2 SINGULAR-SET ENVELOPES 245
14.3 DISCRIMINANT ENVELOPES 255
14.4 DIFFERENT DEFINITIONS AND SINGULARITIES OF ENVELOPES 259
14.5 LIMITING-POSITION ENVELOPES 260
14.6 ORTHOTOMICS AND CAUSTICS 264
14.7 THE RELATION BETWEEN ORTHOTOMICS AND CAUSTICS 266
14.8 ORTHOTOMICS OF A CIRCLE 266
14.9 CAUSTICS OF A CIRCLE 268
IMAGE 5
X CONTENTS
15 SINGULAR POINTS OF ALGEBRAIC CURVES 271
15.1 INTERSECTION MULTIPLICITY WITH A GIVEN LINE 271
15.2 HOMOGENEOUS POLYNOMIALS 273
15.3 MULTIPLICITY OF A POINT 274
15.4 SINGULAR LINES AT THE ORIGIN 276
15.5 ISOLATED SINGULAR POINTS 278
15.6 TANGENTS AND BRANCHES AT NON-ISOLATED SINGULAR POINTS 280
15.7 BRANCHES FOR NON-REPEATED LINEAR FACTORS 282
15.8 BRANCHES FOR REPEATED LINEAR FACTORS 286
15.9 CUBIC CURVES 291
15.10 CURVATURE AT SINGULAR POINTS 295
16 PROJECTIVE CURVES 297
16.1 THE PROJECTIVE LINE 297
16.2 THE PROJECTIVE PLANE 298
16.3 PROJECTIVE CURVES 302
16.4 THE PROJECTIVE CURVE DETERMINED BY A PLANE CURVE 303
16.5 AFFINE VIEWS OF A PROJECTIVE CURVE 304
16.6 PLANE CURVES AS VIEWS OF A PROJECTIVE CURVE 305
16.7 TANGENT LINES TO PROJECTIVE CURVES 307
16.8 BOUNDEDNESS OF THE ASSOCIATED AFFINE CURVE 309
16.9 SUMMARY OF THE ANALYTIC VIEWPOINT 311
16.10 ASYMPTOTES 312
16.11 SINGULAR POINTS AND INFLEXIONS OF PROJECTIVE CURVES 314
16.12 EQUIVALENCE OF CURVES 316
16.13 EXAMPLES OF ASYMPTOTIC BEHAVIOUR 318
16.14 WORKED EXAMPLE 320
17 PRACTICAL WORK 329
18 DRAWN CURVES 345
18.1 PERSONALISING MATLAB FOR METRIC PRINTING 345
18.2 ELLIPSE 1 AND ELLIPSE 2 346
18.3 ELLIPSE 3 347
18.4 PARABOLA 1 347
18.5 PARABOLA 2 348
18.6 PARABOLA 3 348
18.7 HYPERBOLA 349
18.8 SEMICUBICAL PARABOLA 350
18.9 POLAR GRAPH PAPER 350
19 FURTHER READING 353
INDEX 355
|
| any_adam_object | 1 |
| author | Rutter, John W. 1935- |
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| dewey-full | 516.3/52 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/52 |
| dewey-search | 516.3/52 |
| dewey-sort | 3516.3 252 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-09T18:45:00Z |
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| isbn | 1584881666 |
| language | English |
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| spelling | Rutter, John W. 1935- Verfasser (DE-588)115360514 aut Geometry of curves John W. Rutter Boca Raton [u.a.] Champan & Hall/CRC 2000 XVII, 361 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chapman & Hall CRC mathematics Algebraïsche krommen gtt Courbes planes Curves, Plane Kurve (DE-588)4033824-1 gnd rswk-swf Kurve (DE-588)4033824-1 s DE-604 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009133920&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rutter, John W. 1935- Geometry of curves Algebraïsche krommen gtt Courbes planes Curves, Plane Kurve (DE-588)4033824-1 gnd |
| subject_GND | (DE-588)4033824-1 |
| title | Geometry of curves |
| title_auth | Geometry of curves |
| title_exact_search | Geometry of curves |
| title_full | Geometry of curves John W. Rutter |
| title_fullStr | Geometry of curves John W. Rutter |
| title_full_unstemmed | Geometry of curves John W. Rutter |
| title_short | Geometry of curves |
| title_sort | geometry of curves |
| topic | Algebraïsche krommen gtt Courbes planes Curves, Plane Kurve (DE-588)4033824-1 gnd |
| topic_facet | Algebraïsche krommen Courbes planes Curves, Plane Kurve |
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