Lines and surfaces in three-dimensional affine space:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English Russian |
| Veröffentlicht: |
Jerusalem
Israel Program for Scientific Transl.
1964
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | EST: Linii i poverchnosti v affinnom trechmernom prostranstve (engl.) |
| Beschreibung: | IX,92 S. graph. Darst. |
Internformat
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| 245 | 1 | 0 | |a Lines and surfaces in three-dimensional affine space |c Leonid Kondrat'evič Tutaev* |
| 264 | 1 | |a Jerusalem |b Israel Program for Scientific Transl. |c 1964 | |
| 300 | |a IX,92 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 500 | |a EST: Linii i poverchnosti v affinnom trechmernom prostranstve (engl.) | ||
| 650 | 4 | |a Géométrie différentielle | |
| 650 | 4 | |a Geometry, Differential | |
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Datensatz im Suchindex
| _version_ | 1804119878227984384 |
|---|---|
| adam_text | CONTENTS
FOREWORD vii
INTRODUCTION 1
§ 1. Affine space 1
§ 2. Pfaffian forms 1
§ 3. Grassman ring 2
§ 4. Cartan s ring 3
§ 5. Exterior differentiation 4
§ 6. Pfaffian system of differential equations 4
§ 7. Moving frame 5
§ 8. System of exterior differential equations 5
§ 9. Integral manifolds 7
Part One
LINES ON A PLANE
Chapter I. THE CANONICAL FRAME 9
§ 1. Fundamental relations 9
§ 2. Frames of first order 9
§ 3. Continuation of equation (2.3) 10
§ 4. Choice of canonical frame 10
§ 5. Natural parameter 11
§ 6. Frenet Cartan formulas 11
§ 7. Equation of a line in its canonical frame 11
§ 8. Evolute 12
§ 9. Lines of zero curvature 13
§ 10. Osculating line of second order (quadric) 13
§ 11. Exercises 15
Chapter II. RELATION BETWEEN THE CANONICAL FRAME AND
THE LINE 15
§ 1. Affine normal 15
§ 2. Position of the canonical frame relative to the line and
a point of the hyperbolic or elliptic type on it 16
§ 3. Position of the canonical frame relative to the line and a
point of the parabolic type on it 16
§ 4. Geometric interpretation of curvature 16
§ 5. Exercises 17
Part Two
LINES IN SPACE
Chapter III. THE CANONICAL FRAME 19
§ 1. Fundamental relations 19
§ 2. Frames of the first order 19
§ 3. Continuation of equations (2.2) 20
§ 4. Choice of a canonical frame 20
§ 5. Natural parameter. Frenet Cartan formulas 21
§ 6. Equations of a line in its canonical frame 22
§ 7. Osculating line of the third order 22
Chapter IV. RELATION BETWEEN THE CANONICAL FRAME
AND THE LINE 23
§ 1. Osculating plane 23
§ 2. Principal normal 23
§ 3. Rectifying plane 25
§ 4. Binormal. Normal plane 26
§ 5. Coordinate vectors 27
§ 6. Curvatures 28
§ 7. Exercises 29
Chapter V. THE CANONICAL FRAME 30
§ 1. Fundamental relations 30
§ 2. Frames of the first order 30
§ 3. Continuation of the equation 30
§ 4. Conjugate and asymptotic directions 31
§ 5. Invariance of point, plane, quadric 32
§ 6. Fundamental quantity of the third order 33
§ 7. Osculating quadrics 34
§ 8. Choice of the canonical frame 35
§ 9. Darboux Cartan formulas 37
§ 10. Canonical expansions of the coordinates of the
generic point of the surface 38
Chapter VI. RELATION BETWEEN THE CANONICAL FRAME
AND THE SURFACE 39
§ 1. Osculating paraboloid 39
§ 2. Pencil of Darboux quadrics 39
§ 3. Affine normal 40
§ 4. Conjugate normal planes 40
§ 5. Lines on the surface 40
§ 6. Family of planes conjugate to the tangents to the
normal section 41
§ 7. Diagram of the indicatrix of the characteristics of
the normal planes 43
§ 8. Geometric interpretation of the choice of the axes of
the canonical frame 44
§ 9. Geometric interpretation of the choice of coordinate vectors . . 45
Chapter VII. GEOMETRIC INTERPRETATION OF
DIFFERENTIAL INVARIANTS 46
§ 1. Displacement of a point along the line u)l 46
§ 2. Displacement of a point along the line u 2 46
§ 3. Tangents to the coordinate lines of one family at points
of the coordinate line of another family 47
Chapter VIII. CERTAIN LINES ON THE SURFACE 48
§ 1. Conjugate directions. Asymptotic lines 48
§ 2. Straight lines 48
§ 3. Plane lines 49
§ 4. Lines with osculating planes normal to the surface 49
§ 5. Displacement of a point belonging to the normal 50
§ 6. Lines of curvature 51
§ 7. Normal sections of the surface 51
§ 8. Lines having normal planes normal to the surface 54
§ 9. Lines having principal normals which coincide with the
normals to the surface 55
§ 10. Lines with binormals coincident with the normals to the surface . 56
§ 11. Lines having rectifying planes normal to the surface .... 56
§ 12. Lines having rectifying planes tangent to the surface .... 56
§ 13. Darboux lines 57
§ 14. Equations for infinitesimal displacements of the chosen frames
of the surface and of the line of this surface 59
§ 15. Exercises 60
Chapter IX. CERTAIN SURFACES 61
§ 1. Surfaces of the second order (quadrics) 61
§ 2. Affine sphere 61
§ 3. Surface with parallel affine normals 62
§ 4. Surfaces with coplanar normals 62
§ 5. The surfaces (D), (S) and (D, S) 63
§ 6. Surfaces and certain conditions of motion of unit points of
the canonical frame 63
Chapter X. RULED SURFACES 64
§ 1. Choice of a canonical frame 64
§ 2. Darboux Cartan formulas 68
§ 3. Canonical expansion of the coordinates of the generic point
of the surface 69
§ 4. Pencil of Darboux quadrics. Affine normal 71
§ 5. The family of planes conjugate to the tangents to the normal
section of the surface 71
§ 6. Indicatrix of characteristics of normal planes 73
§ 7. Osculating linear complex of straight lines 74
§ 8. Geometric interpretation of the choice of canonical frame . . 77
§ 9. Geometric interpretation of differential invariants 77
Chapter XI. CERTAIN LINES ON A RULED SURFACE 79
§ 1. Canonical frames of the surface and lines on it 79
§ 2. Normal sections of the surface 81
§ 3. Lines with osculating planes normal to the surface 82
§ 4. Lines with principal normals normal to the surface 82
§ 5. Lines of curvature 83
§ 6. Darboux lines 83
Chapter XII. CERTAIN RULED SURFACES 84
§ 1. Affine ruled sphere 84
§ 2. Surfaces with parallel affine normals 84
§ 3. Surfaces with coplanar normals 85
§ 4. Examples and problems 85
Chapter XIII. CONCERNING THE EXISTENCE OF CERTAIN SURFACES . . 88
§ 1. Surfaces with all differential invariants constant 88
§2. Unruled surfaces for which p2 = 0, / 3 = p; 90
§ 3. Unruled surfaces with parallel normals 91
§ 4. Ruled surfaces with parallel normals 91
BIBLIOGRAFHY 92
|
| any_adam_object | 1 |
| author | Tutaev, Leonid K. |
| author_facet | Tutaev, Leonid K. |
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| author_sort | Tutaev, Leonid K. |
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| dewey-full | 516.7 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.7 |
| dewey-search | 516.7 |
| dewey-sort | 3516.7 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV005690439 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:33:11Z |
| institution | BVB |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-003554177 |
| oclc_num | 332020 |
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| owner_facet | DE-703 DE-355 DE-BY-UBR DE-20 DE-91G DE-BY-TUM DE-83 DE-188 |
| physical | IX,92 S. graph. Darst. |
| publishDate | 1964 |
| publishDateSearch | 1964 |
| publishDateSort | 1964 |
| publisher | Israel Program for Scientific Transl. |
| record_format | marc |
| spelling | Tutaev, Leonid K. Verfasser aut Linii i poverchnosti v affinnom trechmernom prostranstve Lines and surfaces in three-dimensional affine space Leonid Kondrat'evič Tutaev* Jerusalem Israel Program for Scientific Transl. 1964 IX,92 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier EST: Linii i poverchnosti v affinnom trechmernom prostranstve (engl.) Géométrie différentielle Geometry, Differential Fläche (DE-588)4129864-0 gnd rswk-swf Dreidimensionaler Raum (DE-588)4220292-9 gnd rswk-swf Kurve (DE-588)4033824-1 gnd rswk-swf Fläche (DE-588)4129864-0 s DE-604 Kurve (DE-588)4033824-1 s Dreidimensionaler Raum (DE-588)4220292-9 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003554177&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tutaev, Leonid K. Lines and surfaces in three-dimensional affine space Géométrie différentielle Geometry, Differential Fläche (DE-588)4129864-0 gnd Dreidimensionaler Raum (DE-588)4220292-9 gnd Kurve (DE-588)4033824-1 gnd |
| subject_GND | (DE-588)4129864-0 (DE-588)4220292-9 (DE-588)4033824-1 |
| title | Lines and surfaces in three-dimensional affine space |
| title_alt | Linii i poverchnosti v affinnom trechmernom prostranstve |
| title_auth | Lines and surfaces in three-dimensional affine space |
| title_exact_search | Lines and surfaces in three-dimensional affine space |
| title_full | Lines and surfaces in three-dimensional affine space Leonid Kondrat'evič Tutaev* |
| title_fullStr | Lines and surfaces in three-dimensional affine space Leonid Kondrat'evič Tutaev* |
| title_full_unstemmed | Lines and surfaces in three-dimensional affine space Leonid Kondrat'evič Tutaev* |
| title_short | Lines and surfaces in three-dimensional affine space |
| title_sort | lines and surfaces in three dimensional affine space |
| topic | Géométrie différentielle Geometry, Differential Fläche (DE-588)4129864-0 gnd Dreidimensionaler Raum (DE-588)4220292-9 gnd Kurve (DE-588)4033824-1 gnd |
| topic_facet | Géométrie différentielle Geometry, Differential Fläche Dreidimensionaler Raum Kurve |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003554177&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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