Differentiable mapping generated by elliptic paraboloid complexes
In three-dimensional equiaffine space, we consider a differentiable map generated by complexes with three-parameter families of elliptic paraboloids according to the method proposed by the author in the materials of the international scientific conference on geometry and applications in Bulgaria i...
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| Format: | Article |
| Language: | English |
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Immanuel Kant Baltic Federal University
2020-08-01
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| Series: | Дифференциальная геометрия многообразий фигур |
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| Online Access: | https://journals.kantiana.ru/geometry/4686/25780/ |
| _version_ | 1819278860067274752 |
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| author | M.V. Kretov |
| author_facet | M.V. Kretov |
| author_sort | M.V. Kretov |
| collection | DOAJ |
| description | In three-dimensional equiaffine space, we consider a differentiable map generated by complexes with three-parameter families of elliptic paraboloids according to the method proposed by the author in the materials of the international scientific conference on geometry and applications in Bulgaria in 1986, as well as in works published earlier in the scientific collection of Differ. Geom. Mnogoobr. Figur. The study is carried out in the canonical frame, the vertex of which coincides with the top of the generating element of the manifold, the first two coordinate vectors are conjugate and lie in the tangent plane of the elliptic paraboloid at its vertex, the third coordinate vector is directed along the main diameter of the generating element so that the ends are, respectively, the sums of the first and third, and also the sums of the second and third coordinate vectors lay on a paraboloid, while the indicatrixes of all three coordinate vectors describe lines with tangents, parallel to the first coordinate vector. The existence theorem of the mapping under study is proved, according to which it exists and is determined with the arbitrariness of one function of one argument. The systems of equations of the indicatrix and the main directions of the mapping under consideration are obtained. The indicatrix and the cone of the main directions of the indicated mapping are geometrically characterized.
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| first_indexed | 2024-12-24T00:18:43Z |
| format | Article |
| id | doaj.art-12b9de75e34340a0bb8cf2e34d7ed5c5 |
| institution | Directory Open Access Journal |
| issn | 0321-4796 2782-3229 |
| language | English |
| last_indexed | 2024-12-24T00:18:43Z |
| publishDate | 2020-08-01 |
| publisher | Immanuel Kant Baltic Federal University |
| record_format | Article |
| series | Дифференциальная геометрия многообразий фигур |
| spelling | doaj.art-12b9de75e34340a0bb8cf2e34d7ed5c52022-12-21T17:24:39ZengImmanuel Kant Baltic Federal UniversityДифференциальная геометрия многообразий фигур0321-47962782-32292020-08-0151768010.5922/0321-4796-2020-51-9Differentiable mapping generated by elliptic paraboloid complexes M.V. Kretov0Immanuel Kant Baltic Federal UniversityIn three-dimensional equiaffine space, we consider a differentiable map generated by complexes with three-parameter families of elliptic paraboloids according to the method proposed by the author in the materials of the international scientific conference on geometry and applications in Bulgaria in 1986, as well as in works published earlier in the scientific collection of Differ. Geom. Mnogoobr. Figur. The study is carried out in the canonical frame, the vertex of which coincides with the top of the generating element of the manifold, the first two coordinate vectors are conjugate and lie in the tangent plane of the elliptic paraboloid at its vertex, the third coordinate vector is directed along the main diameter of the generating element so that the ends are, respectively, the sums of the first and third, and also the sums of the second and third coordinate vectors lay on a paraboloid, while the indicatrixes of all three coordinate vectors describe lines with tangents, parallel to the first coordinate vector. The existence theorem of the mapping under study is proved, according to which it exists and is determined with the arbitrariness of one function of one argument. The systems of equations of the indicatrix and the main directions of the mapping under consideration are obtained. The indicatrix and the cone of the main directions of the indicated mapping are geometrically characterized. https://journals.kantiana.ru/geometry/4686/25780/complexequiaffine spacemappingelliptical paraboloidmain directiondisplay indexsystem of pfaff equationscoordinate vectorsreferencecharacteristic variety |
| spellingShingle | M.V. Kretov Differentiable mapping generated by elliptic paraboloid complexes Дифференциальная геометрия многообразий фигур complex equiaffine space mapping elliptical paraboloid main direction display index system of pfaff equations coordinate vectors reference characteristic variety |
| title | Differentiable mapping generated by elliptic paraboloid complexes |
| title_full | Differentiable mapping generated by elliptic paraboloid complexes |
| title_fullStr | Differentiable mapping generated by elliptic paraboloid complexes |
| title_full_unstemmed | Differentiable mapping generated by elliptic paraboloid complexes |
| title_short | Differentiable mapping generated by elliptic paraboloid complexes |
| title_sort | differentiable mapping generated by elliptic paraboloid complexes |
| topic | complex equiaffine space mapping elliptical paraboloid main direction display index system of pfaff equations coordinate vectors reference characteristic variety |
| url | https://journals.kantiana.ru/geometry/4686/25780/ |
| work_keys_str_mv | AT mvkretov differentiablemappinggeneratedbyellipticparaboloidcomplexes |