Cyclic surfaces accompanying non-ruled quadrics of rotation
The paper considers the shaping of cyclic surfaces based on nonlinear rotation, in which the axis of rotation and the generatrix in the general case are three-dimensional smooth curves. As a tool for shaping surfaces of non-linear rotation, the method of the accompanying Frenet trihedron, known i...
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| Format: | Article |
| Language: | English |
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Omsk State Technical University, Federal State Autonoumos Educational Institution of Higher Education
2023-09-01
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| Series: | Омский научный вестник |
| Subjects: | |
| Online Access: | https://www.omgtu.ru/general_information/media_omgtu/journal_of_omsk_research_journal/files/arhiv/2023/%E2%84%963%20(187)%20(%D0%9E%D0%9D%D0%92)/23-29%20%20%D0%9F%D0%B0%D0%BD%D1%87%D1%83%D0%BA%20%D0%9A.%20%D0%9B.,%20%20%D0%9C%D1%8F%D1%81%D0%BE%D0%B5%D0%B4%D0%BE%D0%B2%D0%B0%20%D0%A2.%20%D0%9C.,%20%D0%9B%D1%8E%D0%B1%D1%87%D0%B8%D0%BD%D0%BE%D0%B2%20%D0%95.%20%D0%92..pdf |
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| author | K. L. Panchuk T. M. Myasoedova E. V. Lyubchinov |
| author_facet | K. L. Panchuk T. M. Myasoedova E. V. Lyubchinov |
| author_sort | K. L. Panchuk |
| collection | DOAJ |
| description | The paper considers the shaping of cyclic surfaces based on nonlinear rotation, in
which the axis of rotation and the generatrix in the general case are three-dimensional
smooth curves. As a tool for shaping surfaces of non-linear rotation, the method of
the accompanying Frenet trihedron, known in the differential geometry of curved
lines, is used. The geometric scheme of surface shaping is based on a construction
that includes: a curvilinear axis of rotation and a one-parameter set of its normal
planes; a generatrix whose points describe in normal planes circular trajectories
centered on a curvilinear axis. A mathematical model of shaping the surface of
non-linear rotation for the general case of specifying the axis of rotation and the
generatrix is given. On the basis of this model, test examples of the formation of
surfaces of nonlinear rotation, which are cyclic surfaces, each of which accompanies
the corresponding nonlinear quadric of rotation, are considered. In the examples
of shaping, the original rectilinear axis of a non-linear quadric of revolution and its
generating line, a second-order curve, are functionally interchanged: the secondorder curve becomes the rotation axis, and the rectilinear axis becomes the
generatrix.
The resulting family of surfaces of non-linear rotation belongs to the well-known
class in the theory of analytic surfaces "Normal cyclic surfaces". It complements this
class and fundamentally differs in the method of shaping. |
| format | Article |
| id | doaj-art-54281c962a3042b8afa8f0dcca29f9e4 |
| institution | Kabale University |
| issn | 1813-8225 2541-7541 |
| language | English |
| publishDate | 2023-09-01 |
| publisher | Omsk State Technical University, Federal State Autonoumos Educational Institution of Higher Education |
| record_format | Article |
| series | Омский научный вестник |
| spelling | doaj-art-54281c962a3042b8afa8f0dcca29f9e42025-02-03T05:18:08ZengOmsk State Technical University, Federal State Autonoumos Educational Institution of Higher EducationОмский научный вестник1813-82252541-75412023-09-013 (187)232910.25206/1813-8225-2023-187-23-29Cyclic surfaces accompanying non-ruled quadrics of rotationK. L. Panchuk0https://orcid.org/0000-0001-9302-8560T. M. Myasoedova1https://orcid.org/0000-0002-9641-9417E. V. Lyubchinov2https://orcid.org/0000-0003-2499-4866Omsk State Technical UniversityOmsk State Technical UniversityOmsk State Technical UniversityThe paper considers the shaping of cyclic surfaces based on nonlinear rotation, in which the axis of rotation and the generatrix in the general case are three-dimensional smooth curves. As a tool for shaping surfaces of non-linear rotation, the method of the accompanying Frenet trihedron, known in the differential geometry of curved lines, is used. The geometric scheme of surface shaping is based on a construction that includes: a curvilinear axis of rotation and a one-parameter set of its normal planes; a generatrix whose points describe in normal planes circular trajectories centered on a curvilinear axis. A mathematical model of shaping the surface of non-linear rotation for the general case of specifying the axis of rotation and the generatrix is given. On the basis of this model, test examples of the formation of surfaces of nonlinear rotation, which are cyclic surfaces, each of which accompanies the corresponding nonlinear quadric of rotation, are considered. In the examples of shaping, the original rectilinear axis of a non-linear quadric of revolution and its generating line, a second-order curve, are functionally interchanged: the secondorder curve becomes the rotation axis, and the rectilinear axis becomes the generatrix. The resulting family of surfaces of non-linear rotation belongs to the well-known class in the theory of analytic surfaces "Normal cyclic surfaces". It complements this class and fundamentally differs in the method of shaping.https://www.omgtu.ru/general_information/media_omgtu/journal_of_omsk_research_journal/files/arhiv/2023/%E2%84%963%20(187)%20(%D0%9E%D0%9D%D0%92)/23-29%20%20%D0%9F%D0%B0%D0%BD%D1%87%D1%83%D0%BA%20%D0%9A.%20%D0%9B.,%20%20%D0%9C%D1%8F%D1%81%D0%BE%D0%B5%D0%B4%D0%BE%D0%B2%D0%B0%20%D0%A2.%20%D0%9C.,%20%D0%9B%D1%8E%D0%B1%D1%87%D0%B8%D0%BD%D0%BE%D0%B2%20%D0%95.%20%D0%92..pdfnon-linear rotationcyclic surfacesnon-ruled quadrics of rotationaxis of rotationgeneratrix |
| spellingShingle | K. L. Panchuk T. M. Myasoedova E. V. Lyubchinov Cyclic surfaces accompanying non-ruled quadrics of rotation Омский научный вестник non-linear rotation cyclic surfaces non-ruled quadrics of rotation axis of rotation generatrix |
| title | Cyclic surfaces accompanying non-ruled quadrics of rotation |
| title_full | Cyclic surfaces accompanying non-ruled quadrics of rotation |
| title_fullStr | Cyclic surfaces accompanying non-ruled quadrics of rotation |
| title_full_unstemmed | Cyclic surfaces accompanying non-ruled quadrics of rotation |
| title_short | Cyclic surfaces accompanying non-ruled quadrics of rotation |
| title_sort | cyclic surfaces accompanying non ruled quadrics of rotation |
| topic | non-linear rotation cyclic surfaces non-ruled quadrics of rotation axis of rotation generatrix |
| url | https://www.omgtu.ru/general_information/media_omgtu/journal_of_omsk_research_journal/files/arhiv/2023/%E2%84%963%20(187)%20(%D0%9E%D0%9D%D0%92)/23-29%20%20%D0%9F%D0%B0%D0%BD%D1%87%D1%83%D0%BA%20%D0%9A.%20%D0%9B.,%20%20%D0%9C%D1%8F%D1%81%D0%BE%D0%B5%D0%B4%D0%BE%D0%B2%D0%B0%20%D0%A2.%20%D0%9C.,%20%D0%9B%D1%8E%D0%B1%D1%87%D0%B8%D0%BD%D0%BE%D0%B2%20%D0%95.%20%D0%92..pdf |
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