Sweeping Surfaces Due to Conjugate Bishop Frame in 3-Dimensional Lie Group
In this work, we present a new Bishop frame for the conjugate curve of a curve in the 3-dimensional Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-04-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/4/910 |
| _version_ | 1797603369562406912 |
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| author | Awatif Al-Jedani Rashad Abdel-Baky |
| author_facet | Awatif Al-Jedani Rashad Abdel-Baky |
| author_sort | Awatif Al-Jedani |
| collection | DOAJ |
| description | In this work, we present a new Bishop frame for the conjugate curve of a curve in the 3-dimensional Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">G</mi><mn>3</mn></msub></semantics></math></inline-formula>. With the help of this frame, we derive a parametric representation for a sweeping surface and show that the parametric curves on this surface are curvature lines. We then examine the local singularities and convexity of this sweeping surface and establish the sufficient and necessary conditions for it to be a developable ruled surface. Additionally, we provide detailed explanations and examples of its applications. |
| first_indexed | 2024-03-11T04:29:12Z |
| format | Article |
| id | doaj.art-8a090c80ccc64b45b28f376401388071 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T04:29:12Z |
| publishDate | 2023-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-8a090c80ccc64b45b28f3764013880712023-11-17T21:34:28ZengMDPI AGSymmetry2073-89942023-04-0115491010.3390/sym15040910Sweeping Surfaces Due to Conjugate Bishop Frame in 3-Dimensional Lie GroupAwatif Al-Jedani0Rashad Abdel-Baky1Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah 23890, Saudi ArabiaDepartment of Mathematics, Faculty of Science, University of Assiut, Assiut 71516, EgyptIn this work, we present a new Bishop frame for the conjugate curve of a curve in the 3-dimensional Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">G</mi><mn>3</mn></msub></semantics></math></inline-formula>. With the help of this frame, we derive a parametric representation for a sweeping surface and show that the parametric curves on this surface are curvature lines. We then examine the local singularities and convexity of this sweeping surface and establish the sufficient and necessary conditions for it to be a developable ruled surface. Additionally, we provide detailed explanations and examples of its applications.https://www.mdpi.com/2073-8994/15/4/910curvature lineprofile curvedevelopable surfaceparabolic points |
| spellingShingle | Awatif Al-Jedani Rashad Abdel-Baky Sweeping Surfaces Due to Conjugate Bishop Frame in 3-Dimensional Lie Group Symmetry curvature line profile curve developable surface parabolic points |
| title | Sweeping Surfaces Due to Conjugate Bishop Frame in 3-Dimensional Lie Group |
| title_full | Sweeping Surfaces Due to Conjugate Bishop Frame in 3-Dimensional Lie Group |
| title_fullStr | Sweeping Surfaces Due to Conjugate Bishop Frame in 3-Dimensional Lie Group |
| title_full_unstemmed | Sweeping Surfaces Due to Conjugate Bishop Frame in 3-Dimensional Lie Group |
| title_short | Sweeping Surfaces Due to Conjugate Bishop Frame in 3-Dimensional Lie Group |
| title_sort | sweeping surfaces due to conjugate bishop frame in 3 dimensional lie group |
| topic | curvature line profile curve developable surface parabolic points |
| url | https://www.mdpi.com/2073-8994/15/4/910 |
| work_keys_str_mv | AT awatifaljedani sweepingsurfacesduetoconjugatebishopframein3dimensionalliegroup AT rashadabdelbaky sweepingsurfacesduetoconjugatebishopframein3dimensionalliegroup |