Direction Curves Associated with Darboux Vectors Fields and Their Characterizations
In this paper, we consider the Darboux frame of a curve α lying on an arbitrary regular surface and we use its unit osculator Darboux vector D¯o, unit rectifying Darboux vector D¯r, and unit normal Darboux vector D¯n to define some direction curves such as D¯o-direction curve, D¯r-direction curve, a...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2021/3814032 |
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| author | Nidal Echabbi Amina Ouazzani Chahdi |
| author_facet | Nidal Echabbi Amina Ouazzani Chahdi |
| author_sort | Nidal Echabbi |
| collection | DOAJ |
| description | In this paper, we consider the Darboux frame of a curve α lying on an arbitrary regular surface and we use its unit osculator Darboux vector D¯o, unit rectifying Darboux vector D¯r, and unit normal Darboux vector D¯n to define some direction curves such as D¯o-direction curve, D¯r-direction curve, and D¯n-direction curve, respectively. We prove some relationships between α and these associated curves. Especially, the necessary and sufficient conditions for each direction curve to be a general helix, a spherical curve, and a curve with constant torsion are found. In addition to this, we have seen the cases where the Darboux invariants δo, δr, and δn are, respectively, zero. Finally, we enrich our study by giving some examples. |
| format | Article |
| id | doaj-art-6df4e6c69eda49b3b86fc2092d9fdd55 |
| institution | Kabale University |
| issn | 1687-0425 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-6df4e6c69eda49b3b86fc2092d9fdd552025-02-03T05:53:26ZengWileyInternational Journal of Mathematics and Mathematical Sciences1687-04252021-01-01202110.1155/2021/3814032Direction Curves Associated with Darboux Vectors Fields and Their CharacterizationsNidal Echabbi0Amina Ouazzani Chahdi1Departement of Mathematics and InformaticsDepartement of Mathematics and InformaticsIn this paper, we consider the Darboux frame of a curve α lying on an arbitrary regular surface and we use its unit osculator Darboux vector D¯o, unit rectifying Darboux vector D¯r, and unit normal Darboux vector D¯n to define some direction curves such as D¯o-direction curve, D¯r-direction curve, and D¯n-direction curve, respectively. We prove some relationships between α and these associated curves. Especially, the necessary and sufficient conditions for each direction curve to be a general helix, a spherical curve, and a curve with constant torsion are found. In addition to this, we have seen the cases where the Darboux invariants δo, δr, and δn are, respectively, zero. Finally, we enrich our study by giving some examples.http://dx.doi.org/10.1155/2021/3814032 |
| spellingShingle | Nidal Echabbi Amina Ouazzani Chahdi Direction Curves Associated with Darboux Vectors Fields and Their Characterizations International Journal of Mathematics and Mathematical Sciences |
| title | Direction Curves Associated with Darboux Vectors Fields and Their Characterizations |
| title_full | Direction Curves Associated with Darboux Vectors Fields and Their Characterizations |
| title_fullStr | Direction Curves Associated with Darboux Vectors Fields and Their Characterizations |
| title_full_unstemmed | Direction Curves Associated with Darboux Vectors Fields and Their Characterizations |
| title_short | Direction Curves Associated with Darboux Vectors Fields and Their Characterizations |
| title_sort | direction curves associated with darboux vectors fields and their characterizations |
| url | http://dx.doi.org/10.1155/2021/3814032 |
| work_keys_str_mv | AT nidalechabbi directioncurvesassociatedwithdarbouxvectorsfieldsandtheircharacterizations AT aminaouazzanichahdi directioncurvesassociatedwithdarbouxvectorsfieldsandtheircharacterizations |