Generalized Bertrand Curve Pairs in Euclidean Four-Dimensional Space
In this study, the existence of Bertrand curves (in the classical sense, i.e., curves with a common principal normal vector field) in four-dimensional Euclidean space is demonstrated using a novel approach. The necessary conditions for a regular curve to be a Bertrand curve pair are obtained. Furthe...
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MDPI AG
2025-03-01
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| Online Access: | https://www.mdpi.com/2075-1680/14/4/253 |
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| author | Yanlin Li Osman Keçilioğlu Kazım İlarslan |
| author_facet | Yanlin Li Osman Keçilioğlu Kazım İlarslan |
| author_sort | Yanlin Li |
| collection | DOAJ |
| description | In this study, the existence of Bertrand curves (in the classical sense, i.e., curves with a common principal normal vector field) in four-dimensional Euclidean space is demonstrated using a novel approach. The necessary conditions for a regular curve to be a Bertrand curve pair are obtained. Furthermore, the relationship between Bertrand curves and Combescure-related curves (pairs of curves with parallel Frenet vectors) is established, and several geometric properties are derived. Additionally, examples are constructed for both Bertrand curve pairs and Combescure-related curve pairs, and their orthogonal projections onto three-dimensional subspaces of four-dimensional space are visualized. |
| format | Article |
| id | doaj-art-ebb16c7eabb2472f9ee74867b3a69375 |
| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-ebb16c7eabb2472f9ee74867b3a693752025-08-20T02:24:39ZengMDPI AGAxioms2075-16802025-03-0114425310.3390/axioms14040253Generalized Bertrand Curve Pairs in Euclidean Four-Dimensional SpaceYanlin Li0Osman Keçilioğlu1Kazım İlarslan2School of Mathematics, Hangzhou Normal University, Hangzhou 311121, ChinaDepartment of Mathematics, Faculty of Engineering and Natural Sciences, Kırıkkale University, 71450 Kırıkkale, TurkeyDepartment of Mathematics, Faculty of Engineering and Natural Sciences, Kırıkkale University, 71450 Kırıkkale, TurkeyIn this study, the existence of Bertrand curves (in the classical sense, i.e., curves with a common principal normal vector field) in four-dimensional Euclidean space is demonstrated using a novel approach. The necessary conditions for a regular curve to be a Bertrand curve pair are obtained. Furthermore, the relationship between Bertrand curves and Combescure-related curves (pairs of curves with parallel Frenet vectors) is established, and several geometric properties are derived. Additionally, examples are constructed for both Bertrand curve pairs and Combescure-related curve pairs, and their orthogonal projections onto three-dimensional subspaces of four-dimensional space are visualized.https://www.mdpi.com/2075-1680/14/4/253bertrand curvescombescure-related space curvesparallel frenet frame |
| spellingShingle | Yanlin Li Osman Keçilioğlu Kazım İlarslan Generalized Bertrand Curve Pairs in Euclidean Four-Dimensional Space Axioms bertrand curves combescure-related space curves parallel frenet frame |
| title | Generalized Bertrand Curve Pairs in Euclidean Four-Dimensional Space |
| title_full | Generalized Bertrand Curve Pairs in Euclidean Four-Dimensional Space |
| title_fullStr | Generalized Bertrand Curve Pairs in Euclidean Four-Dimensional Space |
| title_full_unstemmed | Generalized Bertrand Curve Pairs in Euclidean Four-Dimensional Space |
| title_short | Generalized Bertrand Curve Pairs in Euclidean Four-Dimensional Space |
| title_sort | generalized bertrand curve pairs in euclidean four dimensional space |
| topic | bertrand curves combescure-related space curves parallel frenet frame |
| url | https://www.mdpi.com/2075-1680/14/4/253 |
| work_keys_str_mv | AT yanlinli generalizedbertrandcurvepairsineuclideanfourdimensionalspace AT osmankecilioglu generalizedbertrandcurvepairsineuclideanfourdimensionalspace AT kazımilarslan generalizedbertrandcurvepairsineuclideanfourdimensionalspace |