{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T02:49:24Z","timestamp":1774320564021,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,3,27]],"date-time":"2025-03-27T00:00:00Z","timestamp":1743033600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/axioms14040253","type":"journal-article","created":{"date-parts":[[2025,3,28]],"date-time":"2025-03-28T04:11:40Z","timestamp":1743135100000},"page":"253","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Generalized Bertrand Curve Pairs in Euclidean Four-Dimensional Space"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1614-3228","authenticated-orcid":false,"given":"Yanlin","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5149-5221","authenticated-orcid":false,"given":"Osman","family":"Ke\u00e7ilio\u011flu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Engineering and Natural Sciences, K\u0131r\u0131kkale University, 71450 K\u0131r\u0131kkale, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1708-280X","authenticated-orcid":false,"given":"Kaz\u0131m","family":"\u0130larslan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Engineering and Natural Sciences, K\u0131r\u0131kkale University, 71450 K\u0131r\u0131kkale, Turkey"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,27]]},"reference":[{"key":"ref_1","first-page":"332","article-title":"M\u00e9moire sur la th\u00e9orie des courbes \u00e1 double courbure","volume":"15","author":"Bertrand","year":"1850","journal-title":"Comptes Rendus"},{"key":"ref_2","first-page":"1","article-title":"M\u00e9moire sur les lignes courbes non planes","volume":"18","year":"1845","journal-title":"J. L\u2019Ecole Polytech."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1903","DOI":"10.1016\/j.geomphys.2012.04.007","article-title":"Bertrand curves in the three-dimensional sphere","volume":"62","author":"Lucas","year":"2012","journal-title":"J. Geom. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"883","DOI":"10.3906\/mat-1905-63","article-title":"Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space","volume":"44","author":"Honda","year":"2020","journal-title":"Turk. J. Math."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Li, Y., Eren, K., Ersoy, S., and Savic, A. (2024). Modified Sweeping Surfaces in Euclidean 3-Space. Axioms, 13.","DOI":"10.3390\/axioms13110800"},{"key":"ref_6","first-page":"293","article-title":"Sweeping Surfaces of Polynomial Curves in Euclidean 3-space","volume":"33","author":"Zhu","year":"2025","journal-title":"An. St. Univ. Ovidius Constanta"},{"key":"ref_7","first-page":"1","article-title":"Sweeping surfaces with Darboux frame in Euclidean 3-space","volume":"18","author":"Moraffeh","year":"2021","journal-title":"Aust. J. Math. Anal. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Cheng, Y., Li, Y., Badyal, P., Singh, K., and Sharma, S. (2025). Conformal Interactions of Osculating Curves on Regular Surfaces in Euclidean 3-Space. Mathematics, 13.","DOI":"10.3390\/math13050881"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s13226-020-0385-9","article-title":"Rectifying and osculating curves on a smooth surface","volume":"51","author":"Shaikh","year":"2020","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_10","first-page":"47","article-title":"Some characterization of osculating curves according to Darboux frame in three-dimensional Euclidean space","volume":"7","author":"Isah","year":"2021","journal-title":"Int. J. Adv. Acad. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"883","DOI":"10.1007\/s13226-019-0361-4","article-title":"Rectifying curves on a smooth surface immersed in the Euclidean space","volume":"50","author":"Shaikh","year":"2019","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_12","first-page":"77","article-title":"Rectifying curve as centrode and extremal curve","volume":"33","author":"Chen","year":"2005","journal-title":"Bull. Inst. Math. Acad. Sin."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"609","DOI":"10.3906\/mat-1701-52","article-title":"On rectifying curves in Euclidean 3-space","volume":"42","author":"Deshmukh","year":"2018","journal-title":"Turk. J. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1112\/jlms\/s1-10.2.180","article-title":"Bertrand curves in Riemannian space","volume":"1","author":"Pears","year":"1935","journal-title":"J. London Math. Soc."},{"key":"ref_15","first-page":"41","article-title":"Notes on Bertrand curves","volume":"50","author":"Matsuda","year":"2003","journal-title":"Yokohama Math. J."},{"key":"ref_16","first-page":"33","article-title":"On the Generalized Bertrand Curves in Euclidean N-spaces","volume":"29","author":"Cheng","year":"2009","journal-title":"Note Mat."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s00022-020-00560-5","article-title":"A New Approach to Bertrand Curves in Euclidean 3-Space","volume":"111","year":"2020","journal-title":"J. Geom."},{"key":"ref_18","unstructured":"Kuhnel, W. (1999). Differential Geometry: Curves-Surfaces-Manifolds, American Mathematical Society."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Sun, J., and Zhao, Y. (2021). The Geometrical Characterizations of the Bertrand Curves of the Null Curves in Semi-Euclidean 4-Space. Mathematics, 9.","DOI":"10.3390\/math9243294"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"447","DOI":"10.36890\/iejg.1440270","article-title":"Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-zero Constant Curvatures","volume":"17","author":"Tuncer","year":"2024","journal-title":"Int. Electron. J. Geom."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"699","DOI":"10.1080\/00029890.1966.11970818","article-title":"Higher curvatures of curves in Euclidean space","volume":"73","author":"Gluck","year":"1966","journal-title":"Am. Math. Mon."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"J\u00e4ntschi, L. (2019). The eigenproblem translated for alignment of molecules. Symmetry, 11.","DOI":"10.3390\/sym11081027"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/BF03014031","article-title":"Combescure ed altre analoghe per le curve storte (Translated by D. H. Delphenich)","volume":"20","author":"Sannia","year":"1905","journal-title":"Rend. Circ. Mat. Palermo"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"233","DOI":"10.2307\/2370293","article-title":"On two related transformations of space curves","volume":"39","author":"Graustein","year":"1917","journal-title":"Am. J. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2230","DOI":"10.1016\/j.physleta.2012.05.044","article-title":"Null Cartan Bertrand curves of AW (k)-type in Minkowski 4-space","volume":"376","author":"Sun","year":"2012","journal-title":"Phys. Lett. A"},{"key":"ref_26","first-page":"11333","article-title":"Directional developable surfaces and their singularities in Euclidean 3-space","volume":"38","author":"Zhu","year":"2024","journal-title":"Filomat"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Wu, L., Zhou, A., Yao, K., and Pei, D. (2024). Generalized Bertrand Curves of Non-Light-like Framed Curves in Lorentz\u2013Minkowski 3-Space. Mathematics, 12.","DOI":"10.3390\/math12162593"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/253\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:03:23Z","timestamp":1760029403000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/253"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,27]]},"references-count":27,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["axioms14040253"],"URL":"https:\/\/doi.org\/10.3390\/axioms14040253","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,3,27]]}}}