{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:10:37Z","timestamp":1760145037143,"version":"build-2065373602"},"reference-count":52,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,12]],"date-time":"2024-06-12T00:00:00Z","timestamp":1718150400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004242","name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia","doi-asserted-by":"publisher","award":["PNURSP2024R27"],"award-info":[{"award-number":["PNURSP2024R27"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A curve on a surface is a geodesic curve if its principal normal vector is anywhere aligned with the surface normal. Using the Serret\u2013Frenet frame, a timelike surface couple (TLSC) with the symmetry of a Bertrand couple (BC) can be specified in terms of linear combinations of the components of the local frames in Minkowski 3-space E13. With these parametric representations, the necessary and sufficient conditions for the specified BC are derived to be the geodesic curves defining these surfaces. Afterward, the definition of a TL ruled surface (RS) is also provided. Furthermore, the application of the method to some significant models is given.<\/jats:p>","DOI":"10.3390\/sym16060732","type":"journal-article","created":{"date-parts":[[2024,6,12]],"date-time":"2024-06-12T06:47:30Z","timestamp":1718174850000},"page":"732","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Timelike Surface Couple with Bertrand Couple as Joint Geodesic Curves in Minkowski 3-Space"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2116-7382","authenticated-orcid":false,"given":"Fatemah","family":"Mofarreh","sequence":"first","affiliation":[{"name":"Mathematical Science Department, Faculty of Sciences, Princess Nourah Bint Abdulrahman, Riyadh 11546, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,12]]},"reference":[{"key":"ref_1","unstructured":"Spivak, M.A. (1979). 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