{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:36:02Z","timestamp":1760150162425,"version":"build-2065373602"},"reference-count":49,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,10,27]],"date-time":"2023-10-27T00:00:00Z","timestamp":1698364800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The main outcome of this work is the construction of a surface pencil with a similarity to Bertrand curves in Euclidean 3-space E3. Then, by exploiting the Serret\u2013Frenet frame, we deduce the sufficient and necessary conditions for a surface pencil with Bertrand curves as joint curvature lines. Consequently, the expansion to the ruled surface pencil is also designed. As demonstrations of our essential findings, we illustrate some models to emphasize the process.<\/jats:p>","DOI":"10.3390\/sym15111986","type":"journal-article","created":{"date-parts":[[2023,10,27]],"date-time":"2023-10-27T09:42:52Z","timestamp":1698399772000},"page":"1986","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Surface Pencil with Bertrand Curves as Joint Curvature Lines in Euclidean Three-Space"],"prefix":"10.3390","volume":"15","author":[{"given":"Sahar H.","family":"Nazra","sequence":"first","affiliation":[{"name":"Mathematics Department, Faculty of Sciences, Umm Al-Qura University, lMecca 24382, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rashad A.","family":"Abdel-Baky","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Assiut, Assiut 71516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,27]]},"reference":[{"key":"ref_1","unstructured":"Do Carmo, M.P. 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