{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T16:18:54Z","timestamp":1782317934342,"version":"3.54.5"},"reference-count":27,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,2,15]],"date-time":"2024-02-15T00:00:00Z","timestamp":1707955200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University Researchers Supporting Project","award":["PNURSP2024R337"],"award-info":[{"award-number":["PNURSP2024R337"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we investigate and specify the Bertrand offsets of slant ruled and developable surfaces in Euclidean 3-space E3. This is accomplished by utilizing the symmetry of slant curves. As a consequence of this, we present the parameterization of the Bertrand offsets for any slant ruled and developable surfaces. In addition to this, we investigate the monarchies of these ruled surfaces and assign them their own unique classification. Also, we illustrate some examples of slant ruled surfaces.<\/jats:p>","DOI":"10.3390\/sym16020235","type":"journal-article","created":{"date-parts":[[2024,2,15]],"date-time":"2024-02-15T03:57:29Z","timestamp":1707969449000},"page":"235","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Bertrand Offsets of Slant Ruled Surfaces in Euclidean 3-Space"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7041-3730","authenticated-orcid":false,"given":"Areej A.","family":"Almoneef","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rashad A.","family":"Abdel-Baky","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Assiut, Assiut 71516, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,15]]},"reference":[{"key":"ref_1","unstructured":"Gugenheimer, H.W. (1956). Differential Geometry, Graw-Hill."},{"key":"ref_2","unstructured":"Bottema, O., and Roth, B. (1979). Theoretical Kinematics, North-Holland Press."},{"key":"ref_3","unstructured":"Karger, A., and Novak, J. (1985). 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