{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T05:41:29Z","timestamp":1769233289027,"version":"3.49.0"},"reference-count":22,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,6,29]],"date-time":"2023-06-29T00:00:00Z","timestamp":1687996800000},"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>Dual representations of the Bertrand offset-surfaces are specified and several new results are gained in terms of their integral invariants. A new description of Bertrand offsets of developable surfaces is given. Furthermore, several relationships through the striction curves of Bertrand offsets of ruled surfaces and their integral invariants are obtained.<\/jats:p>","DOI":"10.3390\/axioms12070649","type":"journal-article","created":{"date-parts":[[2023,6,30]],"date-time":"2023-06-30T01:14:12Z","timestamp":1688087652000},"page":"649","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Bertrand Offsets of Ruled Surfaces with Blaschke Frame in Euclidean 3-Space"],"prefix":"10.3390","volume":"12","author":[{"given":"Sahar H.","family":"Nazra","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, College of Applied 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":[[2023,6,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/0094-114X(76)90006-9","article-title":"On the use of Dual numbers, Vectors, and Matrices in Instantaneous Spatial Kinematics","volume":"11","author":"Veldkamp","year":"1976","journal-title":"Mech. 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