{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:34:18Z","timestamp":1760060058660,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,7,28]],"date-time":"2025-07-28T00:00:00Z","timestamp":1753660800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University","award":["PNURSP2025R27"],"award-info":[{"award-number":["PNURSP2025R27"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper presents a detailed investigation into the isometric properties of osculating and rectifying curves on smooth immersed surfaces in E3. We examine the geometric interactions between these curves, specifically when the osculating curve is associated with one surface and the rectifying curve with another. The main objective of this study is to identify the conditions under which these curves exhibit isometric behavior, preserving their intrinsic geometric properties along their respective Frenet frames. Our findings demonstrate that these curves retain isometric characteristics along the tangent, normal, and binormal directions, offering new insights into their structural invariance. This research makes a significant contribution to the broader field of differential geometry, with potential applications in surface theory.<\/jats:p>","DOI":"10.3390\/axioms14080586","type":"journal-article","created":{"date-parts":[[2025,7,28]],"date-time":"2025-07-28T14:50:03Z","timestamp":1753714203000},"page":"586","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Study on the Behavior of Osculating and Rectifying Curves on Smooth Immersed Surfaces in E3"],"prefix":"10.3390","volume":"14","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 Science, Princess Nourah bint Abdulrahman University, Riyadh 11546, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-9927-5154","authenticated-orcid":false,"given":"Ahmer","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Narowal, Narowal 51600, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-4791-6606","authenticated-orcid":false,"given":"Farah","family":"Naz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Narowal, Narowal 51600, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4899-5313","authenticated-orcid":false,"given":"Muhammad","family":"Hanif","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Narowal, Narowal 51600, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,28]]},"reference":[{"key":"ref_1","first-page":"65","article-title":"Quaternionic osculating curves in Euclidean and semi-Euclidean space","volume":"14","year":"2016","journal-title":"J. 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