{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T05:26:35Z","timestamp":1781587595498,"version":"3.54.5"},"reference-count":24,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,9]],"date-time":"2024-09-09T00:00:00Z","timestamp":1725840000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Sakarya University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we investigate the spatial quaternionic expressions of partner-ruled surfaces. Moreover, we formulate the striction curves and dralls of these surfaces by use of the quaternionic product. Furthermore, the pitches and angles of pitches are interpreted for the spatial quaternionic ruled surfaces that are closed. Additionally, we calculate the integral invariants of these surfaces using quaternionic formulas. Finally, the partner-ruled surfaces of a given spatial quaternionic ruled surface are demonstrated as an example, and their graphics are drawn.<\/jats:p>","DOI":"10.3390\/axioms13090612","type":"journal-article","created":{"date-parts":[[2024,9,9]],"date-time":"2024-09-09T07:14:04Z","timestamp":1725866044000},"page":"612","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Characterizations of Spatial Quaternionic Partner-Ruled Surfaces"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7183-7081","authenticated-orcid":false,"given":"Soley","family":"Ersoy","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences, Sakarya University, 54050 Sakarya, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5273-7897","authenticated-orcid":false,"given":"Kemal","family":"Eren","sequence":"additional","affiliation":[{"name":"Sakarya University Technology Developing Zones Manager Company, 54050 Sakarya, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1512-2452","authenticated-orcid":false,"given":"Abdussamet","family":"\u00c7al\u0131\u015fkan","sequence":"additional","affiliation":[{"name":"Fatsa Vocational School, Department of Accounting and Tax Applications, Ordu University, 52100 Ordu, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,9]]},"reference":[{"key":"ref_1","unstructured":"Hamilton, W.R. (1969). Elements of Quaternions, Chelsea Publishing Co.. [3rd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1088\/0143-0807\/5\/1\/007","article-title":"The quaternion group and modern physics","volume":"5","author":"Girard","year":"1984","journal-title":"Eur. J. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1109\/2945.468403","article-title":"Quaternion frame approach to streamline visualization","volume":"1","author":"Hanson","year":"1984","journal-title":"IEEE Trans. Vis. Comput. Graph."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1111\/sapm.12211","article-title":"Linear quaternion differential equations: Basic theory and fundamental results","volume":"141","author":"Kou","year":"2018","journal-title":"Stud. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1145\/325165.325242","article-title":"Animating rotation with quaternionic curves","volume":"9","author":"Shoemake","year":"1985","journal-title":"ACM SIGGRAPH Comput. Graph."},{"key":"ref_6","first-page":"741","article-title":"Quaternion valued function of a real Serret-Frenet formulae","volume":"16","author":"Bharathi","year":"1985","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_7","first-page":"194","article-title":"On the quaternionic curves according to parallel transport frame","volume":"4","author":"Soyfidan","year":"2013","journal-title":"TWMS J. Pure Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"67","DOI":"10.5831\/HMJ.2014.36.1.67","article-title":"Some characterizations of quaternionic rectifying curves in the semi-Euclidean space E24","volume":"36","year":"2014","journal-title":"Honam Math. J."},{"key":"ref_9","first-page":"36","article-title":"On the quaternionic curves in the semi-Euclidean space E24","volume":"7","year":"2018","journal-title":"Casp. J. Math. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"14428","DOI":"10.1002\/mma.7709","article-title":"A primer on the differential geometry of quaternionic curves","volume":"44","author":"Giardino","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2250087","DOI":"10.1142\/S0219887822500876","article-title":"Winding number and homotopy for quaternionic curves","volume":"19","author":"Giardino","year":"2022","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1007\/s00006-011-0284-6","article-title":"On the quaternionic B2-slant helices in the Euclidean space E4","volume":"21","author":"Okuyucu","year":"2011","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_13","first-page":"8","article-title":"Quaternionic curves which lie on the special planes in 4-dimensional Euclidean space E4","volume":"2024","year":"2024","journal-title":"J. Math."},{"key":"ref_14","first-page":"91","article-title":"Motion of inextensible quaternionic curves and modified Korteweg-de Vries equation","volume":"30","author":"Eren","year":"2022","journal-title":"An. \u015etiint. Univ. \u201cOvidius\u201d Constan\u0163a Ser. Mat."},{"key":"ref_15","first-page":"375","article-title":"Quaternionic maps and minimal surfaces","volume":"4","author":"Chen","year":"2005","journal-title":"Ann. Scuola Norm. Sup. Pisa Cl. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"100004","DOI":"10.1016\/j.jaca.2023.100004","article-title":"Special syzygies of rational surfaces generated by dual quaternions","volume":"3","author":"Hoffman","year":"2022","journal-title":"J. Algebra Comput. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"20213","DOI":"10.3934\/math.20221106","article-title":"Simultaneous characterizations of partner ruled surfaces using Flc frame","volume":"7","author":"Li","year":"2022","journal-title":"AIMS Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"5753","DOI":"10.2298\/FIL1816753S","article-title":"The quaternionic expression of ruled surfaces","volume":"32","year":"2018","journal-title":"Filomat"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"403","DOI":"10.2298\/TSCI181125053C","article-title":"The dual spatial quaternionic expression of ruled surfaces","volume":"23","year":"2019","journal-title":"Therm. Sci."},{"key":"ref_20","first-page":"406","article-title":"The quaternionic ruled surfaces in terms of alternative frame","volume":"11","year":"2022","journal-title":"Palest. J. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1175","DOI":"10.1007\/s00006-016-0703-9","article-title":"Quaternionic approach of canal surfaces constructed by some new ideas","volume":"27","year":"2017","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s00006-015-0602-5","article-title":"Canal surfaces with quaternions","volume":"26","author":"Aslan","year":"2016","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1950091","DOI":"10.1142\/S1793557119500918","article-title":"The construction of circular surfaces with quaternions","volume":"12","year":"2019","journal-title":"Asian-Eur. J. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"104048","DOI":"10.1016\/j.geomphys.2020.104048","article-title":"Ruled surfaces constructed by quaternions","volume":"161","author":"Aslan","year":"2021","journal-title":"J. Geom. Phys."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/9\/612\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:52:07Z","timestamp":1760111527000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/9\/612"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,9]]},"references-count":24,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["axioms13090612"],"URL":"https:\/\/doi.org\/10.3390\/axioms13090612","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,9,9]]}}}