{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T00:13:52Z","timestamp":1777508032967,"version":"3.51.4"},"reference-count":66,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T00:00:00Z","timestamp":1684281600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12101168"],"award-info":[{"award-number":["12101168"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["LQ22A010014"],"award-info":[{"award-number":["LQ22A010014"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004731","name":"Zhejiang Provincial Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12101168"],"award-info":[{"award-number":["12101168"]}],"id":[{"id":"10.13039\/501100004731","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004731","name":"Zhejiang Provincial Natural Science Foundation of China","doi-asserted-by":"publisher","award":["LQ22A010014"],"award-info":[{"award-number":["LQ22A010014"]}],"id":[{"id":"10.13039\/501100004731","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, we examine the relationship between Darboux frames along parameter curves and the Darboux frame of the base curve of the ruled surface. We derive the equations of the quaternionic shape operators, which can rotate tangent vectors around the normal vector, and find the corresponding rotation matrices. Using these operators, we examine the Gauss curvature and mean curvature of the ruled surface. We explore how these properties are found by the use of Frenet vectors instead of generator vectors. We provide illustrative examples to better demonstrate the concepts and results discussed.<\/jats:p>","DOI":"10.3390\/axioms12050486","type":"journal-article","created":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T07:35:50Z","timestamp":1684395350000},"page":"486","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Quaternionic Shape Operator and Rotation Matrix on Ruled Surfaces"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1614-3228","authenticated-orcid":false,"given":"Yanlin","family":"Li","sequence":"first","affiliation":[{"name":"Key Laboratory of Cryptography of Zhejiang Province, School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1512-2452","authenticated-orcid":false,"given":"Abdussamet","family":"\u00c7al\u0131\u015fkan","sequence":"additional","affiliation":[{"name":"Fatsa Vocational School, Accounting and Tax Applications, Ordu Universty, Ordu 52200, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,17]]},"reference":[{"key":"ref_1","unstructured":"Burstall, F.E., Ferus, D., Leschke, K., Pedit, F., and Pinkall, U. (2004). Conformal Geometry of Surfaces in S4 and Quaternions, Springer. [1st ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1507","DOI":"10.1103\/PhysRevLett.57.1507","article-title":"Exact solution to localized-induction-approximation equation modeling smoke ring motion","volume":"57","author":"Cieslinski","year":"1986","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1396","DOI":"10.1023\/A:1026096404956","article-title":"Geometry of submanifolds derived from Spin-valued spectral problems","volume":"137","author":"Cieslinski","year":"2003","journal-title":"Theor. Math. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1007\/3-540-16039-6_6","article-title":"Soliton Surfaces and their Applications in Geometrical Aspects of the Einstein Equations and Integrable Systems","volume":"239","author":"Sym","year":"1985","journal-title":"Lect. Notes Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1016\/S0393-0440(02)00130-4","article-title":"On the integrability of Bertrand curves and Razzaboni surfaces","volume":"45","author":"Schief","year":"2003","journal-title":"J. Geom. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Rogers, C., Rogers, C., and Schief, W.K. (2002). B\u00e4cklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory, Cambridge University Press.","DOI":"10.1017\/CBO9780511606359"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1145\/325165.325242","article-title":"Animating Rotation with Quaternion Curves","volume":"19","author":"Shoemake","year":"1985","journal-title":"Siggraph Comput. Graph."},{"key":"ref_8","first-page":"9567","article-title":"A novel calibration method of SINS\/DVL integration navigation system based on quaternion","volume":"20","author":"Xu","year":"2020","journal-title":"IEEE Sens. J."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Fordy, A., and Wood, J. (1994). Harmonic Maps and Integrable Systems, Vieweg.","DOI":"10.1007\/978-3-663-14092-4"},{"key":"ref_10","first-page":"126359","article-title":"A new approach to constant slope surfaces with quaternions","volume":"2012","author":"Babaarslan","year":"2012","journal-title":"ISRN Geom."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.gmod.2014.04.002","article-title":"Quaternion rational surfaces: Rational surfaces generated from the quaternion product of two rational space curves","volume":"81","author":"Wang","year":"2015","journal-title":"Adv. Appl. Graph. Model."},{"key":"ref_12","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_13","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1109\/TRO.2010.2040201","article-title":"A Darboux-Frame-Based formulation of spin-rolling motion of rigid objects with point contact","volume":"26","author":"Cui","year":"2010","journal-title":"IEEE Trans. Robot."},{"key":"ref_14","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_15","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_16","doi-asserted-by":"crossref","first-page":"2921","DOI":"10.1007\/s00006-017-0804-0","article-title":"Quaternionic shape operator","volume":"27","author":"Aslan","year":"2017","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ryuh, B.S. (1989). Robot Trajectory Planning Using the Curvature Theory of Ruled Surfaces. [Ph.D. Thesis, Purdue University].","DOI":"10.1115\/DETC1989-0034"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Li, Y., \u015eenyurt, S., \u00d6zduran, A., and Canl\u0131, D. (2022). The Characterizations of Parallel q-Equidistant Ruled Surfaces. Symmetry, 14.","DOI":"10.3390\/sym14091879"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1007\/s00009-021-01843-0","article-title":"Ruled surfaces of generalized self-similar solutions of the mean curvature flow","volume":"18","year":"2021","journal-title":"Mediterr. J. Math."},{"key":"ref_20","first-page":"11","article-title":"On ruled surfaces according to quasi-frame in Euclidean 3-space","volume":"17","author":"Saad","year":"2020","journal-title":"Aust. J. Math. Anal. Appl."},{"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","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_23","doi-asserted-by":"crossref","first-page":"103833","DOI":"10.1016\/j.geomphys.2020.103833","article-title":"Non-lightlike constant angle ruled surfaces in Minkowski 3-space","volume":"157","author":"Ali","year":"2020","journal-title":"J. Geom. Phys."},{"key":"ref_24","first-page":"69","article-title":"A constant angle ruled surfaces","volume":"7","author":"Ali","year":"2018","journal-title":"Int. J. Geom."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1850068","DOI":"10.1142\/S0219887818500688","article-title":"Non-lightlike ruled surfaces with constant curvatures in Minkowski 3-space","volume":"15","author":"Ali","year":"2018","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_26","first-page":"537","article-title":"Surfaces foliated by ellipses with constant Gaussian curvature in Euclidean 3-space","volume":"25","author":"Ali","year":"2017","journal-title":"Korean J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"593","DOI":"10.5831\/HMJ.2016.38.3.593","article-title":"On some geometric properties of quadric surfaces in Euclidean space","volume":"38","author":"Ali","year":"2016","journal-title":"Honam Math. J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.joems.2014.02.007","article-title":"On curvatures and points of the translation surfaces in Euclidean 3-space","volume":"23","author":"Ali","year":"2015","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"206","DOI":"10.3390\/sym15010206","article-title":"The Invariants of Dual Parallel Equidistant Ruled Surfaces","volume":"15","author":"Grilli","year":"2023","journal-title":"Symmetry"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5735","DOI":"10.2298\/FIL2317735G","article-title":"Geometric properties of timelike surfaces in Lorentz-Minkowski 3-space","volume":"37","year":"2023","journal-title":"Filomat"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1062","DOI":"10.3390\/sym14051062","article-title":"The Dual Expression of Parallel Equidistant Ruled Surfaces in Euclidean 3-Space","volume":"14","author":"Grilli","year":"2022","journal-title":"Symmetry"},{"key":"ref_32","first-page":"684","article-title":"Curves and ruled surfaces according to alternative frame in dual space","volume":"69","year":"2020","journal-title":"Commun. Fac. Sci. Univ."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1750118","DOI":"10.1142\/S0219887817501183","article-title":"Spacelike surface geometry","volume":"14","year":"2017","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"587289","DOI":"10.1155\/2013\/587289","article-title":"Some Characteristic Properties of Parallel-Equidistant Ruled Surfaces","volume":"2013","author":"As","year":"2013","journal-title":"Math. Probl. Eng."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1007\/s00006-012-0327-7","article-title":"On Some Characterizations of Ruled Surface of a Closed Timelike Curve in Dual Lorentzian Space","volume":"22","year":"2012","journal-title":"Adv. Appl. Clifford Al."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1515\/ADVGEOM.2007.017","article-title":"Circular surfaces","volume":"7","author":"Izumiya","year":"2007","journal-title":"Adv. Geom."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/j.difgeo.2011.02.005","article-title":"Great circular surfaces in the three-sphere","volume":"29","author":"Izumiya","year":"2011","journal-title":"Differ. Geom. Its Appl."},{"key":"ref_38","first-page":"203","article-title":"Special curves and ruled surfaces","volume":"44","author":"Izumiya","year":"2003","journal-title":"Cotributions Algebra Geom."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"725","DOI":"10.2298\/FIL1504725G","article-title":"Circular surfaces CS(\u03b1, p)","volume":"29","author":"Gorjanc","year":"2015","journal-title":"Filomat"},{"key":"ref_40","first-page":"50","article-title":"On the curvatures of spacelike circular surfaces","volume":"43","author":"Unluturk","year":"2016","journal-title":"Kuwait J. Sci."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2050074","DOI":"10.1142\/S0219887820500747","article-title":"A study on timelike circular surfaces in Minkowski 3-space","volume":"17","author":"Alluhaibi","year":"2020","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s00006-018-0883-6","article-title":"Circular surfaces with split quaternionic representations in Minkowski 3-space","volume":"28","author":"Tuncer","year":"2018","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_43","first-page":"931","article-title":"Some characterizations of osculating curves in the Euclidean spaces","volume":"41","year":"2008","journal-title":"Demonstr. Math."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/0010-4485(91)90005-H","article-title":"Bertrand offsets of ruled and developable surfaces","volume":"23","author":"Ravani","year":"1991","journal-title":"Comput. Aided Des."},{"key":"ref_45","first-page":"763","article-title":"On the invariants of Bertrand trajectory surface offsets","volume":"151","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s40574-015-0025-1","article-title":"On the Bertrand offsets for ruled and developable surfaces","volume":"8","author":"Aldossary","year":"2015","journal-title":"Boll. Unione Mat. Ital."},{"key":"ref_47","first-page":"37","article-title":"Integral invariants of the pairs of the Bertrand ruled surface","volume":"21","author":"Kasap","year":"2002","journal-title":"Bull. Pure Appl. Sci. Sect. E Math."},{"key":"ref_48","first-page":"39","article-title":"The Bertrand offsets of ruled surfaces in R13","volume":"31","author":"Kasap","year":"2006","journal-title":"Acta Math. Vietnam"},{"key":"ref_49","first-page":"195","article-title":"The involute-evolute offsets of ruled surfaces","volume":"33","author":"Kasap","year":"2009","journal-title":"Iran. J. Sci. Tech. Trans. A"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"160917","DOI":"10.1155\/2009\/160917","article-title":"Mannheim offsets of ruled surfaces","volume":"2009","author":"Orbay","year":"2009","journal-title":"Math. Probl. Eng."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"O\u2019Neill, B. (2006). Elementary Differential Geometry, Elsevier.","DOI":"10.1016\/B978-0-12-088735-4.50011-0"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Hanson, J.A. (2006). Visualing Quaternions, Elsevier.","DOI":"10.1145\/1281500.1281634"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Kuipers, J.B. (1999). Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality, Princeton University Press.","DOI":"10.1515\/9780691211701"},{"key":"ref_54","unstructured":"Do Carmo, M.P. (1976). Differential Geometry of Curves and Surfaces, Prentice-Hall."},{"key":"ref_55","unstructured":"Hac\u0131saliho\u011flu, H.H. (1983). Differential Geometry, \u0130n\u00f6n\u00fc University."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"11157","DOI":"10.1002\/mma.9173","article-title":"On (contra)pedals and (anti)orthotomics of frontals in de Sitter 2-space","volume":"46","author":"Li","year":"2023","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Li, Y., Abolarinwa, A., Alkhaldi, A., and Ali, A. (2022). Some Inequalities of Hardy Type Related to Witten-Laplace Operator on Smooth Metric Measure Spaces. Mathematics, 10.","DOI":"10.3390\/math10234580"},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Li, Y., Aldossary, M.T., and Abdel-Baky, R.A. (2023). Spacelike Circular Surfaces in Minkowski 3-Space. Symmetry, 15.","DOI":"10.3390\/sym15010173"},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Li, Y., Chen, Z., Nazra, S.H., and Abdel-Baky, R.A. (2023). Singularities for Timelike Developable Surfaces in Minkowski 3-Space. Symmetry, 15.","DOI":"10.3390\/sym15020277"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"13875","DOI":"10.3934\/math.2023709","article-title":"Investigation of ruled surfaces and their singularities according to Blaschke frame in Euclidean 3-space","volume":"8","author":"Li","year":"2023","journal-title":"AIMS Math."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"2226","DOI":"10.3934\/math.2023115","article-title":"The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space","volume":"8","author":"Li","year":"2023","journal-title":"AIMS Math."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s00009-023-02396-0","article-title":"Kenmotsu Metric as Conformal \u03b7-Ricci Soliton","volume":"20","author":"Li","year":"2023","journal-title":"Mediterr. J. Math."},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Li, Y., Srivastava, S.K., Mofarreh, F., Kumar, A., and Ali, A. (2023). Ricci Soliton of CR-Warped Product Manifolds and Their Classifications. Symmetry, 15.","DOI":"10.3390\/sym15050976"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"16278","DOI":"10.3934\/math.2023833","article-title":"Zermelo\u2019s navigation problem for some special surfaces of rotation","volume":"8","author":"Li","year":"2023","journal-title":"AIMS Math."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"2386","DOI":"10.3934\/math.2023123","article-title":"Primitivoids of curves in Minkowski plane","volume":"8","author":"Li","year":"2023","journal-title":"AIMS Math."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"114","DOI":"10.15672\/hujms.1052831","article-title":"Differential Geometric Approach of Betchow-Da Rios Soliton Equation","volume":"52","author":"Li","year":"2023","journal-title":"Hacet. J. Math. Stat."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/5\/486\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:37:09Z","timestamp":1760125029000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/5\/486"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,17]]},"references-count":66,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["axioms12050486"],"URL":"https:\/\/doi.org\/10.3390\/axioms12050486","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,17]]}}}