{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T08:21:45Z","timestamp":1773908505112,"version":"3.50.1"},"reference-count":53,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,13]],"date-time":"2022-09-13T00:00:00Z","timestamp":1663027200000},"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\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["PNURSP2022R27"],"award-info":[{"award-number":["PNURSP2022R27"]}],"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"}]},{"DOI":"10.13039\/501100004731","name":"Zhejiang Provincial Natural Science Foundation of China","doi-asserted-by":"publisher","award":["PNURSP2022R27"],"award-info":[{"award-number":["PNURSP2022R27"]}],"id":[{"id":"10.13039\/501100004731","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Princess Nourah bint Abdulrahman University","award":["12101168"],"award-info":[{"award-number":["12101168"]}]},{"name":"Princess Nourah bint Abdulrahman University","award":["LQ22A010014"],"award-info":[{"award-number":["LQ22A010014"]}]},{"name":"Princess Nourah bint Abdulrahman University","award":["PNURSP2022R27"],"award-info":[{"award-number":["PNURSP2022R27"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The present paper is focused on time-like circular surfaces and singularities in Minkowski 3-space. The timelike circular surface with a constant radius could be swept out by moving a Lorentzian circle with its center while following a non-lightlike curve called the spine curve. In the present study, we have parameterized timelike circular surfaces and examined their geometric properties, such as singularities and striction curves, corresponding with those of ruled surfaces. After that, a different kind of timelike circular surface was determined and named the timelike roller coaster surface. Meanwhile, we support the results of this work with some examples.<\/jats:p>","DOI":"10.3390\/sym14091914","type":"journal-article","created":{"date-parts":[[2022,9,13]],"date-time":"2022-09-13T22:37:28Z","timestamp":1663108648000},"page":"1914","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":["Timelike Circular Surfaces and Singularities in Minkowski 3-Space"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1614-3228","authenticated-orcid":false,"given":"Yanlin","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics, Hangzhou Normal University, Hangzhou 311120, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2116-7382","authenticated-orcid":false,"given":"Fatemah","family":"Mofarreh","sequence":"additional","affiliation":[{"name":"Mathematical Science Department, Faculty of Science, Princess Nourah bint Abdulrahman University, Riyadh 11546, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7016-9280","authenticated-orcid":false,"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":[[2022,9,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Izumiya, S., and Takeuchi, N. (2001). Singularities of ruled surfaces in \ud835\udd3c3. Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press.","DOI":"10.1017\/S0305004100004643"},{"key":"ref_2","unstructured":"Izumiya, S., and Takeuchi, N. (2003). Geometry of ruled surfaces. Applicable Mathematics in the Golden Age, Narosa Publishing House."},{"key":"ref_3","first-page":"203","article-title":"Special curves and ruled surfaces","volume":"44","author":"Izumiya","year":"2003","journal-title":"Contrib. Algebra Geom."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/BFb0088679","article-title":"Circles on surfaces in the Euclidean 3-space","volume":"Volume 792","author":"Blum","year":"1980","journal-title":"Geometry and Differential Geometry"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/j.cam.2005.08.002","article-title":"Analytic and algebraic properties of canal surfaces","volume":"195","author":"Xu","year":"2006","journal-title":"J. Comput. Appl. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"101009","DOI":"10.1115\/1.3212679","article-title":"Kinematic geometry of circular surfaces with a fixed radius based on Euclidean invariants","volume":"131","author":"Cui","year":"2009","journal-title":"ASME J. Mech."},{"key":"ref_7","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_8","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. Appl."},{"key":"ref_9","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_10","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_11","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_12","doi-asserted-by":"crossref","first-page":"12671","DOI":"10.3934\/math.2022701","article-title":"Circular surfaces and singularities in Euclidean 3-space E3","volume":"7","author":"Alluhaibi","year":"2022","journal-title":"AIMS Math."},{"key":"ref_13","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_14","unstructured":"O\u2019Neil, B. (1983). Semi-Riemannian Geometry with Applications to Relativity, Academic Press."},{"key":"ref_15","unstructured":"Mc-Nertney, L.V. (1980). One-Parameter Families of Surfaces with Constant Curvature in Lorentz Three-Space. [Ph.D. Thesis, Brown University]."},{"key":"ref_16","unstructured":"Walrave, J. (1995). Curves and Surfaces in Minkowski Space. [Ph.D. Thesis, K.U. Leuven, Faculty of Science]."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Sharma, J.R., Kumar, S., and J\u00e4ntschi, L. (2019). On a class of optimal fourth order multiple root solvers without using derivatives. Symmetry, 11.","DOI":"10.3390\/sym11121452"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Joita, D.M., Tomescu, M.A., B\u00e0lint, D., and J\u00e4ntschi, L. (2021). An Application of the Eigenproblem for Biochemical Similarity. Symmetry, 13.","DOI":"10.3390\/sym13101849"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"J\u00e4ntschi, L. (2022). Introducing Structural Symmetry and Asymmetry Implications in Development of Recent Pharmacy and Medicine. Symmetry, 14.","DOI":"10.3390\/sym14081674"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"J\u00e4ntschi, L. (2022). Binomial Distributed Data Confidence Interval Calculation: Formulas, Algorithms and Examples. Symmetry, 14.","DOI":"10.3390\/sym14061104"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"J\u00e4ntschi, L. (2021). Formulas, Algorithms and Examples for Binomial Distributed Data Confidence Interval Calculation: Excess Risk, Relative Risk and Odds Ratio. Mathematics, 9.","DOI":"10.3390\/math9192506"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2855","DOI":"10.3390\/math9222855","article-title":"Comparison of Molecular Geometry Optimization Methods Based on Molecular Descriptors","volume":"9","author":"Donatella","year":"2021","journal-title":"Mathematics"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1419","DOI":"10.3390\/math9121419","article-title":"Figures of Graph Partitioning by Counting, Sequence and Layer Matrices","volume":"9","author":"Mihaela","year":"2021","journal-title":"Mathematics"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Kumar, S., Kumar, D., Sharma, J.R., and J\u00e4ntschi, L. (2020). A Family of Derivative Free Optimal Fourth Order Methods for Computing Multiple Roots. Symmetry, 12.","DOI":"10.3390\/sym12121969"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1494","DOI":"10.3390\/sym12091494","article-title":"A Novel Family of Efficient Weighted-Newton Multiple Root Iterations","volume":"12","author":"Deepak","year":"2020","journal-title":"Symmetry"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1091","DOI":"10.3390\/math8071091","article-title":"On Derivative Free Multiple-Root Finders with Optimal Fourth Order Convergence","volume":"8","author":"Janak","year":"2020","journal-title":"Mathematics"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"J\u00e4ntschi, L. (2020). Detecting Extreme Values with Order Statistics in Samples from Continuous Distributions. Mathematics, 8.","DOI":"10.3390\/math8020216"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"919","DOI":"10.3390\/math7100919","article-title":"Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method","volume":"7","author":"Deepak","year":"2019","journal-title":"Mathematics"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"25614","DOI":"10.1002\/qua.25614","article-title":"Conformational study of C24 cyclic polyyne clusters","volume":"118","year":"2018","journal-title":"Int. J. Quantum Chem."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"\u00dcnl\u00fct\u00fcrk, Y., and Yilmaz, S. (2019). A new approach to timelike surfaces which contain inclined curves as geodesics. Differ. Equ. Dyn. Syst., 1\u20138.","DOI":"10.1007\/s12591-019-00485-9"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"104513","DOI":"10.1016\/j.geomphys.2022.104513","article-title":"Evolving evolutoids and pedaloids from viewpoints of envelope and singularity theory in Minkowski plane","volume":"176","author":"Yang","year":"2022","journal-title":"J. Geom. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"5408","DOI":"10.3934\/math.2022300","article-title":"Conformal \u03b7-Ricci solitons within the framework of indefinite Kenmotsu manifolds","volume":"7","author":"Li","year":"2022","journal-title":"AIMS Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"12077","DOI":"10.3934\/math.2022671","article-title":"Yamabe constant evolution and monotonicity along the conformal Ricci flow","volume":"7","author":"Li","year":"2022","journal-title":"AIMS Math."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Li, Y., Khatri, M., Singh, J.P., and Chaubey, S.K. (2022). Improved Chen\u2019s Inequalities for Submanifolds of Generalized Sasakian-Space-Forms. Axioms, 11.","DOI":"10.3390\/axioms11070324"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Li, Y., U\u00e7um, A., \u0130larslan, K., and Camc\u0131, \u00c7. (2022). A New Class of Bertrand Curves in Euclidean 4-Space. Symmetry, 14.","DOI":"10.3390\/sym14061191"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1186\/s13660-022-02838-5","article-title":"Reilly-type inequality for the \u03a6-Laplace operator on semislant submanifolds of Sasakian space forms","volume":"2022","author":"Li","year":"2022","journal-title":"J. Inequal. Appl."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Li, Y., Mofarreh, F., Dey, S., Roy, S., and Ali, A. (2022). General Relativistic Space-Time with \u03b71-Einstein Metrics. Mathematics, 10.","DOI":"10.3390\/math10142530"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"574","DOI":"10.1515\/math-2022-0048","article-title":"Geometry of conformal \u03b7-Ricci solitons and conformal \u03b7-Ricci almost solitons on Paracontact geometry","volume":"20","author":"Li","year":"2022","journal-title":"Open Math."},{"key":"ref_39","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_40","doi-asserted-by":"crossref","first-page":"6605127","DOI":"10.1155\/2022\/6605127","article-title":"LP-Kenmotsu manifolds admitting \u03b7-Ricci solitons and spacetime","volume":"2022","author":"Li","year":"2022","journal-title":"J. Math."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Anti\u0107, M., Moruz, M., and Van, J. (2020). H-Umbilical Lagrangian Submanifolds of the Nearly K\u00e4hler S3 \u00d7 S3. Mathematics, 8.","DOI":"10.3390\/math8091427"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Anti\u0107, M., and Djordje, K. (2022). Non-Existence of Real Hypersurfaces with Parallel Structure Jacobi Operator in S6(1). Mathematics, 10.","DOI":"10.3390\/math10132271"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1007\/s00025-022-01621-8","article-title":"Characterization of Warped Product Lagrangian Submanifolds in Cn","volume":"77","year":"2022","journal-title":"Results Math."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1641","DOI":"10.1007\/s11425-021-1897-0","article-title":"Conformally flat, minimal, Lagrangian submanifolds in complex space forms","volume":"65","author":"Vrancken","year":"2022","journal-title":"Sci. China Math."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2286","DOI":"10.1002\/mana.201900376","article-title":"Surfaces of the nearly K\u00e4hler S3 \u00d7 S3 preserved by the almost product structure","volume":"294","author":"Hu","year":"2021","journal-title":"Math. Nachr."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1129","DOI":"10.1515\/ms-2017-0175","article-title":"A class of four-dimensional CR submanifolds in six dimensional nearly K\u00e4hler manifolds","volume":"68","year":"2018","journal-title":"Math. Slovaca"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/j.geomphys.2016.07.014","article-title":"A class of four dimensional CR submanifolds of the sphere S6(1)","volume":"110","year":"2016","journal-title":"J. Geom. Phys."},{"key":"ref_48","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_49","first-page":"69","article-title":"A constant angle ruled surfaces","volume":"7","author":"Ali","year":"2018","journal-title":"Int. J. Geom."},{"key":"ref_50","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_51","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_52","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_53","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. Egyptian Math. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1914\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:30:12Z","timestamp":1760142612000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1914"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,13]]},"references-count":53,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["sym14091914"],"URL":"https:\/\/doi.org\/10.3390\/sym14091914","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,13]]}}}