{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T00:13:15Z","timestamp":1777507995133,"version":"3.51.4"},"reference-count":36,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,20]],"date-time":"2024-11-20T00:00:00Z","timestamp":1732060800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we give rotational motions on any straight line or any parabola in a scalar product space. To achieve this goal, we first define the generalized Galilean scalar product and determine the generalized Galilean skew symmetric and orthogonal matrices. Then, using the well-known Rodrigues, Cayley, and Householder maps, we produce the generalized Galilean rotation matrices. Finally, we show that these rotation matrices can also be used to determine parabolic rotational motion.<\/jats:p>","DOI":"10.3390\/sym16111553","type":"journal-article","created":{"date-parts":[[2024,11,20]],"date-time":"2024-11-20T03:57:04Z","timestamp":1732075024000},"page":"1553","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Generalized Galilean Rotations"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5559-9768","authenticated-orcid":false,"given":"Harun Bar\u0131\u015f","family":"\u00c7olako\u011flu","sequence":"first","affiliation":[{"name":"Department of Computer Technologies, Akdeniz University, Antalya 07070, T\u00fcrkiye"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"\u0130skender","family":"\u00d6zt\u00fcrk","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Akdeniz University, Antalya 07070, T\u00fcrkiye"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5331-5100","authenticated-orcid":false,"given":"O\u011fuzhan","family":"\u00c7elik","sequence":"additional","affiliation":[{"name":"Department of Banking and Finance, Pamukkale University, Denizli 20160, T\u00fcrkiye"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mustafa","family":"\u00d6zdemir","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Akdeniz University, Antalya 07070, T\u00fcrkiye"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"114766","DOI":"10.1016\/j.cam.2022.114766","article-title":"Non-parabolic conical rotations","volume":"420","year":"2023","journal-title":"J. Comput. Appl. Math."},{"key":"ref_2","first-page":"115","article-title":"Occurrence of Galilean geometry","volume":"2","author":"Kurudirek","year":"2022","journal-title":"Appl. Comput."},{"key":"ref_3","unstructured":"Yaglom, I.M. (1979). A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity, Springer."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"104885","DOI":"10.1016\/j.rinp.2021.104885","article-title":"Galilean transformation in polar coordinates and Doppler effect","volume":"31","author":"Klinaku","year":"2021","journal-title":"Results Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"105719","DOI":"10.1016\/j.rinp.2022.105719","article-title":"Geometric representation of the Galilean transformation","volume":"39","author":"Klinaku","year":"2022","journal-title":"Results Phys."},{"key":"ref_6","first-page":"94","article-title":"Generalized Galilean transformations and dual quaternions","volume":"5","year":"2009","journal-title":"Sci. Magna"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Kisil, V. (2012). Geometry of Mobius Transformations: Elliptic, Parabolic and Hyperbolic Actions of SL2(R), Imperial College Press.","DOI":"10.1142\/p835"},{"key":"ref_8","unstructured":"Yaglom, I.M. (1968). Complex Numbers in Geometry, Academic Press."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1112\/plms\/s1-4.1.381","article-title":"Preliminary sketch of biquaternions","volume":"1","author":"Clifford","year":"1871","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_10","unstructured":"Kotelnikov, A.P. (1895). Screw Calculus and Some Applications to Geometry and Mechanics, Scientific notes of Kazan University; URSS."},{"key":"ref_11","unstructured":"Study, E. (1903). Geometrie der Dynamen: Die Zusammensetzung von Kraften und Verwandte Gegenstande der Geometrie, Cornell University Library."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Behr, N., Dattoli, G., Lattanzi, A., and Licciardi, S. (2019). Dual numbers and operational umbral methods. Axioms, 8.","DOI":"10.3390\/axioms8030077"},{"key":"ref_13","unstructured":"Bongardt, B. (2019, January 3\u20137). An analysis of the dual-complex unit circle with applications to line geometry. Proceedings of the Conference on Geometry: Theory and Applications, Innsbruck, Austria."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"152","DOI":"10.31349\/RevMexFisE.65.152","article-title":"Some applications in classical mechanics of the double and the dual numbers","volume":"65","year":"2019","journal-title":"Rev. Mex. F\u00edsica E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"418","DOI":"10.31349\/RevMexFis.66.418","article-title":"Double and dual numbers. SU (2) groups, two-component spinors and generating functions","volume":"66","year":"2020","journal-title":"Rev. Mex. F\u00edsica"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"151","DOI":"10.36753\/mathenot.421424","article-title":"A new construction of the Sierpinski triangles with Galilean transformations","volume":"4","year":"2016","journal-title":"Math. Sci. Appl. E-Notes"},{"key":"ref_17","first-page":"79","article-title":"Kuruo\u011flu N. One-parameter planar motion on the Galilean plane","volume":"6","author":"Akar","year":"2013","journal-title":"Int. Electron. J. Geom."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1007\/s00006-015-0530-4","article-title":"One-parameter planar motions in generalized complex number plane","volume":"25","year":"2015","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Valverde, A., and Tsiotras, P. (2018). Spacecraft robot kinematics using dual quaternions. Robotics, 7.","DOI":"10.3390\/robotics7040064"},{"key":"ref_20","first-page":"9","article-title":"Quaternion formulation of the Galilean space-time transformation","volume":"56","author":"Majernik","year":"2006","journal-title":"Acta Phys. Slovaca"},{"key":"ref_21","first-page":"373","article-title":"Dual quaternions in spatial kinematics in an algebraic sense","volume":"32","author":"Akyar","year":"2008","journal-title":"Turk. J. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"9323490","DOI":"10.1155\/2022\/9323490","article-title":"Time-like ruled surface in one-parameter hyperbolic dual spherical motions","volume":"2022","author":"Naghi","year":"2022","journal-title":"Abstr. Appl. Anal."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"8542","DOI":"10.3934\/math.2022476","article-title":"Geometry of the line space associated to a given dual ruled surface","volume":"7","author":"Hussein","year":"2022","journal-title":"AIMS Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1080\/0025570X.2004.11953236","article-title":"Geometry of generalized complex numbers","volume":"77","author":"Harkin","year":"2004","journal-title":"Math. Mag."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1007\/s00006-018-0833-3","article-title":"Introduction to hybrid numbers","volume":"28","year":"2018","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_26","first-page":"463","article-title":"Dual plane and kinematics","volume":"41","author":"Akar","year":"2014","journal-title":"Chiang Mai J. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Rooney, J. (2014). Generalised complex numbers in mechanics. Advances on Theory and Practice of Robots and Manipulators, Springer.","DOI":"10.1007\/978-3-319-07058-2_7"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"013509","DOI":"10.1063\/1.2161072","article-title":"The decomposition of an orthogonal transformation as a product of reflections","volume":"47","year":"2006","journal-title":"J. Math. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/j.laa.2003.07.009","article-title":"G-reflectors: Analogues of householder transformations in scalar product spaces","volume":"385","author":"Mackey","year":"2004","journal-title":"Linear Algebra Its Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1238","DOI":"10.1016\/j.laa.2010.11.005","article-title":"An algorithm for the Cartan\u2013Dieudonn\u2019e theorem on generalized scalar product spaces","volume":"434","year":"2011","journal-title":"Linear Algebra Appl."},{"key":"ref_31","first-page":"10","article-title":"Computing exponentials of skew-symmetric matrices and logarithms of orthogonal matrices","volume":"1","author":"Gallier","year":"2003","journal-title":"Int. J. Robot. Autom."},{"key":"ref_32","unstructured":"Gallier, J. (2006). Remarks on the Cayley representation of orthogonal matrices and on perturbing the diagonal of a matrix to make it invertible. arXiv."},{"key":"ref_33","first-page":"119","article-title":"Sur quelques propri\u00e9t\u00e9s des d\u00e9terminants gauches","volume":"32","author":"Cayley","year":"1846","journal-title":"J. F\u00fcR Die Reine Und Angew. Math."},{"key":"ref_34","first-page":"1550058","article-title":"Cayley formula in Minkowski space-time","volume":"12","year":"2015","journal-title":"Rep. Math. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1177\/1081286507077982","article-title":"Euler-Rodrigues and Cayley formulae for rotation of elasticity tensors","volume":"13","author":"Norris","year":"2008","journal-title":"Math. Mech. Solids"},{"key":"ref_36","unstructured":"Selig, J.M. (2005). Geometric Fundamentals of Robotics, Springer."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/11\/1553\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:35:44Z","timestamp":1760114144000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/11\/1553"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,20]]},"references-count":36,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["sym16111553"],"URL":"https:\/\/doi.org\/10.3390\/sym16111553","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,20]]}}}