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Later, we generalize the Lorentzian rotation about a lightlike axis, and determine matrices of these transformations using the Cartan frame and the well-known Rodrigues formula, then using the Cayley map, and finally using the generalized split quaternions. We see that such transformations give parabolic rotational motions on general cones or general hyperboloids of one or two sheets, while they also give linear rotational motions on general hyperboloids of one sheet.<\/jats:p>","DOI":"10.3390\/axioms14080612","type":"journal-article","created":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T15:09:53Z","timestamp":1754492993000},"page":"612","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Parabolic and Linear Rotational Motions on Cones and Hyperboloids"],"prefix":"10.3390","volume":"14","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, 07070 Antalya, T\u00fcrkiye"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-5827-4333","authenticated-orcid":false,"given":"Mehmet","family":"Duru","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Akdeniz University, 07070 Antalya, T\u00fcrkiye"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8051-2879","authenticated-orcid":false,"given":"Ay\u015fe","family":"Y\u0131lmaz Ceylan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Akdeniz University, 07070 Antalya, T\u00fcrkiye"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Xie, G., Yang, H., Deng, H., Shi, Z., and Chen, G. 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