{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:09:14Z","timestamp":1760231354071,"version":"build-2065373602"},"reference-count":55,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,15]],"date-time":"2022-09-15T00:00:00Z","timestamp":1663200000000},"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","LQ22A010014"],"award-info":[{"award-number":["12101168","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","LQ22A010014"],"award-info":[{"award-number":["12101168","LQ22A010014"]}],"id":[{"id":"10.13039\/501100004731","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>E. Study map is one of the most basic and powerful mathematical tools to study lines in line geometry, it has symmetry property. In this paper, based on the E. Study map, clear expressions were developed for the differential properties of one-parameter Lorentzian dual spherical movements that are coordinate systems independent. This eliminates the requirement of demanding coordinates transformations necessary in the determination of the canonical systems. With the proposed technique, new proofs for Euler\u2013Savary, and Disteli\u2019s formulae were derived.<\/jats:p>","DOI":"10.3390\/sym14091930","type":"journal-article","created":{"date-parts":[[2022,9,16]],"date-time":"2022-09-16T03:48:04Z","timestamp":1663300084000},"page":"1930","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["One-Parameter Lorentzian Dual Spherical Movements and Invariants of the Axodes"],"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\/0000-0001-7135-0559","authenticated-orcid":false,"given":"Nadia","family":"Alluhaibi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Science and Arts College, King Abdulaziz University, Rabigh 21589, 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":[[2022,9,15]]},"reference":[{"key":"ref_1","unstructured":"Bottema, O., and Roth, B. 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