{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,31]],"date-time":"2026-07-31T01:10:31Z","timestamp":1785460231561,"version":"3.56.0"},"reference-count":38,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,15]],"date-time":"2023-05-15T00:00:00Z","timestamp":1684108800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University Researchers Supporting Project","award":["PNURSP2023R337"],"award-info":[{"award-number":["PNURSP2023R337"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The association between the instantaneous invariants of a one-parameter Lorentzian spatial movement and the spacelike lines with certain trajectories is considered in this study. To be more precise, we present a theoretical formulation of a Lorentzian inflection line congruence, which is the spatial symmetrical of the inflection circle of planar kinematics. Finally, we establish novel Lorentzian explanations for the Disteli and Euler\u2013Savary formulae. Our results add to a better understanding of the interaction between axodes and Lorentzian spatial movements, with potential implications in fields such as robotics and mechanical engineering.<\/jats:p>","DOI":"10.3390\/sym15051087","type":"journal-article","created":{"date-parts":[[2023,5,16]],"date-time":"2023-05-16T02:20:13Z","timestamp":1684203613000},"page":"1087","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Spacelike Lines with Special Trajectories and Invariant Axodes"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7041-3730","authenticated-orcid":false,"given":"Areej A.","family":"Almoneef","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rashad A.","family":"Abdel-Baky","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Assiut, Assiut 71516, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,15]]},"reference":[{"key":"ref_1","unstructured":"Bottema, O., and Roth, B. (1979). Theoretical Kinematics, North-Holland Press."},{"key":"ref_2","unstructured":"Karger, A., and Novak, J. (1985). Space Kinematics and Lie Groups, Gordon and Breach Science Publishers."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/S0167-8396(97)00032-0","article-title":"Geometric continuity of ruled surfaces","volume":"15","author":"Schaaf","year":"1998","journal-title":"Comput. Aided Geom. Des."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Pottman, H., and Wallner, J. 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