{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:30:36Z","timestamp":1760149836025,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,9,24]],"date-time":"2023-09-24T00:00:00Z","timestamp":1695513600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University Researchers","award":["PNURSP2023R337"],"award-info":[{"award-number":["PNURSP2023R337"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this study, we establish a novel Lorentzian interpretation of the Euler\u2013Savary (E\u2212S) and Disteli (Dis) formulae. Subsequently, we proceed to establish a theoretical structure for a Lorentzian torsion line congruence which is the spatial symmetry of the Lorentzian circling-point dual curve, in accordance with the principles of the kinematic theory of spherical locomotions. Further, a timelike (Tlike) torsion line congruence is defined and its spatial equivalence is examined. The findings contribute to an enhanced comprehension of the interplay between axodes and Lorentzian spatial movements, which has possible significance in various disciplines, such as the fields of robotics and mechanical engineering.<\/jats:p>","DOI":"10.3390\/sym15101816","type":"journal-article","created":{"date-parts":[[2023,9,24]],"date-time":"2023-09-24T10:40:22Z","timestamp":1695552022000},"page":"1816","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Study on a Spacelike Line Trajectory in Lorentzian Locomotions"],"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":[{"role":"author","vocabulary":"crossref"}]},{"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":[[2023,9,24]]},"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","unstructured":"Schaaf, J.A. (1988). Curvature Theory of Line Trajectories in Spatial Kinematics. [Ph.D. Thesis, University of California]."},{"key":"ref_4","unstructured":"Stachel, H. (1996). Instantaneous Spatial Kinematics and the Invariants of the Axodes, Institute fur Geometrie, TU Wien. Technical Report."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Pottman, H., and Wallner, J. (2001). Computational Line Geometry, Springer.","DOI":"10.1007\/978-3-642-04018-4"},{"key":"ref_6","unstructured":"Garnier, R. (1956). Cours de Cin\u00e9matique, Tome II: Roulement et Vibration-La Formule de Savary et son Extension a l\u2019Espace, Gauthier-Villars."},{"key":"ref_7","first-page":"267","article-title":"On the theorem of three axes in the spatial motion of three bodies","volume":"15","author":"Phillips","year":"1964","journal-title":"J. Appl. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/0022-2569(66)90017-6","article-title":"A study of the geometry and the kinematics of Instantaneous spatial motion","volume":"1","author":"Skreiner","year":"1966","journal-title":"J. Mech."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/0094-114X(74)90042-1","article-title":"Einfacbe Herleitung der Euler-Savaryschen Konstruktion der riiumlichen Bewegung","volume":"9","author":"Dizioglu","year":"1974","journal-title":"Mech. Mach. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1007\/s10665-007-9139-5","article-title":"A new geometrical approach to one-parameter spatial motion","volume":"60","year":"2008","journal-title":"J. Eng. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1007\/s00170-007-1133-6","article-title":"Cylindrical milling of ruled surfaces","volume":"38","author":"Sprott","year":"2008","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Kecskem\u00e9thy, A., and M\u00fcller, A. (2009). Computational Kinematics, Proceedings of the 5th International Workshop on Computational Kinematics, Springer.","DOI":"10.1007\/978-3-642-01947-0"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.cagd.2010.09.007","article-title":"On the one-parameter dual spherical motions","volume":"28","year":"2011","journal-title":"Comput. Aided Geom. Des."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3597","DOI":"10.1007\/s12206-015-0803-9","article-title":"Kinematic geometry of a line trajectory in spatial motion","volume":"29","year":"2015","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_15","first-page":"333","article-title":"On the curvature theory of a line trajectory in spatial kinematics","volume":"34","year":"2019","journal-title":"Commun. Korean Math. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"820","DOI":"10.1016\/j.euromechsol.2009.03.007","article-title":"Dual Lorentzian spherical movements and dual Euler-Savary formula","volume":"28","author":"Gungor","year":"2009","journal-title":"Eur. J. Mech.-A\/Solids"},{"key":"ref_17","first-page":"223","article-title":"On instantaneous invariants in dual Lorentzian space kinematics","volume":"62","author":"Ayyilidiz","year":"2010","journal-title":"Arch. Mech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1016\/j.ijmachtools.2015.12.003","article-title":"An accurate prediction method of cutting forces in 5-axis flank milling of sculptured surface","volume":"104","author":"Zhang","year":"2016","journal-title":"Int. J. Mach. Tools Manuf."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1605","DOI":"10.1007\/s00170-018-2033-7","article-title":"Highly accurate 5-axis flank CNC machining with conical tools","volume":"97","author":"Calleja","year":"2018","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_20","first-page":"091011-2","article-title":"Flank-Milling of Integral Blade Rotors Made in Ti6Al4V Using Cryo CO2 and Minimum Quantity Lubrication","volume":"143","author":"Pereira","year":"2021","journal-title":"J. Manuf. Sci. Eng."},{"key":"ref_21","unstructured":"Hamdi, M., Aifaoui, N., and Abdelmajid, B. (2007, January 28\u201331). Idealization of cadgeometry using design and analysis integration features models. Proceedings of the ICED 2007, the 16th International Conference on Engineering Design, Paris, France."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"905","DOI":"10.1016\/0010-4485(96)00018-8","article-title":"Midsurface abstraction from 3d solid models: General theory and applications","volume":"28","author":"Rezayat","year":"1996","journal-title":"Comput. Aided Des."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"135","DOI":"10.36890\/iejg.545870","article-title":"On a spacelike line congruence which has the parameter ruled surfaces as principal ruled surfaces","volume":"12","author":"Ferhat","year":"2019","journal-title":"Int. Electron. J. Geom."},{"key":"ref_24","first-page":"551","article-title":"A new construction of timelike ruled surfaces with constant Disteli-axis","volume":"42","author":"Unluturk","year":"2020","journal-title":"Honam Math. J."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1950197","DOI":"10.1142\/S0219887819501974","article-title":"On the one-parameter Lorentzian spatial motions","volume":"16","author":"Alluhaibi","year":"2019","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"7721","DOI":"10.1007\/s00170-022-09812-x","article-title":"On the kinematic-geometry of one-parameter Lorentzian spatial movement","volume":"121","author":"Alluhaibi","year":"2022","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_27","unstructured":"O\u2019Neil, B. (1983). Semi-Riemannian Geometry Geometry, with Applications to Relativity, Academic Press."},{"key":"ref_28","unstructured":"Walfare, J. (1995). Curves and Surfaces in Minkowski Space. [Ph.D. Thesis, K.U. Leuven, Faculty of Science]."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/10\/1816\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:57:02Z","timestamp":1760129822000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/10\/1816"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,24]]},"references-count":28,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["sym15101816"],"URL":"https:\/\/doi.org\/10.3390\/sym15101816","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,9,24]]}}}