{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:09:39Z","timestamp":1760148579541,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,14]],"date-time":"2023-05-14T00:00:00Z","timestamp":1684022400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University Researchers Supporting","award":["PNURSP2023R337"],"award-info":[{"award-number":["PNURSP2023R337"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper investigates the kinematic differential geometry of a line trajectory in spatial movement. Specifically, we provide a theoretical expression of inflection line congruence, which is the spatial equivalent of the inflection circle of planar kinematics. Additionally, we introduce new proofs for the Euler\u2013Savary and Disteli formulae and thoroughly analyze their spatial equivalence.<\/jats:p>","DOI":"10.3390\/axioms12050472","type":"journal-article","created":{"date-parts":[[2023,5,15]],"date-time":"2023-05-15T08:33:01Z","timestamp":1684139581000},"page":"472","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Kinematic Differential Geometry of a Line Trajectory in Spatial Movement"],"prefix":"10.3390","volume":"12","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":[[2023,5,14]]},"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. 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