{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:30:54Z","timestamp":1760149854046,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,9,26]],"date-time":"2023-09-26T00:00:00Z","timestamp":1695686400000},"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":["Axioms"],"abstract":"<jats:p>This study utilizes the axodes invariants to derive novel hyperbolic proofs of the Euler\u2013Savary and Disteli formulae. The inflection circle, which is widely recognized, is situated on the hyperbolic dual unit sphere, in accordance with the principles of the kinematic theory of spherical locomotions. Subsequently, a timelike line congruence is defined and its spatial equivalence is thoroughly studied. The formulated assertions degenerate into a quadratic form, which facilitates a comprehensive understanding of the geometric features of the inflection line congruence.<\/jats:p>","DOI":"10.3390\/axioms12100915","type":"journal-article","created":{"date-parts":[[2023,9,26]],"date-time":"2023-09-26T08:58:17Z","timestamp":1695718697000},"page":"915","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Kinematic Geometry of a Timelike Line Trajectory in Hyperbolic Locomotions"],"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":[[2023,9,26]]},"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","unstructured":"Pottman, H., and Wallner, J. (2001). Computational Line Geometry, Springer.","DOI":"10.1007\/978-3-642-04018-4"},{"key":"ref_4","unstructured":"Stachel, H. (2004, January 27\u201331). On spatial involute gearing, TU Wien, Geometry Preprint No 119. Proceedings of the 6th International Conference on Applied Informatics, Eger, Hungary."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"105427","DOI":"10.1016\/j.mechmachtheory.2023.105427","article-title":"On spatial relations to the Euler\u2013Savary formula","volume":"189","author":"Dooner","year":"2023","journal-title":"Mech. Mach. Theory"},{"key":"ref_6","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_7","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_8","doi-asserted-by":"crossref","unstructured":"Kecskem\u00e9thy, A., and M\u00fcller, A. (2009). Computational Kinematics: Proceedings of the 5th International Workshop on Computational Kinematics, Duisburg, Germany, 6\u20138 May 2009, Springer.","DOI":"10.1007\/978-3-642-01947-0"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"808","DOI":"10.19113\/sdufbed.76191","article-title":"A study on geometry of spatial kinematics in Lorentzian space","volume":"21","author":"Turhan","year":"2017","journal-title":"S\u00fcleyman Demirel \u00dcniversitesi Fen Bilimleri Enstit\u00fcs\u00fc Dergisi"},{"key":"ref_10","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 \u00e1ank milling of sculptured surface","volume":"104","author":"Zhang","year":"2016","journal-title":"Int. J. Mach. Tools Manuf."},{"key":"ref_11","first-page":"95","article-title":"Blaschke approach to Euller-Savary formulae","volume":"4","author":"Ekinci","year":"2016","journal-title":"Konuralp J. Math."},{"key":"ref_12","first-page":"333","article-title":"Some results on space-like line congruences and their space-like parameter ruled surface","volume":"23","year":"1999","journal-title":"Turk. J. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"972","DOI":"10.1115\/1.2722775","article-title":"On the One-Parameter Lorentzian Spherical Motions and Euler-Savary Formula","volume":"74","author":"Tosun","year":"2007","journal-title":"ASME J. Appl. Mech."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"820","DOI":"10.1016\/j.euromechsol.2009.03.007","article-title":"Dual Lorentzian spherical motions and dual Euler\u2013Savary formula","volume":"28","author":"Gungor","year":"2009","journal-title":"Eur. J. Mech. A\/Solids"},{"key":"ref_15","first-page":"39","article-title":"Dual ruled surface constructed by the pole curve of the involute curve","volume":"15","author":"Palavar","year":"2022","journal-title":"Int. J. Open Probl. Compt. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"8542","DOI":"10.3934\/math.2022476","article-title":"Geometry of the line space associated to a given dual ruled surface","volume":"7","author":"Rawya","year":"2022","journal-title":"AIMS Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"207","DOI":"10.37394\/23206.2021.20.22","article-title":"Spacelike surfaces with a common line of curvature in Lorentz-Minkowski 3-space","volume":"20","author":"Saad","year":"2021","journal-title":"Wseas Trans. Math."},{"key":"ref_18","first-page":"50","article-title":"Ball and Burmester points in Lorentzian sphere kinematics","volume":"42","author":"Inalcik","year":"2015","journal-title":"Kuwait J. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Alluhaibi, N.S., Abdel-Baky, R.A., and Naghi, M.F. (2022). On the Bertrand offsets of timelike ruled surfaces in Minkowski 3-space. Symmetry, 14.","DOI":"10.3390\/sym14040673"},{"key":"ref_20","first-page":"261","article-title":"Uber des Analogon der Savaryschen Formel und Konstruktion in der kinematischen Geometrie des Raumes","volume":"62","author":"Disteli","year":"1914","journal-title":"Z. Math. Phys."},{"key":"ref_21","first-page":"163","article-title":"The Euler\u2013Savary analogue equations of a point trajectory in Lorentzian spatial motion","volume":"83","author":"Caliskan","year":"2013","journal-title":"Proc. Natl. Acad. Sci. India Sect. Phys. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/s00009-022-02252-7","article-title":"Singularities of non-lightlike developable surfaces in Minkowski 3-space","volume":"20","author":"Nazra","year":"2023","journal-title":"Mediterr. J. Math."},{"key":"ref_23","first-page":"1","article-title":"On (contra) pedals and (anti)orthotomics of frontals in de Sitter 2-space","volume":"1","author":"Li","year":"2023","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Li, Y., Aldossary, M.T., and Abdel-Baky, R.A. (2023). Spacelike circular surfaces in Minkowski 3-Space. Symmetry, 15.","DOI":"10.3390\/sym15010173"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Li, Y., Chen, Z., Nazra, S.H., and Abdel-Baky, R.A. (2023). Singularities for timelike developable surfaces in Minkowski 3- Space. Symmetry, 15.","DOI":"10.3390\/sym15020277"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/10\/915\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:58:37Z","timestamp":1760129917000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/10\/915"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,26]]},"references-count":25,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["axioms12100915"],"URL":"https:\/\/doi.org\/10.3390\/axioms12100915","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,9,26]]}}}