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Volume 14
Issue 8
10.3390/axioms14080612
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Parabolic and Linear Rotational Motions on Cones and Hyperboloids

by
Harun Barış Çolakoğlu
1,
Mehmet Duru
2,* and
Ayşe Yılmaz Ceylan
2
1
Department of Computer Technologies, Akdeniz University, 07070 Antalya, Türkiye
2
Department of Mathematics, Akdeniz University, 07070 Antalya, Türkiye
*
Author to whom correspondence should be addressed.
Axioms 2025, 14(8), 612; https://doi.org/10.3390/axioms14080612
Submission received: 8 July 2025 / Revised: 2 August 2025 / Accepted: 4 August 2025 / Published: 6 August 2025
(This article belongs to the Section Geometry and Topology)
Versions Notes

Abstract

In this study, we consider the Lorentzian rotation about a lightlike axis. First, we introduce a geometric characterization for the rotation angle between two vectors that can overlap each other under a Lorentzian rotation about a lightlike axis. Then, we give a definition for the angle measurement between two spacelike vectors whose vector product is lightlike. Later, we generalize the Lorentzian rotation about a lightlike axis, and determine matrices of these transformations using the Cartan frame and the well-known Rodrigues formula, then using the Cayley map, and finally using the generalized split quaternions. We see that such transformations give parabolic rotational motions on general cones or general hyperboloids of one or two sheets, while they also give linear rotational motions on general hyperboloids of one sheet.
Keywords: Lorentzian scalar product; generalized Lorentzian rotation matrix; Cartan’s frame; Rodrigues rotation formula; Cayley map; split quaternion Lorentzian scalar product; generalized Lorentzian rotation matrix; Cartan’s frame; Rodrigues rotation formula; Cayley map; split quaternion

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MDPI and ACS Style

Çolakoğlu, H.B.; Duru, M.; Yılmaz Ceylan, A. Parabolic and Linear Rotational Motions on Cones and Hyperboloids. Axioms 2025, 14, 612. https://doi.org/10.3390/axioms14080612

AMA Style

Çolakoğlu HB, Duru M, Yılmaz Ceylan A. Parabolic and Linear Rotational Motions on Cones and Hyperboloids. Axioms. 2025; 14(8):612. https://doi.org/10.3390/axioms14080612

Chicago/Turabian Style

Çolakoğlu, Harun Barış, Mehmet Duru, and Ayşe Yılmaz Ceylan. 2025. "Parabolic and Linear Rotational Motions on Cones and Hyperboloids" Axioms 14, no. 8: 612. https://doi.org/10.3390/axioms14080612

APA Style

Çolakoğlu, H. B., Duru, M., & Yılmaz Ceylan, A. (2025). Parabolic and Linear Rotational Motions on Cones and Hyperboloids. Axioms, 14(8), 612. https://doi.org/10.3390/axioms14080612

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