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Journals / Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics / 2026 / Volume: 71 Issue: 1

Associated curves of a Frenet curve in the dual Lorentzian space

Bahar Abalı(SÜLEYMAN DEMİREL ÜNİVERSİTESİ, FEN BİLİMLERİ ENSTİTÜSÜ)0000-0001-5900-6595
Ahmet Yücesan(Süleyman Demirel Üniversitesi)0000-0002-5419-925X
Pages
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Publication Date
2022-03-30
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AI Summary & Key Takeaways

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⚡ TL;DR(Too Long; Didn't Read)

Finally, according to causal character of the principal donor curve of principal directed rectifying curve in $\mathbb{D}_{1}^{3}$, we show this curve to correspond to any timelike or spacelike ruled surface in Minkowski 3−space R 3 1 R13 .

📌 3 Key Takeaways

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Objective & Hypothesis:In this work, we firstly introduce notions of principal directed curves and principal donor curves which are associated curves of a Frenet curve in the dual Lorentzian space D 3 1 D13 .
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Key Findings & Data:Then we show that the principal donor curve of a circle in $\mathbb{D}^{2}$ or a hyperbola in $\mathbb{D}_{1}^{2}$ and the principal directed curve of a slant helix in $\mathbb{D}_{1}^{3}$ are a helix and general helix, respectively.
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Conclusion & Impact:Finally, according to causal character of the principal donor curve of principal directed rectifying curve in $\mathbb{D}_{1}^{3}$, we show this curve to correspond to any timelike or spacelike ruled surface in Minkowski 3−space R 3 1 R13 .
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Dual Lorentzian space · associated curves · dual general helix

Abstract

In this work, we firstly introduce notions of principal directed curves and principal donor curves which are associated curves of a Frenet curve in the dual Lorentzian space D 3 1 D13 . We give some relations between the curvature and the torsion of a dual principal directed curve and the curvature and the torsion of a dual principal donor curve. We show that the dual principal directed curve of a dual general helix is a plane curve and obtain the equation of dual general helix by using position vector of plane curve. Then we show that the principal donor curve of a circle in $\mathbb{D}^{2}$ or a hyperbola in $\mathbb{D}_{1}^{2}$ and the principal directed curve of a slant helix in $\mathbb{D}_{1}^{3}$ are a helix and general helix, respectively. We explain with an example for the second case. Finally, according to causal character of the principal donor curve of principal directed rectifying curve in $\mathbb{D}_{1}^{3}$, we show this curve to correspond to any timelike or spacelike ruled surface in Minkowski 3−space R 3 1 R13 .

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Cite This Article

Abalı, B., Yücesan, A. (2022). Associated curves of a Frenet curve in the dual Lorentzian space. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Volume: 71 Issue: 1. https://doi.org/10.31801/cfsuasmas.877170

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