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Journals / International Electronic Journal of Geometry / 2026 / Volume: 19 Issue: 1

Characterizations of $f$-Osculating Curves in $3$ and $4$ Dimensional Euclidean Spaces

Absos Ali Shaikh(The University of Burdwan)0000-0001-6312-2564
Pushpinder Badyal(Shri Mata Vaishno Devi University)0009-0008-0899-8457
Sandeep Sharma(Shri Mata Vaishno Devi University)0000-0003-0101-5610
Pages
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Publication Date
2026-04-22
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

We also examine special cases in which one or more curvature functions are constant, analyzing the behaviour of the remaining curvature functions.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, we introduce and study the concept of $f$-osculating curves in both three and four dimensional Euclidean spaces ($\mathbb{E}^{3}$ and $\mathbb{E}^{4}$).
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Key Findings & Data:We establish necessary and sufficient conditions for a unit-speed curve in $\mathbb{E}^3$ and $\mathbb{E}^4$ to be an $f$-osculating curve and derive relations among their curvature functions.
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Conclusion & Impact:We also examine special cases in which one or more curvature functions are constant, analyzing the behaviour of the remaining curvature functions.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Serret-Frenet frame · $f$-osculating curve · curvature

Abstract

In this paper, we introduce and study the concept of $f$-osculating curves in both three and four dimensional Euclidean spaces ($\mathbb{E}^{3}$ and $\mathbb{E}^{4}$). These curves are characterized by the condition that their $f$-position vectors lie in the osculating plane. We establish necessary and sufficient conditions for a unit-speed curve in $\mathbb{E}^3$ and $\mathbb{E}^4$ to be an $f$-osculating curve and derive relations among their curvature functions. We also examine special cases in which one or more curvature functions are constant, analyzing the behaviour of the remaining curvature functions.

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Cite This Article

Shaikh, A. A., Badyal, P., Sharma, S. (2026). Characterizations of $f$-Osculating Curves in $3$ and $4$ Dimensional Euclidean Spaces. International Electronic Journal of Geometry, Volume: 19 Issue: 1. https://doi.org/10.36890/iejg.1762742

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