Trajectories Generated by Special Smarandache Curves According to Positional Adapted Frame
Diferansiyel geometride eğriler teorisi önemli bir yere sahiptir. Eğriler üzerinde tanımlanan hareketli çatı kavramı bu teorinin önemli bir parçasıdır. Yakın geçmişte, Özen ve Tosun, 3 boyutlu Öklid uzayında sıfırlanmayan açısal momentuma sahip yörüngeler için yeni bir hareketli çatı tanıttı (J. Math. Sci. Model. 4(1), 2021). Bu çatı $\left\{\mathbf{T}, \mathbf{M}, \mathbf{Y} \right\}$ile gösterilir ve konumsal uyarlanmış çatı olarak adlandırılır. Bu çalışmada konumsal uyarlanmış çatıya göre $\mathbf{TM}$, $\mathbf{TY}$ ve $\mathbf{MY}-$Smarandache eğrilerinin ürettiği özel yörüngeleri ${{E}^{3}}$ de araştırdık ve bu yörüngelerin Serret-Frenet elemanlarını hesapladık. Daha sonra, spesifik bir eğriyi ele aldık ve bu eğri için, daha önce belirtilen özel yörüngelerin parametrik denklemlerini elde ettik. Son olarak elde edilen bu özel yörüngelerin mathematica programıyla çizilmiş grafiklerini verdik. Burada elde edilen sonuçlar alana yeni birer katkıdır. Bu sonuçların gelecekte diferansiyel geometri ve parçacık kinematiğinin bazı özel uygulamalarında faydalı olacağını umuyoruz.
Anahtar Kelimeler:
Açısal momentum, parçacık kinematiği, hareketli çatı, Smarandache eğrileri
Trajectories Generated by Special Smarandache Curves According to Positional Adapted Frame
In differential geometry, the theory of curves has an important place. The concept of moving frame defined on curves is an important part of this theory. Recently, \"Ozen and Tosun have introduced a new moving frame for the trajectories with non-vanishing angular momentum in 3-dimensional Euclidean space (J. Math. Sci. Model. 4(1), 2021). This frame is denoted by $\left\{\mathbf{T}, \mathbf{M}, \mathbf{Y} \right\}$ and called as positional adapted frame. In the present study, we investigate the special trajectories generated by $\mathbf{TM}$, $\mathbf{TY}$ and $\mathbf{MY}-$Smarandache curves according to positional adapted frame in ${{E}^{3}}$ and we calculate the Serret-Frenet apparatus of these trajectories. Later, we consider a specific curve and obtain the parametric equations of the aforesaid special trajectories for this curve. Finally, we give the graphics of these obtained special trajectories which were drawn with the mathematica program. The results obtained here are new contributions to the field. We expect that these results will be useful in some specific applications of differential geometry and particle kinematics in the future.
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- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2019
- Yayıncı: Karamanoğlu Mehmetbey Üniversitesi
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Trajectories Generated by Special Smarandache Curves According to Positional Adapted Frame