Smarandache Curves According to Alternative Frame in E^3

Abstract

In this study, we focus on Smarandache curves which are a special class of curves. These curves have previously been studied by many authors in different spaces. We will re-characterize these curves with the help of an alternative frame different from Frenet frame. Also, we will obtain frame vectors curvature and torsion of these curves.

On Bishop Frames of Any Regular Curve in Euclidean 3-Space

“…As can be understood from their definitions, tangential vector fields of type-1 Bishop and Frenet frames, binormal vector fields of type-2 Bishop and Frenet frames, and principal normal vector fields of N-Bishop and Frenet frames are common. There are many studies on this new types of Bishop and alternative frame (Alıç and Yılmaz 2021, Çakmak and Şahin 2022, Damar et al 2017, Kızıltuğ et al 2013, Masal and Azak 2015, Ourab et al 2018, Samancı and Sevinç 2022, Şenyurt 2018, Şenyurt et al 2023, Yılmaz and Has 2022, Şenyurt and Kaya 2018. In these studies, the relationships between Frenet and various Bishop frames of a curve are given.…”

Some Applications on Spherical Indicatrices of the Helix Curve

“…When the Frenet vectors of a differentiable curve are taken as position vectors, the regular curves that are drawn by these vectors are called Smarandache curves (Taşköprü & Tosun, 2014). Some properties of Smarandache curves obtained by using different frames and different curves were examined (Alıç & Yılmaz, 2021;Ali, 2010;Bektaş &Yüce, 2013;Çetin et al, 2014;Çetin &Kocayiğit, 2013;Şenyurt, 2018;Şenyurt & Canlı, 2023;Şenyurt & Çalışkan, 2015;Şenyurt & Öztürk, 2018;Şenyurt & Sivas, 2013;Şenyurt et al, 2019;Şenyurt et. al, 2020;Şenyurt et.…”

Smarandache Curves According to Flc-frame in Euclidean 3-space

“…Turgut and Yılmaz, described the Smarandache curves in Minkowski space [15]. Later, at either Euclidean Süleyman S ¸enyurt, Kebire Hilal Ayvacı and Davut Canlı / FCMS or Minkowski space, some features of the Smarandache curves are investigated according to the Darboux frame, Bishop frame, alternative frame, q frame and Sabban frame, [1,3,4,6,7,9,[11][12][13][14]. In this study, we introduce special Smarandache curves according to the new Flc frame in Euclidean 3-space.…”