On the Smarandache Curves of Spatial Quaternionic Involute Curve

A primer on the differential geometry of quaternionic curves

“…There are several former studies on this theme, [16][17][18][19][20] and their development is much more complicated than ours. In this section, we provide simple solutions for these solutions using a generalization to quaternionic curves that are analogous to the complex formulation for plane real curves.…”

“…To the best of our knowledge, this approach has never been reported, although two-dimensional curves within a real space have a simple parametrization with complex numbers, 15 and here we closely follow this approach. The example of evolutes and evolvents presented here shows that our proposal is much simpler than the current approach, [16][17][18][19][20] based in Bharathi and Nagaraj. 1 Before developing the differential geometric approach, in the next section, we present the most important features of quaternions that will be used in this article.…”

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 of Involute-Evolute Curve According to Frenet Frame

“…By using the Darboux frame, Bektaş and Yüce obtained the results about Smarandache curves [4]. In another study, they studied the spatial quaternionic curve and the relationship between Frenet frames of the involute curve of the spatial quaternionic curve which are expressed by using the angle between the Darboux vector and binormal vector [15]. S ¸enyurt et al used special curves as a base to create Smarandache curves, and then studied their geometric properties [12][13][14].…”