In this paper, we investigate Smarandache curves according to type-2 Bishop frame in Euclidean 3- space and we give some differential geometric properties of Smarandache curves. Also, some characterizations of Smarandache breadth curves in Euclidean 3space are presented. Besides, we illustrate examples of our results.

A third order vectorial differential equation of position vector of Smarandache breadth curves has been obtained in Minkowski 3-space.

In this paper, we introduce some special Smarandache curves according to Bishop frame in Euclidean 3-space E3.

In this paper, we investigate special Smarandache curves according to Bishop frame in Euclidean 3-space and we give some differential geometric properties of Smarandache curves. Also we find the centers of the osculating spheres and curvature spheres of Smarandache curves.

In differential geometry, the theory of curves has an important place. The concept of moving frames 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 called positional adapted frame. In the present study, we investigate the special trajectories generated by Smarandache curves according to positional adapted frame in E3 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.

In the light of great importance of curves and their frames in many different branches of science, especially differential geometry as well as geometric properties and the uses in various fields, we are interested here to study a special kind of curves called Smarandache curves in Lorentz 3-space.

The mathematical combinatorics is a subject that applying combinatorial notion to all mathematics and all sciences for understanding the reality of things in the universe. The International J. Mathematical Combinatorics is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly, which publishes original research papers and survey articles in all aspects of mathematical combinatorics, Smarandache multi-spaces, Smarandache geometries, non-Euclidean geometry, topology and their applications to other sciences.