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In this article, spatial quaternionic Bertrand curve pairs in the 3-dimensional Euclidean space are examined. Algebraic properties of quaternions, basic definitions and theorems are given. Later, some characterizations of spatial quaternionic Bertrand curve pairs are obtained in the 3-dimensional Euclidean space.

In this study, the effect of fractional derivatives, whose application area is increasing day by day, on curve pairs is investigated. As it is known, there are not many studies on a geometric interpretation of fractional calculus. When examining the effect of fractional analysis on a curve, the Conformable fractional derivative that fits the algebraic structure of differential geometry derivative is used. This effect is examined with the help of examples consistent with the theory and visualized for different values of the Conformable fractional derivative. The difference of this study from others is the use of Conformable fractional derivatives and integrals in calculations. Fractional calculus has applications in many fields such as physics, engineering, mathematical biology, fluid mechanics,signal processing, etc. Fractional derivatives and integrals have become an extremely important and new mathematical method in solving various problems in many sciences.

In this present paper, rectifying curves are re-characterized in a shorter and simpler way using harmonic curvatures and some relations between rectifying curves and focal curves are found in terms of their harmonic curvature functions in $n-$dimensional Euclidean space. Then, a rectifying Salkowski curve, which is the focal curve of a given space curve is investigated. Finally, some figures related to the theory are given in the case $n=3$.

The aim of the paper is to find polynomial parametric equations of rectifying Salkowski curves in Minkowski 3-space, via a serial approach. These curves are characterized by according to their curvature; in particular those curves with constant curvature functions and linear harmonic curvature functions are fully characterized. Then, the equations of the rectifying Salkowski curves are obtained as serial solutions of differential equations with third-order polynomial coefficients.

In the present paper, a new type of special curve couple which are called WC^{∗}-partner curves are introduced according to alternative moving frame {N,C,W}. The distance function between the corresponding points of reference curve and its partner curve is obtained. Besides, the angle function between the vector fields of alternative frame of the curves is expressed by means of alternative curvatures f and g. In addition to these, various characterizations are obtained related to these curves.

In this study, we focus on Smarandache curves which is 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 elements, curvature and torsion of these curves.

In this article, quaternionic curves in 3-dimensional Euclidean space have examined. Firstly, algebraic properties of quaternions and their basic definitions and theorems are given. Later, some characterizations of the quaternionic Mannheim curves in the 3-dimensional Euclidean space have obtained.

In the present paper, we investigate the determinative conditions for the isothermic Hopf cylinders and Hopf tori of spherical helices which are the tangent indicatrices of slant helices. Based on the relationship between geodesic curvature and principal curvatures, the characterizations are obtained by blending the results of inspired references.

Rectifying, normal and osculating curves have been worked on many times in dierent spaces. The aim of this paper is to characterize these curves from another point of view in three dimensional Dual space.

In this paper, we consider nonhelical rectifying curves using the any orthonormal moving frame in Minkowski 3−space. Then, we give some relations between nonhelical rectifying curves and their Darboux vectors. Also, we demonstrate that the modified Darboux vectors of the curves are rectifying curves. So, this study is a general expression for the known rectifying curves. In the last section, we give some examples and draw their pictures.