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 the present paper, a new type of special curve couple which are called W C partner curves are introduced according to alternative moving frame fN; C; W g. The distance function between the corresponding points of reference curve and its partner curve is obtained. Besides, the angle function between the vector …elds 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 present paper, the effect of fractional analysis on magnetic curves is researched. A magnetic field is defined by the property that its divergence is zero in a three dimensional Riemannian manifold. We investigate the trajectories of the magnetic fields called as t-magnetic, n-magnetic and b-magnetic curves according to fractional derivative and integral. 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 magnetic curve, the conformable fractional derivative that best 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 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 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 this study, for the first time, a method is given for a developable ruled surface to be a constant angle ruled surface. The general equations of constant angle surfaces have been shown in the studies done so far. In this study, a new method is given on how to obtain a constant angled surface when any constant direction is given in Minkowski $3-$space.

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