İlkay Arslan Güven

This work examined the direction curves of principal normal indicatrix curves regarding a regular curve in Euclidean space of three dimension. The {N,C,W} frame was assigned as a new alternative frame for the mentioned curve. Investigating relationships among the main curve and direction curves using such novel alternative frame is an innovative approach. The Frenet members, curvature, torsion, harmonic curvature and geodesic curvature of direction curves were determined. Direction curve characterizations in the forms of the C-slant helix , slant helix and general helix were provided. In the end, the relevant figures for these curves were displayed.

In this paper, we introduce a new quadruple number sequence by means of Leonardo numbers, which we call ordered Leonardo quadruple numbers. We determine the properties of ordered Leonardo quadruple numbers including relations with Leonardo, Fibonacci, and Lucas numbers. Symmetric and antisymmetric properties of Fibonacci numbers are used in the proofs. We attain some well-known identities, the Binet formula, and a generating function for these numbers. Finally, we provide illustrations of the identities.

This paper investigates curves of stationary acceleration by using alternative frame which includes the principal normal vector, the derivative of principal normal vector and the Darboux vector. Mentioned curves are studied by the way of rigid body motions, that is to say a point in the moving body follows the curve and the alternative frame in the moving body stays aligned with the members of frame. It is determined that in which condition this special motion becomes to stationary acceleration motion. The matrix representations of a constant vector related to velocity vector of the motion which is used to characterize stationary acceleration is obtained by means of alternative frame curvatures. Some examinations are developed with some solutions of differential equations. The main result is attained as: general helix curves with linear curvature and torsion functions are curves of stationary acceleration which are curves in the rigid body motions group SE(3) correlated with robotics. The paths designed as stationary acceleration curves can lead the way to control the end-effectors of robots. Finally, some explanatory examples are imputed.

In this paper, we introduce new adjoint curves which are associated curves in Euclidean space of three dimension. They are generated with the help of integral curves of special Smarandache curves. We attain some connections between Frenet apparatus of these new adjoint curves and main curve. We characterize these curves in which conditions they are general helix and slant helix. Finally, we exemplify them with figures.

In this study, direction curves of spherical indicatrix curves (primarily tangent indicatrix) for a curve in three dimensional Euclidean space were investigated. Mentioned curve was given with a new alternative frame which is {N,C,W} frame. It is a new aspect that researching relations between direction curves and the main curve via that new alternative frame. The harmonic curvatures and geodesic curvatures of spherical image of principal normal indicatrix of direction curves were obtained. The general helix, slant helix and C-slant helix characterizations of direction curves were given. Finally these curves with their figures were exemplified.

Bu makalede, bir düzlemsel eğriye hiperbolik yükseltme dönüşümü uygulayarak, regle zarf yüzeyi inşa ettik. Daha sonra bu yüzeyin üç boyutlu Öklid uzayında açılabilir yüzey olduğunu gösterdik ve bu yüzeyin minimal yüzey ve sabit ortalama eğrilikli yüzey olabilmesi için koşulları verdik. Son olarak, bazı örnekler verdik ve örneklerdeki eğrilerin ve yüzeylerin grafiklerini çizdik. Bu makalede, bir düzlemsel eğriye hiperbolik yükseltme dönüşümü uygulayarak, regle zarf yüzeyi inşa ettik. Daha sonra bu yüzeyin üç boyutlu Öklid uzayında açılabilir yüzey olduğunu gösterdik ve bu yüzeyin minimal yüzey ve sabit ortalama eğrilikli yüzey olabilmesi için koşulları verdik. Son olarak, bazı örnekler verdik ve örneklerdeki eğrilerin ve yüzeylerin grafiklerini çizdik.

In this study, some special ruled surfaces were determined which were constituted by using the base curve as the W-direction curves of a Frenet curve. We gave the characterizations of developability and minimality of these ruled surfaces. We investigated the relation between the primary curve and the base curve of ruled surfaces, we also inquired the occurence geodesic curve, asymptotic line and principal line of the W-direction curve on the ruled surface. Terminally, some examples were given.

In this paper, some new integral curves are defined in three-dimensional Euclidean space by using a new frame of a polynomial spatial curve. The Frenet vectors, curvature and torsion of these curves are obtained by means of new frame and curvatures. We give the characterizations and properties of these integral curves under which conditions they are general helix. Also, the relationships between these curves in terms of being some kinds of associated curves are introduced. Finally, an example is illustrated.

In this study, we define a new type of Fibonacci and Lucas num- bers which are called bicomplex Fibonacci and bicomplex Lucas numbers. We obtain the well-known properties e.g. Docagnes, Cassini, Catalan for these new types. We also give the identities of negabicomplex Fibonacci and negabi- complex Lucas numbers, Binet formulas and relations of them.