Levent Kula

In this paper, we investigate three classes of f-weighted direction curves, namely tangent, principal, and binormal direction curves, obtained by integrating positive smooth multiples of the corresponding Frenet vector fields of a generating Frenet curve in Euclidean three-space. We derive curvature and torsion relations for these associated curves and investigate their connections with general helices, slant helices, clad helices, and g-clad helices. In particular, the weighted tangent direction construction preserves certain helix invariants of the generating curve, while suitable choices of the weight function modify its curvature and torsion. We establish conditions on the weight functions under which the resulting curves satisfy the characterizations of rectifying, Mannheim, and Bertrand curves. We also investigate conditions for constructing spherical weighted direction curves and introduce g-weighted normal, rectifying, and osculating direction curves associated with them. Explicit examples illustrate successive weighted constructions, including the generation of a rectifying clad helix and a Salkowski curve. The main contribution is a common construction framework in which the choice of a Frenet direction field and a positive weight function allows several classes of special space curves to be investigated through their geometric characterizations, subject to the required regularity and nonvanishing conditions.

In this study, we introduce a new type of space curve which is called ac-slant curve whose inner product of its acceleration vector and a fixed direction is a constant. Then, we give a characterization according to the curvatures of ac-slant curve. Later on, we obtain relationship between an ac-slant curve and some special curves (spherical curve, helix, slant helix, Salkowski curve, rectifying curve). Similarly to concept of ac-slant curves, we consider normalized ac-slant curve which has a unit acceleration vector and give a characterization with respect to its curvatures. Especially, we give some results for ac-slant or normalized ac-slant curves with respect to the magnitude of its velocity and acceleration vector.

In this study, we define two types of mappings that preserve the constant angle between the tangent vector field and the axis of a given helix in Euclidean spaces. The first type generates helices in the n‐dimensional Euclidean space from helices in the same space. The second type generates helices in the (n+1)‐dimensional Euclidean space from helices in the n‐dimensional Euclidean space. In addition, we give invariants of these mappings and study polynomial, rational, conical, ellipsoidal, and hyperboloidal helices supported by examples.

In this article, we investigate polynomial helices in the n-dimensional semi-Euclidean space with index two for n ≥ 4. We obtain some families of spacelike and timelike polynomial helices. These helices have spacelike or timelike or null axes. After that, we give some examples of the spacelike and the timelike polynomial helices in the n-dimensional semi-Euclidean space with index two for n = 4,5 and 6.

In this work, we study timelike rectifying slant helices in E 3 1. First, we find general equations of the curvature and the torsion of timelike rectifying slant helices. After that, by solving second order linear differential equations, we obtain families of timelike rectifying slant helices that lie on cones.

In this paper, we are investigating that under which conditions of the geodesic curvature of unit speed curve γ that lies on the unit sphere, the curve c which is obtained by using γ, is a spherical helix or slant helix.

In this paper, we investigate tangent indicatrix, principal normal indicatrix and binormal indicatrix of a spacelike curve with spacelike, timelike and null principal normal vector in Minkowski 3-space E³ ₁ and we construct their Frenet equations and curvature functions. Moreover, we obtain some differential equations which characterize for a spacelike curve to be a slant helix by using the Frenet apparatus of spherical indicatrix of the curve. Also related examples and their illustrations are given. Mathematics Subject Classification 2010: 53A04, 53C50.

In this paper, we are investigating that under which conditions of the geodesic curvature of unit speed curve gamma that lies on the unit sphere, the curve c which is obtained by using gamma, is a spherical helix or slant helix.