Cumali Ekici

In this work, using q-frame Tzitzeica (T-curve), spherical (s-curve), and spherical Tzitzeica (sT-curve) curves are studied in three-dimensional Euclidean space. There is a T-curve when its torsion has a non-zero constant ratio to the square of the distance from the origin to the plane generated by tangent and normal vectors at any point on the curve. Using q-frame in 3-dimensional Euclidean space, after defining T-curve and s-curve and investigating related theorems, the theorems related to sT-curves are proved.

This paper examines a tubular surface, a specific example of a canal surface, in 4-dimensional Euclidean space. In the plane stretched by the quasi-frame vectors B_q and C_q, this surface is established by the motion of a circle with a constant radius that uses each point on the curve a(t) as its center. Using the general equation provided in Euclidean 4-space, the first and second partial derivatives are determined. The Gram-Schmidt technique was used to derive the surface's first unit normal vector field U_1, and second unit normal vector field U_2, using the acquired partial derivatives. Using quasi-vectors, the tubular surface's first and second fundamental form coefficients were found. Furthermore, the shape operator matrices for the tubular surface's the unit normal vector fields U_1 and U_2 were acquired. We have found algebraic invariants of the shape operator, Gaussian curvature, and mean curvature. For a thorough understanding of the obtained theoretical calculations, an example of a directional tubular surface, the equation of the tubular surface has been parametrized using quasi-frame vectors and quasi-frame curvatures for a given space curve in 4-dimensional Euclidean space.

In this paper, we first provide a summary of certain results related to the differential geometry of ruled surfaces. Next, we introduce the signature curve for ruled surfaces in Euclidean three-space. Additionally, we present a straightforward algorithm for the reconstruction of a ruled surface, which is both efficient and entirely local, requiring only the initial motion direction and starting point. Finally, we illustrate the efficiency and accuracy of the algorithm through several examples. Bu makalede, öncelikle regle yüzeylerin diferesiyel geometrisi ile ilgili bazı sonuçların bir özetini sunuyoruz. Daha sonra, Öklid üç uzayında regle yüzeyler için işaret eğrisini tanıtıyoruz. Ek olarak, bir regle yüzeyin yeniden yapılandırılması için hem verimli hem de tamamen yerel olan, yalnızca ilk hareket yönü ve başlangıç noktası gerektiren basit bir algoritma sunuyoruz. Son olarak, algoritmanın verimliliğini ve doğruluğunu birkaç örnekle gösteriyoruz.

In this examination, we take q-helices into consideration. By q-helices, we mean curves due to the quasi-frame (abbv. q-frame) whose vector fields make constant angles with a non-zero fixed axis. One by one, all types of these q-helices we study in the work are therefore classified in three dimensional Euclidean space. Additionally, we study Darboux q-helices by using Darboux vector obtained with respect to q-frames fields of a curve. For a curve enclosed with q-frame as a general case, we reach some results for Darboux q-helices.

In this work, both timelike and spacelike Tzitzeica, spherical, and spherical Tzitzeica curves are analyzed in 3-dimensional Minkowski space by using q-frame. Tzitzeica and spherical curves are characterized using spacelike and timelike q-frames within the context of Minkowski three-space, and the theorems concerning spherical Tzitzeica curves are established.