Ruled surfaces generated by q-frame vectors in 4-dimensional space

This thesis aims to investigate 2-dimensional regular surfaces generated by quasi-vectors in 4-dimensional Euclidean space using a quasi-frame which is an alternative to the Frenet frame. Our study consists of 4 parts. In the first part of this study, definitions and theorems related to the Frenet frame and quasi-frame for a space curve in 3-dimensional Euclidean space are presented. Then, the Frenet frame, Frenet vectors, Frenet derivative equations, and curvatures of a curve in 4-dimensional Euclidean space are given. Similarly, the quasi-frame and quasi-curvatures of a space curve in 4-dimensional Euclidean space, for example, for the projection vectors k_{x} and k_{y} in the xy-plane, are given by using the unit vectors T tangent, N_{q} quasi-normal, B_{q} first quasi-binormal and C_{q} second-quasi binormal. Some theories for surfaces and ruled surfaces in 4-dimensional Euclidean space are stated. The ruled surfaces generated by q-frame vectors in Euclidean 3-space are given. In the fourth section, parametric expressions of 2-dimensional ruled surfaces constructed with T tangent, N_{q} quasi-normal, B_{q} first quasi-binormal and C_{q} second quasi-binormal vectors, which are units instead of the directrix vector in 4-dimensional Euclidean space, are given. In addition, the partial derivatives up to the second order and the first fundamental form coefficients are obtained with the help of the parametric expressions of the surfaces. By using these fundamental form coefficients, the striction lines, Gaussian curvatures, and mean curvatures of 2-dimensional ruled surfaces are calculated. In addition, valid relations between the obtained results are also given. To make these calculations clearer and more understandable, an example is presented to show how quasi-vectors and quasi-curvatures are calculated for a suitable space curve in 4-dimensional Euclidean space and how parametric equations of 2-dimensional ruled surfaces are obtained using this information. In addition, the curvatures of these surfaces are calculated using the fundamental form coefficients, and the shapes of the ruled surface sample in projection spaces are plotted together with the striction lines.