Translation hypersurfaces according to quasi frame in 4-dimensional space

Differential geometry, particularly the study of curves and surfaces, holds significance in defining and representing special surfaces such as translation surfaces. Given their widespread application in computer graphics and design, understanding and representing these surfaces has become an intriguing area of research. A translation surface in 3-dimensional space is a rational surface formed by parallel displacement of one regular space curve along another, thereby generating a curve where each point undergoes a translation along the second curve. Noteworthy directional systems for curves on translation surfaces, such as the Frenet frame and Bishop frame, have been explored. Additionally, a more general system termed "quasi-frame" has been introduced in 3-dimensional space. The quasi-frame is constructed using a unit quasi-normal vector perpendicular to the tangent and projection vectors along the space curve. This study focuses on 4-dimensional space and investigates translational hypersurfaces using quasi-frame. The construction of a quasi-frame along a space curve involves forming a quasi-normal vector as a unit vector perpendicular to the tangent vector and curvature projection vector. Subsequently, unit tangent, unit quasi-normal, and unit quasi-binormal vectors collectively constitute the quasi-frame. In addition, the fundamental form coefficient matrices were calculated by using the first and second partial derivatives of the translation hypersurfaces for the cases where the unit normal vector field is located in three different planes. The shape operator matrices, Gaussian curvatures and mean curvatures of the translational hypersurfaces are obtained from the fundamental form coefficients matrices for each case. In addition, the asymptotic lines of the minimal and generating curves of the quasi-cratched hypersurface are analysed. Examples are given in which quasi-vectors and quasi-curvatures are computed for a suitable space curve in 4-dimensional Euclidean space and the equations of the quasi-frame translational hypersurface using them are given. In addition, the curvatures of this hypersurface are obtained by finding the basic form coefficients and shape operator matrices and the shapes of the hypersurface in projection spaces are drawn. Key Words: Quasi-frame, translation hypersurfaces, shape operator, curvatures