Alternative moving frame approach to the problem of determination of position vector of a curve in 3-dimensional Euclidean space

This thesis consists of five chapters. The problem of determining the position vector of a curve in 3-dimensional Euclidean space is introduced, the importance of the problem is given and the existing studies in the literature related to this problem are summarized in the first chapter. In addition general information about the concept of thesis is given. In chapter two, the basic definitions and theorems about curves in 3 - dimensional Euclidean space are given. The Frenet frame and alternative moving frame of a curve are introduced and the relations between these frames are obtained. Geometric interpretations of the curvatures are given by using relations between curvatures of the Frenet frame and alternative moving frame. In chapter three, the problem of finding the position vector of a curve with determined curvature and torsion functions is introduced. For the solution of the problem, a method based on the Frenet frame, which is known in the literature, is summarized. In chapter four, which is the original part of the thesis, firstly by using alternative moving frame derivative formulae, a differential equation with vectorial form according to principal normal vector of curve is constructed. Since the equation has coefficients as variables, it has been seen that its solution is difficult for an arbitrary curve. Then, this vector differential equation for a general helix curve is solved. Thus, the position vector of a general helix for given curvature and torsion functions, is obtained in parametric form. In addition, the position vector of the circular helix by this method based on the use of alternative moving frame. Using the above method, general helix examples for some of the given curvature and torsion functions are obtained. In chapter five, which is another original part of the thesis, by using alternative moving frame derivative formulae, a vector differential equation according to the unit principal normal vector is constructed for a slant helix curve. By using this equations, the position vector of a slant helix curve, given the curvature and torsion functions is determined. Slant helix examples is obtained for some curvature and torsion functions. In chapter six, the results obtained in the thesis and the comparison between these results and some existing studies in the literature are given.