Polynomial space curves

This thesis consists of four chapters. In the first chapter, brief information is given about the previous studies on Pythagoras hodograph curves, Frenet frame and Frenet-like frames. In the second chapter, the Pythagorean hodograph curves and the Frenet frame and basic concepts about Frenet vectors are given. In the third chapter, Pythagoras hodograph curves are studied. Some theorems that make it easier for us to find the degree of polynomial type curves formed in the cross product of these curves are mentioned. The relation of Pythagorean hodograph curves with helices has been studied. In the fourth chapter, we assume that the polynomial curve is given in Euclidean space. The basic idea behind the proposed frame is the initial version of the new binormal vector and its computation using the highest derivatives of the cross curve. We call this new frame the Frenet-like curved frame or Flc frame for short. This new roof reduces the number of single points that we cannot identify in the Frenet frame. This frame also reduces unwanted rotation around the tangent vector of the curve. Additionally we introduce three new curvatures of a polynomial space curve. Finally we give the geometric interpretation of the new curvatures.