Generalized osculator and rectifying curve pairs

The aim of this thesis is to give a new pair of curves definition that generalizes some famous curve pairs such as Bertrand, Mannheim and Backlund, parallel curve defined in the Euclidean space. For this, a new definition is given with the help of the intersection of the osculating and rectifying planes at the reciprocal points of the curves. In the first chapter, studies in Euclidean space related to each of the parallel, involute, evolute, pedal, Bertrand, Mannheim and Backlund curve pairs are investigated and information about these curves is given, and the definitions and theorems to be used in the following sections of the thesis are given. In the second chapter, osculating curve pair is defined in the Euclidean space and Frenet vector fields, curvature and torsion of this curve pair are given. It has been expressed which well-known curve pair corresponds to the curve resulting from the specific values of the Ɣ and Ɵ, where Ɣ is the angle between the tangents of the curves and the intersection line of the osculator planes and, Ɵ is the angle between corresponding points of the binormals. In the third chapter, the rectifying curve pair is defined and the cases that are reduced to well-known curve pairs are discussed by making similar examinations.