Characterizations of some special curves according to the modified adapted orthogonal frame

In this thesis, the modified adapted orthogonal frame is defined. Some special curves and ruled surfaces are examined according to this frame. There are five main sections in the thesis. The first section discussed significance and use of the curve and surface theory in differential geometry. The literature review is given in the second section. Some basic definitions and theorems that are used in this thesis are given in the third section. The modified orthogonal frame is initially analyzed in the fourth section. Additionally, alternative moving frame and their theorems are addressed. Thirdly, C-partner curves and their relationships with some special curves are introduced. The fifth section introduced a newly constructed modified alternative orthogonal frame {N*, C*, W*} that characterizes curves based on where their alternative curvature points are zero. N*-slant helix and C*-slant helix definitions are provided with respect to this frame. C*-partner curves are defined, and then their relationships with some special curves are investigated based on the modified alternative orthogonal frame. In the last part of the fifth section, the ruled surface has been examined according to the modified adapted orthogonal frame, and the conditions of the developable ruled surface are discussed.