Alternative approaches and applications of direction curves

This thesis consists of five chapters. The first chapter includes an introduction that includes literature reviews, the thesis's purpose, and its motivation. The second chapter covers the fundamental definitions and concepts of curve theory in three-dimensional Euclidean space. In curve theory, the Frenet and Alternating Frame Formulas defined along a curve, helix types such as helices, slant helices, and C-slant helices, as well as special curve pairs such as Bertrand curves, Mannheim curves, and C-Partner curves, and their characterizations are discussed. The third chapter reviews studies on the Normal, Rectifying, and Osculator directions and donor curves defined by the Frenet planes, which form the motivation behind the thesis's research topic, and adds new results that advance theories. The fourth chapter defines the CW, NW, and NC directions and donor curves, respectively, using planes defined by Alternating Frame Vectors, and examines their geometric properties. The relationships between direction-donor curve pairs are obtained using various space curves. The relationship between the defined curves and the rectifying curve is investigated within subsections. Finally, in the fifth chapter, conclusions regarding the thesis and suggestions for future studies are included.