Framed general and framed slant helices in the Euclidean −space

This thesis consists of 5 sections. In the first section, real-life applications of helix curves in our daily life, the relationship between curvature and torsion for a curve to be a helix, and a literature review about the concepts of the general helix, slant helix, and B_2-slant helix are given. The second section is divided into four subsections. In the first subsection, basic definitions and notions are given in Euclidean space. After that, a curve in this space, the curvatures and the Frenet frame of a curve are mentioned. In this subsection, the definitions related to higher-dimensional Euclidean space are given. However, in the other subsections, the necessary definitions and theorems are given for 4-dimensional space since the concepts used in this thesis depend on 4-dimensional space. In the second subsection, a framed curve (γ,η) in 4-dimensional Euclidean space is defined, and the generalized tangent v, generalized principal normal η_1, generalized binormal η_2 and generalized second binormal η_3 vectors at each point of this generalized frame are introduced. In the third subsection, the definition of general helices, which are widely used in 3-dimensional space, in 4-dimensional space, and the necessary and sufficient conditions for a curve to be a general helix are examined. In the last subsection of the second part, B_2-slant helix is defined as a curve formed by the relation that the second unit binormal vector B_2- of this curve makes a constant angle with a constant vector. Moreover, the relations between the curvatures are presented while the curve is a B_2-slant helix. In the third section, the notion of framed general helices in 4-dimensional Euclidean space are discussed. It is defined that the framed general helix in this space is the framed curve formed as a result of the generalized tangent vector v making a constant angle θ with a constant vector U. Then, the theorem expressing the necessary and sufficient condition for a framed curve to be a framed general helix is given as a constant value p^2/q^2 +(1/r (p/q)')^2 equation consisting of the generalized curvatures p,q,r of the framed curve is provided. The necessary and sufficient conditions are given for this framed curve to be a framed general helix as the equation (p/q) (p/q)'+(1/r (p/q)')(1/r (p/q)')'=0 which is found by taking the derivative of the equation with the help of this theorem, is satisfied. Finally, the necessary and sufficient conditions are given for this framed curve to be a framed general helix, such as the equality p/q=C_1cost+C_2sint obtained with the help of first and second-order differentiable functions p(s),q(s) and t(s) are provided. In the fourth section, B_2-slant helices in 4-dimensional Euclidean space are examined with framed curves that can have singular points in order to generalize the subject of B_2-slant helices, to increase their usage areas, and prepare the base for new studies. The definition of framed η_3-slant helices in 4-dimensional Euclidean space is given considering a study on B_2-slant helices in 4-dimensional Euclidean space presented by Önder et al., and a study on the framed curves in 4-dimensional Euclidean curve discussed by Akyiğit and Yıldız. Here the framed η_3-slant helix in 4-dimensional Euclidean space is defined as the framed curve formed as a result of the generalized second binormal vector η_3 making a constant angle θ with a fixed direction U. Then, if a constant value (r/q)^2+(1/p^2)((r/q)')^2 equation consisting of the generalized curvatures p,q,r of the framed curve is provided, the theorem expressing the necessary and sufficient condition for this framed curve to be a framed η_3-slant helix is given. Starting from the equation obtained with the help of this theorem, a quadratic differentiable function f is found and if this function satisfies the f(s)p=d/ds (r/q) and d/ds f(s)=-p r/q equations, the necessary and sufficient condition is given for the framed curve to be a framed η_3-slant helix. Finally, necessary and sufficient conditions are given for this framed curve to be a framed η_3-slant helix if the equality r/q=Acosβ(s)+Bsinβ(s) obtained with the help of first and second-order differentiable functions q(s),r(s) and β(s) are provided. In the conclusion section, in the third section, a characterization is given for framed generalized helices in 4-dimensional space in terms of their generalized curvatures. Then, a different characterization obtained if this characterization is provided is given. Finally, a characterization is given for framed general helices in 4-dimensional space with the help of properly selected first and second-order differentiable functions. In the fourth section, a characterization of framed η_3-slant helices found in terms of generalized curvatures is given. Then, based on this data, a different characterization is given for the framed η_3-slant helices, which is obtained with the help of a second-order differentiable function. Finally, another characterization is given for the framed η_3-slant helix with the help of properly selected first and second-order differentiable functions. At the end of this section, suggestions for advanced research are given.