The geometry of the spirals on the hyperboloid model H^+(r)

This thesis is a study on spirals on the Hyperboloid model H^+(r) in 3-dimensional Minkowski space. The first chapter is devoted to the introduction. In the second chapter, some necessary basic concepts about Minkowski 3-space are given. Vectors, curves and surfaces in Minkowski 3-space may have spacelike, timelike and lightlike characters. The set of timelike vectors r- radius and with center O is the hyperbolic sphere H^+2_0(r). In Euclidean case, H^+2_0(r) is a hyperboloid of two sheets and its positive side is denoted with H^+(r). Since tangent plane at any point on the spacelike surface H^+(r) is spacelike, only spacelike curves exist on this surface. Therefore, the spirals on this surface are spacelike curves. In this study, firstly, the Frenet frame, the Frenet derivative formulas, and the instantaneous rotation vector of the spirals on the H^+(r) model were calculated. Secondly, the Darboux frame, the Darboux derivative formulas and the Darboux instantaneous rotation vector of these curves are obtained. The relationship between the Frenet and Darboux frames is investigated. Finally, the differential geometric properties of the curve were investigated and their graphs were drawn with the Maple program.