A new characterization of rectifiying curves in Minkowski 3-space

A rectifying curve is a twisted curve with the property that all rectifying planes pass through a fixed point. If this point is the origin of the Cartesian coordinate system, the position vector of the rectifying curve always lies in the rectifying plane. A notable feature of these curves is that the ratio between torsion and curvature is a non-constant linear function of the arc length parameter. On the other hand, involute-evolute curve pair are curve concepts that have an important place in mathematics and engineering. For example, they play an important role in the construction of gears, which can make it easier for gears to mesh. Thus, they can ensure that the vehicle turns safely to reduce traffic accidents. In this thesis study, the relationship between rectifying curves and involute-evolute curve pair in Minkowski 3-space is examined and a new characterization of rectifying curves is characterized with the help of this curve pair. Due to the metric structure on Minkowski 3-space, curves have three different causal characters, and in this thesis, spacelike and timelike rectifying curves are considered. First, timelike rectifying curves are discussed. The necessary and sufficient condition for a timelike rectifying curve to lie on the pseudo-sphere is given depending on the curvature and torsion of the curve. It has been shown that any spacelike involute of a timelike rectifying curve lies on hyperquadrics (pseudo-sphere or pseudo-hyperbolic space) that consider the reference point as the center. It is shown that for the spacelike involute of a timelike rectifying curve to also be a rectifying curve, a necessary and sufficient condition is that the curvature of the curve satisfies a Bernoulli differential equation, and this differential equation is solved. Finally, it is given that the evolute of a hyperquadric (located on a pseudo-sphere or pseudo-hyperbolic space) non-planar spacelike curve is a timelike rectifying curve whose reference point is the center of the hyperquadratics. Then, spacelike rectifying curves are discussed and the necessary and sufficient condition for a spacelike rectifying curve to lie on the hyperquadrics is obtained based on curvature and torsion, which includes the sign of the causal character of the principal normal of the curve. It has been shown that any spacelike or timelike involute of a spacelike rectifying curve lies on hyperquadratics centered on the reference point. It is shown that for the spacelike or timelike involute of a spacelike rectifying curve to also be a rectifying curve, a necessary and sufficient condition is that the curvature of the curve should satisfy a Bernoulli differential equation, and this differential equation is solved. Finally, it has been ontained that the evolute of a hyperquadrical non-planar spacelike or timelike curve is a spacelike rectifying curve whose reference point is the center of the hyperquadrics, and an example is given in which a spacelike rectifying curve is constructed which is the evolute of a timelike helix located on the pseudo-sphere.