In this paper, we study the modified orthogonal vector fields with curvature and torsion of a space curve in Minkowski 3-Space.

T.Ikawa obtained the following differential equation$$D_{T}D_{T}D_{T}T-KD_{T}T,K=\kappa ^{2}-\tau ^{2}$$for the c\i rcular helix which corresponds the case that the curvature $ \kappa $ and torsion $ \tau $ of timelike curve $ \alpha $ on the Lorentzian manifold $ M_{1} $ are constant [5]. In this paper, we have defined a slant helix according to Bishop frame of the timelike curve. Furthermore, we have given some necessary and sufficent conditions for the slant helix and T.Ikawa's result is generalized to the case of the general slant helix.

A special Frenet motion with a one parameter has been given by Bottema [5] in E ³. In this study, we have given a generalization of [5] to Bishop motion in Euclidean 3-space. Firstly, Bishop motion is defined for space curve α and then Darboux vector of this motion is calculated for fixed and moving spaces in E ³.The Bishop darboux rotation for space curves in Euclidean space E ³ is decomposed into two simultaneous rotations. The axes of these simultaneous rotations are joined by a simple mechanism. One of these axes is a parallel of the tangent vector of the curve, the direction of the other is the direction of the Bishop darboux vector of the curve. This decomposition of the Bishop darboux rotation yields a necessary condition for the curve to be closed.

In this paper, we classify helicoidal surfaces in the three dimensional simply isotropic space I₃¹ satisfying some algebraic equations in terms of the coordinate functions and the Laplacian operators with respect to the first, the second and the third fundamental form of the surface. We also give explicit forms of these surfaces.