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In this study, we have generalized the involute and evolute curves of the spacelike curve α with a spacelike binormal in Minkowski 3-Space. Firstly, we have shown that, the length between the spacelike curve α and the timelike curve β is constant. Furthermore, the Frenet frame of the involute curve β has been found as depend on curvatures of the curve α. We have determined the curve α is planar in which conditions. Secondly, we have found transformation matrix between the evolute curve β and the curve α. Finally, we have computed the curvatures of the evolute curve β.

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.

In this paper, we study helices and the Bertrand curves. We obtain some of the classification results of these curves with respect to the modified orthogonal frame in Euclidean 3-spaces.2000 Mathematics Subject Classification. 53A04, 53A35.

Abstract. In this study we have defined Bäcklund transformations of curves according to Bishop frame preserving the natural curvatures under certain assumptions in Minkowski 3-space.

We describe a robust method for constructing a tubular surface surrounding a spacelike curve with a spacelike principal normal in Minkowski 3-Space. Our method is designed to eliminate undesirable twists and wrinkles in the tubular surface's skin at points where the curve experiences high torsion. In our construction the tubular surface's twist is bounded by the spacelike curve's curvature and is independent of the spacelike curve's torsion.

In this paper, we classify translation surfaces in the three-dimensional simply isotropic space [Formula: see text] under the condition [Formula: see text] where [Formula: see text] is the Laplace operator with respect to the first and second fundamental forms and [Formula: see text] is a real number. We also give explicit forms of these surfaces.