Abstract. In this paper, we study spinor Frenet equations in three dimensional Lie Groups with a bi-invariant metric. Also, we obtain spinor Frenet equations for special cases of three dimensional Lie groups.

In this paper, we study trajectory ruled surface of a curve with singular points in the Euclidean 3‐space as an application of singularity theory of a space curve with singular points. By considering notion of framed curve, we investigate the trajectory ruled surface and give some results about invariants of these surfaces. Then, we give some examples of trajectory ruled surfaces. Moreover, we determine local diffeomorphic image of these surfaces.

In this paper, kinematics of semi‐real quaternionic curve in semi‐Euclidean space double-struckE24 is obtained in terms of its curvature functions. The evolution equation of Frenet frame and curvatures of quaternionic curve are obtained. Also, examples of evolution equations of curvatures are given.

In this paper, we construct a helicoidal surface of type I+ with prescribed weighted mean curvature and Gaussian curvature in the Minkowski 3−space ${\Bbb R}_1^3$with a positive density function. We get a result for minimal case. Also, we give examples of a helicoidal surface with weighted mean curvature and Gaussian curvature.

We build the structure of the tubular surface which has singular points. In the second section, we give a brief exposition of framed base curves and framed surfaces, respectively. In the third section, our main results are stated and proved. Moreover, in this section, the normal of the tubular surface and its mean and Gauss curvatures are found, and the characterizations of the parameter curves on the surface are given. Finally, we have expressed the tubular surface with a framed base curve with examples.