A Generalization of Curve Mates: Normal Mate of a Curve

This a paper, a new curve pair is defined that generalizes some pairs of curves well known as Mannheim and Bertrand curve pairs. A normal curve pair is defined in such a way that a vector u obtained by overlapping the normal planes of the G and G* curves makes the same angle as the binormals of these curves. The relationship between torsions and curvatures of curve pairs was analyzed. Moreover, The unit quaternion q corresponding to the rotation matrix between the Frenet vectors of the curves was defined. In the conclusion, it is expressed express which famous pairs of curves will be obtained in which particular case.