Before revision:  Later, we extend these expressions for the curvature and torsion of a generic curve by studying its behavior near an osculating sphere, i.e., we describe them in terms of spherical analogs. Finally, we also address in this work the problem of characterizing those curves whose position vector  , up to a translation, lies on a (moving ) plane spanned by the unit tangent and a rotation minimizing vector fieldand prove that they are precisely the plane curves .
After revision:  Later, we characterize plane and spherical curves as curves whose position vector , up to a translation, lies on a (moving ) plane spanned by the unit tangent and a rotation minimizing vector fieldand prove that they are precisely the plane curves .