  The present paper deals with a study of curves on a smooth surface whose
position vector always lies in the tangent plane of the surface and it is
proved that such curves remain invariant under isometry of surfaces. It is also
shown that length of the position vector, tangential component of the position
vector and geodesic curvature of a curve on a surface whose position vector
always lies in the tangent plane are invariant under isometry of surfaces.
