  In this work, we study plane and spherical curves in Euclidean and
Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By
conveniently writing the curvature and torsion for a curve on a sphere, we show
how to find the angle between the principal normal and an RM vector field for
spherical curves. Later, we characterize plane and spherical curves as curves
whose position vector lies, up to a translation, on a moving plane spanned by
their unit tangent and an RM vector field. Finally, as an application, we
characterize Bertrand curves and slant helices as curves whose so-called
natural mates are spherical and general helices, respectively.
