  The properties of two local reference frames based on the magnetic field and
the current density are investigated for magnetized plasmas in toroidal
geometry with symmetric angle. The magnetic field-based local frame of
reference has been well-studied for example by Dewar and colleagues [Phys.
Fluids 27, 1723 (1984)] An analogous frame based on the current density vector
is possible because it is also divergence free and perpendicular to the
gradient of the poloidal flux. The concept of straightness of a vector is
introduced and used to elucidate the Boozer and Hamada coordinate systems. The
relationship of these local frames to the more well-known Frenet frame of
reference, which specifies a curve in terms of curvature and torsion, is given.
As an example of the usefulness of the these formal relationships, we briefly
review the ideal MHD theory and their use. We also present a new annihilation
operator, useful for eliminating shorter time scales than the time scale of
interest, for deriving the inner layer equations of Glasser, Greene, and
Johnson [Phys. Fluids 18, 875 (1975)]. Compared to the original derivation that
is based on the local frame of reference in terms of the magnetic field, the
new annihilation operator that is based on the local frame of reference in
terms of the current density simplifies the derivation.
