The figure shows the steps necessary to transform from a spacecraft-
centered orthogonal Geocentric Solar Ecliptic coordinate system to an orthogonal
magnetic cloud system, in which the x-axis is aligned with the magnetic
cloud axis with positive polarity in the sense of the magnetic field
along the axis (note that the orientation of the cloud axis in GSE
coordinates is given by a longitude angle,_{}, and a latitude angle,
_{}):

(a) The transformation from X_{GSE},Y_{GSE},Z_{GSE}
(GSE, Geocentric Solar Ecliptic) to an
Intermediate coordinate system x_{I},y_{I},z_{I}, in which the z_{I }_{
}-axis is
parallel to Z_{GSE}, and the rotation about Z_{GSE} is through the angle _{A}.

(b) The transformation from the x_{I},y_{I},z_{I} system to a quasi-magnetic
cloud system in which the x_{C}-axis is aligned with the magnetic cloud
axis. This transformation required a single rotation around the y_{I}-
axis through the angle _{A}; this provides for the "tilt" of the cloud
axis with respect to the Ecliptic plane. Note that the light blue
plane is the crosssectional plane of the magnetic cloud. Finally,

(c) The rotation about the cloud axis (x_{C}) through an arbitrary angle
, so that the y_{C}axis is placed along some useful direction. The
magnetic cloud axis system is then, x_{C},y_{C},z_{C}, where z_{C} = x_{C} × y_{C}, as
usual.

Then the matrix that corresponds to the general transformation
(arbitrary ) from GSE coordinate system to the magnetic cloud system
has the following 9 elements (m_{ij}, where the subscripts i and j
represent row and column, respectively):

If one chooses to arrange the y_{C}-axis to be along a line
(perpendicular to the cloud's axis) that goes from
the center of the magnetic cloud to the observing spacecraft at the
time of closest approach to the cloud axis, then ψ is related to _{A} and _{A} in the following way:

but this line can be either
the +y_{c} axis or -y_{c} axis.

_{ }

To choose which sign is appropriate for any given case, we examine the sign of
the matrix element of row 3, column 1 above; we call this element k (shown
underlined above). If k is < 0, then we set:

ψ = ψ + 180°, otherwise ψ remains its original value.

For some studies this particular definition and phasing of the y_{C }-axis is useful.
It ensures, for example, that while the spacecraft travels through the
magnetic cloud it does so at a constant y_{C }_{
}-value at all times, in
the magnetic cloud coordinate frame, and that the projection of the spacecraft's
trajectory (+z_{c}) on the cloud's cross-section will have a POSITIVE
component along X_{GSE}.