Because of the many coordinate systems used in the flight simulator, the simulator makes extensive use of coordinate transformations. Most coordinate systems in the simulator are Cartesian, so that the transformation is linear.

(A word on notation: this paper indicates which coordinate system a
component belongs to by a superscript. For example, the velocity
components of the airplane in body axes are
*u*^{B},*v*^{B},*w*^{B}. In local
axes, the components are
*u*^{L},*v*^{L},*w*^{L}. The superscript is not used
where there is no confusion about what coordinate system a quantity
belongs to.)

To transform a velocity, or some other vector quantity, from one
coordinate system to another, only a rotation is required. For
example, the transformation of the wind velocity vector from local
axes to body axes is simply:

where .

Transforming points requires a rotation and translation. Transforming
points between body and local axes has an additional complication as
well. The body axes originate from the reference point of the
airplane; however, because of dynamic considerations, the airplane's
location is specified by the coordinates of its CG. This requires an
additional translation from the CG to the reference point. The
transformation of a point from body to local coordinates is:

where is the point in question, and is the coordinates of the CG in the indicated coordinate system. The opposite transformation is: