As shown in transform, the mean position of the LEDs as
determined by the OPTOTRAK gives the vector 
. The
position of the tip of the probe in the OPTOTRAK reference system is
given by the vector 
. From the figure, it can be seen
that 
can be determined if the vector 
is known together with the orientation of the coordinate system P with
respect to O. The vector 
(constant in the P
coordinate system) was determined by the following procedure.
The probe was rotated about a fixed endpoint (i.e., 
is
constant), 500 measurements were taken and each sample and the
centroid of the LEDs (
) was calculated. A least squares fit
was then performed to find the radius and center of the sphere that
best fitted this set of points (since that, during the motion, the
centroid of the LEDs moved on a surface of a sphere). The vector
was then taken to be center of this sphere.
The next step was to establish the LED coordinate system P with an
origin at the centroid (
) of LEDs. Unit vectors having
for direction the principal axis of the LEDs, 
, 
and 
, were determined by solving the eigenvalue
problem outlined in [1], and the axis of the probe
coordinate system were taken parallel to these vectors.
Using an homogeneous transformation matrix notation, the transformation
between 
and 
can be written:
where
was then determined from the inverse of
(4.1):
Once the constant vector 
was determined, the position
of the probe tip for any data point was computed using 0vtip.
Note that the transformation matrix in this equation involves the
determination of the axis of the P coordinate system.
