Figure: For each projectile the trajectory is
measured by two position-sensitive parallel plate counters
(PPAC-1 and PPAC-2). For a straight line trajectory the
interaction point at the target and the expected position
at PPAC-3 can be calculated. This information allows the
orientation of a coordinate system along the trajectory
with its origion in the interaction point at the target.
From the measured position (PPAC-3) of the projectile and
the constructed coordinate system one can determine the
scattering angle.
straight line equation:
) and PPAC-3 ( 
)
and
Origin 
of a new coordinate system
Transformation of coordinate system
The z'''-axis of the shifted and rotated coordinate system coincides with the straight-line trajectory calculated from the position of PPAC-1 and PPAC-2.
with 
.
For the calculated position 
one
obtains a scattering angle ( 
, 
)=( 
).
The measured position transformed to the shifted
and rotated coordinate system 
yields the scattering angle ( , ).
As a result of this transformation
the angles ( 
) of each 
-detector
have also to be calculated from the fixed position 
for each projectile.
1. step: expected position at target ( ) and PPAC-3 ( )
2. step: spherical coordinates for
3. step: rotation of the coordinate system around z'-axis
4. step: rotation of the coordinate system around y''-axis
For the calculated position
the scattering angle would be 
and 
.
measured position of scattered projectile ( 
):
result: r=200.73, 
, 
.
position of Ge-detector ( 
):
result: r=99.84, 
, 
.