# Z.2 POSITIONER COORDINATE SYSTEM TRANSFORMATIONS

Any point of the Positioner coordinate system (P
*
*_{
Xp
}
, P
*
*_{
Yp
}
, P
*
*_{
Zp
}
) can be expressed in the Isocenter coordinate system (P
*
*_{
X
}
, P
*
*_{
Y
}
, P
*
*_{
Z
}
) by applying the following transformation:

*
(P
*
*
*_{
X
}
*
, P
*
*
*_{
Y
}
*
, P
*
*
*_{
Z
}
*
)
*
*
*^{
T
}
*
= (R
*
*
*_{
2
}
*
*^{
.
}
*
R
*
*
*_{
1
}
*
)
*
*
*^{
T
}
*
*
*
*^{
.
}
*
( R
*
*
*_{
3
}
*
*^{
T
}
*
*
*
*^{
.
}
*
(P
*
*
*_{
Xp
}
*
, P
*
*
*_{
Yp
}
*
, P
*
*
*_{
Zp
}
*
)
*
*
*^{
T
}
*
)
*

And inversely, any point of the Isocenter coordinate system (P
*
*_{
X
}
, P
*
*_{
Y
}
, P
*
*_{
Z
}
) can be expressed in the Positioner coordinate system (P
*
*_{
Xp
}
, P
*
*_{
Yp
}
, P
*
*_{
Zp
}
) by applying the following transformation:

*
(P
*
*
*_{
Xp
}
*
, P
*
*
*_{
Yp
}
*
, P
*
*
*_{
Zp
}
*
)
*
*
*^{
T
}
*
= R
*
*
*_{
3
}
*
*
*
*^{
.
}
*
( (R
*
*
*_{
2
}
*
*^{
.
}
*
R
*
*
*_{
1
}
*
)
*
*
*^{
.
}
*
(P
*
*
*_{
X
}
*
, P
*
*
*_{
Y
}
*
, P
*
*
*_{
Z
}
*
)
*
*
*^{
T
}
*
)
*

Where
*
R
*
*
*_{
1
}
,
*
R
*
*
*_{
2
}
and
*
R
*
*
*_{
3
}
are defined as follows:

[pic]