# FFF.1.2.4.1 Positioner Movement in the Isocenter Coordinate System

The positioner coordinate system is defined in section C.8.19.6.13.1.2 of PS 3.3 where the positioner angles are represented as (Ap
_{
1
}
, Ap
_{
2
}
, Ap
_{
3
}
).

A point (P
*
*_{
Xp
}
, P
*
*_{
Yp
}
, P
*
*_{
Zp
}
) in the positioner coordinate system can be expressed as a point (P
*
*_{
X
}
, P
*
*_{
Y
}
, P
*
*_{
Z
}
) in the Isocenter coordinate system 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, a point (P
*
*_{
X
}
, P
*
*_{
Y
}
, P
*
*_{
Z
}
) in the Isocenter coordinate system can be expressed as a point (P
*
*_{
Xp
}
, P
*
*_{
Yp
}
, P
*
*_{
Zp
}
) in the positioner coordinate system 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]