# FFF.1.2.4 Positioner Movement

## 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]

## FFF.1.2.4.2 X-Ray Incidence and Image Coordinate System

The following concepts illustrate the model of X-Ray cone-beam projection:

The
*
X-Ray incidence
*
represents the vector going from the X-Ray source to the Isocenter.

The
*
receptor plane
*
represents the plane perpendicular to the X-Ray Incidence, at distance SID from the X-Ray source. Applies for both image intensifier and digital detector. In case of digital detector it is equivalent to the detector plane.

The
*
image coordinate system
*
is represented by (o, u, v), where “o” is the projection of the Isocenter on the receptor plane.

The
*
source to isocenter distance
*
is called ISO. The
*
source image receptor distance
*
is called SID.

The projection of a point (P
*
*_{
Xp
}
, P
*
*_{
Yp
}
, P
*
*_{
Zp
}
) in the positioner coordinate system is represented as a point (P
*
*_{
u
}
, P
*
*_{
v
}
) in the image coordinate system.

[pic]

Figure FFF.1.2-8Projection of a Point of the Positioner Coordinate System

A point (P
*
*_{
Xp
}
, P
*
*_{
Yp
}
, P
*
*_{
Zp
}
) in the positioner coordinate system (O
_{
p
}
, X
_{
p
}
, Y
_{
p
}
, Z
_{
p
}
) can be expressed as a point (P
*
*_{
u
}
, P
_{
v
}
) in the image coordinate system by applying the following transformation:

P
_{
u
}
= (SID / (ISO – P
_{
Yp
}
) )
^{
.
}
P
_{
Xp
}

P
_{
v
}
= (SID / (ISO – P
_{
Yp
}
) )
^{
.
}
P
_{
Zp
}

The ratio SID / (ISO - P
_{
Yp
}
) is also called magnification ratio of this particular point.