# Annex Z X-Ray Isocenter Reference Transformations (Informative)

## Z.1 INTRODUCTION

The Isocenter Reference System Attributes describe the 3D geometry of the X-Ray equipment composed by the X-Ray positioner and the X-Ray table.

These attributes define three coordinate systems in the 3D space:

The Isocenter Reference System attributes describe the relationship between the 3D coordinates of a point in the table coordinate system and the 3D coordinates of such point in the positioner coordinate system (both systems moving in the equipment), by using the Isocenter coordinate system that is fixed in the equipment.

## 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]

## Z.3 TABLE COORDINATE SYSTEM TRANSFORMATIONS

Any point of the table coordinate system (P
*
*_{
Xt
}
, P
*
*_{
Yt
}
, P
*
*_{
Zt
}
) (see Figure Z-1) can be expressed in the Isocenter Reference coordinate system (P
*
*_{
X
}
, P
*
*_{
Y
}
, P
*
*_{
Z
}
) by applying the following transformation:

*
(P
*
*
*_{
X
}
*
, P
*
*
*_{
Y
}
*
, P
*
*
*_{
Z
}
*
)
*
*
*^{
T
}
*
= (R
*
*
*_{
3
}
*
*^{
.
}
*
R
*
*
*_{
2
}
*
*^{
.
}
*
R
*
*
*_{
1
}
*
)
*
*
*^{
T
}
*
*
*
*^{
.
}
*
(P
*
*
*_{
Xt
}
*
, P
*
*
*_{
Yt
}
*
, P
*
*
*_{
Zt
}
*
)
*
*
*^{
T
}
*
+ (T
*
*
*_{
X
}
*
, T
*
*
*_{
Y
}
*
, T
*
*
*_{
Z
}
*
)
*
*
*^{
T
}

And inversely, any point of the Isocenter coordinate system (P
*
*_{
X
}
, P
*
*_{
Y
}
, P
*
*_{
Z
}
) can be expressed in the table coordinate system (P
*
*_{
Xt
}
, P
*
*_{
Yt
}
, P
*
*_{
Zt
}
) by applying the following transformation:

*
(P
*
*
*_{
Xt
}
*
, P
*
*
*_{
Yt
}
*
, P
*
*
*_{
Zt
}
*
)
*
*
*^{
T
}
*
= (R
*
*
*_{
3
}
*
*^{
.
}
*
R
*
*
*_{
2
}
*
*^{
.
}
*
R
*
*
*_{
1
}
*
)
*
*
*^{
.
}
*
( (P
*
*
*_{
X
}
*
, P
*
*
*_{
Y
}
*
, P
*
*
*_{
Z
}
*
)
*
*
*^{
T
}
*
- (T
*
*
*_{
X
}
*
, T
*
*
*_{
Y
}
*
, T
*
*
*_{
Z
}
*
)
*
*
*^{
T
}
*
)
*

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

[pic]

[pic]

*
Figure Z-1
*
*
Coordinates of a Point āPā in the Isocenter and Table coordinate systems
*