# 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
*