Parameters of rototranslation used in coordinates conversion from WGS-84 (actual epoch) to ETRS89 (epoch 1989.0)-IGM95

For converting coordinates is been used the following formula:

X(I) = T + (1+K) * R * X(II)

Where:

• X(I): unknown coordinate in reference frame I;
• X(II): unknown coordinate in reference frame II;
• K: scale factor;
• R: rotation matrix;
• T: vector of translation;

The parameters of roto-translation (coefficents of matrix R, vector T, scale factor K, are been calculated with the software TRASFO developed by Politecnico di Milano.
Data used to calculate parameters are relative to stations ACOM, AFAL, CANV,MDEA, MPRA, TRIE, UDI1, ZOUF.

Parameters of roto-translation calculated for region Friuli Venezia Giulia

ITRF00(2008.0) >>>> ETRF2000(2008.0)

$$
\begin{matrix}
Scale\:factor &
k= &
k &
3.270665e-10 \\
Matrix &
R= &
\begin{pmatrix}
R_{11} & R_{12} & R_{13}\\
R_{21} & R_{22} & R_{23}\\
R_{31} & R_{32} & R_{33}
\end{pmatrix} &
\begin{pmatrix}
1.000000\text{e+}00 & 7.095722e-08 & 4.369945\text{e-}08 \\
-7.095722\text{e-}08 & 1.000000\text{e+}00 & -7.352123\text{e-}09 \\
-4.369945\text{e-}08 & 7.352123\text{e-}09 & 1.000000\text{e+}00
\end{pmatrix} \\
Vector &
T= &
\begin{pmatrix}
T_{1} & T_{2} & T_{3}
\end{pmatrix} &
\begin{pmatrix}
0.0524989162 & 0.0523446096 & -0.0497923456
\end{pmatrix}
\end{matrix}
$$

ETRF2000(2008.0) >>>> ITRF00(2008.0)

$$
\begin{matrix}
Scale\:factor &
k= &
k &
3.270665e-10 \\
Matrix &
R= &
\begin{pmatrix}
R_{11} & R_{12} & R_{13} \\
R_{21} & R_{22} & R_{23} \\
R_{31} & R_{32} & R_{33}
\end{pmatrix} &
\begin{pmatrix}
1.000000\text{e+}00 & -7.095722\text{e-}08 & -4.369945\text{e-}08 \\
7.095722\text{e-}08 & 1.000000\text{e+}00 & 7.352123\text{e-}09 \\
4.369945\text{e-}08 & -7.352123\text{e-}09 & 1.000000\text{e+}00
\end{pmatrix} \\
Vector &
T= &
\begin{pmatrix}
T_{1} & T_{2} & T_{3}
\end{pmatrix} &
\begin{pmatrix}
-0.0524989162 & -0.0523446096 & 0.0497923456
\end{pmatrix}
\end{matrix}
$$

Example of coordinates calculation for station ZOUF

Conversion from geographical coordinates to cartesian coordinates

$$
\begin{matrix}
\begin{matrix}
N_{ITRF00} \\ E_{ITRF00} \\ U_{ITRF00}
\end{matrix} &
>>
\quad
\begin{matrix}
X_{ITRF00} \\ Y_{ITRF00} \\ Z_{ITRF00}
\end{matrix}
\qquad
\begin{matrix}
N\:46°\:33’\:25.9839″ \\ E\:12°\:58’\:24.7889″ \\ U\:1946.489
\end{matrix} &
>>
\quad
\begin{matrix}
4\:282\:710,222 \\ 986\:659,532 \\ 4\:609\:469,588
\end{matrix}
\end{matrix}
$$

Conversione da ITRF00(2008.0) a ETRF2000(2008.0) tramite rototraslazione: X(I)= T + (1+K) * R * X(II)

$$
\begin{matrix}
\begin{pmatrix}
X_{ETRF2000} \\ Y_{ETRF2000} \\ Z_{ETRF2000}
\end{pmatrix} &
= &
\begin{pmatrix}
T_1 \\ T_2 \\ T_3
\end{pmatrix} &
\begin{pmatrix}
1 + k
\end{pmatrix} &
\begin{pmatrix}
R_{11} & R_{12} & R_{13} \\
R_{21} & R_{22} & R_{23} \\
R_{31} & R_{32} & R_{33}
\end{pmatrix} &
\begin{pmatrix}
X_{ITRF00} \\ Y_{ITRF00} \\ Z_{ITRF00}
\end{pmatrix} \\
\begin{pmatrix}
X_{ETRF2000} \\ Y_{ETRF2000} \\ Z_{ETRF2000}
\end{pmatrix} &
= &
\begin{pmatrix}
0.0524989162 \\ 0.0523446096 \\ -0.0497923456
\end{pmatrix} &
\begin{pmatrix}
1 + k
\end{pmatrix} &
\begin{pmatrix}
1.000000\text{e+}00 & 7.095722\text{e-}08 & 4.369945\text{e-}08 \\
-7.095722\text{e-}08 & 1.000000\text{e+}00 & -7.352123\text{e-}09 \\
-4.369945\text{e-}08 & 7.352123\text{e-}09 & 1.000000\text{e+}00
\end{pmatrix} &
\begin{pmatrix}
X_{ITRF00} \\ Y_{ITRF00} \\ Z_{ITRF00}
\end{pmatrix} \\
\begin{pmatrix}
X_{ETRF2000} \\ Y_{ETRF2000} \\ Z_{ETRF2000}
\end{pmatrix} &
= &
\begin{pmatrix}
4\:282\:710,320 \\ 986\:659,200 \\ 4\:609\:469,569
\end{pmatrix}
\end{matrix}
$$

Conversion from cartesian coordinates to geographical coordinates

$$
\begin{matrix}
\begin{matrix}
X_{ETRF2000} \\ Y_{ETRF2000} \\ Z_{ETRF2000}
\end{matrix} &
>>
\quad
\begin{matrix}
N_{ETRF2000} \\ E_{ETRF2000} \\ U_{ETRF2000}
\end{matrix}
\qquad
\begin{matrix}
4\:282\:710,320 \\ 986\:659,200 \\ 4\:609\:469,569
\end{matrix} &
>>
\quad
\begin{matrix}
N\:46°\:33’\:25.9830″ \\ E\:12°\:58’\:24.7727″ \\ U\:1946.489
\end{matrix}
\end{matrix}
$$