> For the complete documentation index, see [llms.txt](https://docs.os.uk/more-than-maps/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.os.uk/more-than-maps/a-guide-to-coordinate-systems-in-great-britain/helmert-transformation-worked-example.md).

# Helmert transformation worked example

All the transformation functions are coded into a spreadsheet available from the OS website.

Transform ETRS89/WGS84 coordinates to OSGB36.

*XA* = 3790644.900 m\
\&#xNAN;*YA* = -110149.210 m\
\&#xNAN;*ZA* = 5111482.970 m

| tX(m)     | tY(m)     | tZ(m)     | s (ppm)   | rX(sec)  | rY(sec)  | rZ(sec)  |
| --------- | --------- | --------- | --------- | -------- | -------- | -------- |
| - 446.448 | + 125.157 | - 542.060 | + 20.4894 | - 0.1502 | - 0.2470 | - 0.8421 |

Coordinate matrix for system “A” (ETRS89):

#### D1

$$\begin{bmatrix} 3790644.900 \ -110149.210 \ 5111482.970 \end{bmatrix}$$

Rotation & scale matrix. Diagonal scale elements = 1+(scale ppm \* 0.000001); rotation elements are rotation parameters converted from seconds to radians:

#### D2

$$
\begin{bmatrix}
1.0000204894 & 4.08261601 \times 10^{-6} & -1.19748979 \times 10^{-6} \\
-4.08261601 \times 10^{-6} & 1.0000204894 & 7.28190149 \times 10^{-7} \\
1.19748979 \times 10^{-6} & -7.28190149 \times 10^{-7} & 1.0000204894
\end{bmatrix}
$$

Coordinate translations matrix:

#### D3

$$
\begin{bmatrix}
-446.448 \\
125.157 \\
-542.060
\end{bmatrix}
$$

#### (D2) \* (D1):

$$
\begin{bmatrix}
3790715.9974 & \\-110163.2205 & \5111592.3207
\end{bmatrix}
$$

#### (D3) + (D4):

$$
\begin{bmatrix}
3790269.549 &\ -110038.064 &\ 5111050.261
\end{bmatrix}
$$

So system “B” (OSGB36) coordinates are:\\

$$
x\_B = 3790269.549 m\\
y\_B = -110038.064 m\\
z\_B = 5111050.261 m
$$

The above OSGB36 XYZ coordinates can be converted to latitude, longitude, height using the equations in '[Converting between 3D Cartesian and ellipsoidal latitude, longitude and height coordinates](/more-than-maps/a-guide-to-coordinate-systems-in-great-britain/converting-between-3d-cartesian-and-ellipsoidal-latitude-longitude-and-height-coordinates.md)'. Remember that the height in this case approximates a height above ODN and not above the Airy 1830 ellipsoid. The latitude and longitude can also be projected to eastings and northings using the equations in '[Converting between grid eastings and northings and ellipsoidal latitude and longitude](/more-than-maps/a-guide-to-coordinate-systems-in-great-britain/converting-between-grid-eastings-and-northings-and-ellipsoidal-latitude-and-longitude.md)'.

53° 36’ 42.2972” N, 001° 39’ 46.5416” W, 249.950 m

422297.792 mE, 412878.741 mN


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