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 YA = -110149.210 m 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:

$$xB = 3790269.549\\ m yB = -110038.064\\ m zB = 5111050.261 m$$

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'. 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'.

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

422297.792 mE, 412878.741 mN