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 Annex B.2. 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 Annex C.1.

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

422297.792 mE, 412878.741 mN