# EPSG:1033

## Position Vector transformation (geocentric domain)

### Attributes

Data source: EPSG

Information source: EPSG guidance note #7-2, http://www.epsg.org

Revision date: 2019-02-21

Remarks: This method is a specific case of the Molodensky-Badekas (PV) method (code 1061) in which the evaluation point is the geocentre with coordinate values of zero. Note the analogy with the Coordinate Frame method (code 1032) but beware of the differences!

### Formula

```Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

The transformation between source and target CRS geocentric coordinates is usually described as a simplified 7-parameter Helmert transformation, expressed in matrix form with 7 parameters, in what is known as the "Bursa-Wolf" formula:

(Xt)          (  1      -rZ     +rY)     (Xs)    (tX)
(Yt)  =  M *  ( +rZ      1      -rX)  *  (Ys)  + (tY)
(Zt)          ( -rY     +rX      1 )     (Zs)    (tZ)

The parameters are commonly referred to defining the transformation "from source coordinate reference system to target coordinate reference system" in which (Xs, Ys, Zs) are the coordinates of the point in the source geocentric coordinate reference system and (Xt, Yt, Zt) are the coordinates of the point in the target geocentric coordinate reference system.  But that does not define the parameters uniquely; neither is the definition of the parameters implied in the formula, as is often believed.  However, the following definition, which is consistent with the 'Position Vector Transformation' convention is common E&P survey practice,

(tX, tY, tZ): Translation vector, to be added to the point's position vector in the source coordinate reference system in order to transform from source system to target system; also: the coordinates of the origin of the source coordinate reference system in the target coordinate reference system.

(rX, rY, rZ): Rotations to be applied to the point's vector.  The sign convention is such that a positive rotation about an axis is defined as a clockwise rotation of the position vector when viewed from the origin of the Cartesian coordinate reference system in the positive direction of that axis; e.g. a positive rotation about the Z-axis only from source system to target system will result in a larger longitude value for the point in the target system.  Although rotation angles may be quoted in any angular unit of measure, the formula as given here requires the angles to be provided in radians.

M: Multiplication factor to be applied to the position vector in the source coordinate reference system in order to obtain the correct scale of the target coordinate reference system. M = (1+dS) where dS is the scale difference. When dS is expressed in parts per million, M = (1+dS*10^-6). When dS is the scale difference expressed in parts per billion, M = (1+dS*10^-9).

<<<<<This text continues in the description of the Coordinate Frame Rotation formula>>>>>```

### Example

```Input point:
Coordinate reference system: WGS 72
Cartesian geocentric coords:
X = 3 657 660.66 (m)
Y =   255 768.55 (m)
Z = 5 201 382.11 (m)

Transformation parameters WGS 72 to WGS 84:
tX (m) = 0.000
tY (m) = 0.000
tZ (m) = +4.5
rX (") = 0.000 = 0.0 radians
rY (") = 0.000 = 0.0 radians
rZ (") = +0.554 = 0.000002685868 radians
dS (ppm) = +0.219

First M = 1 + dS = 1.000000219

Then application of this 7 parameter Position Vector transformation results in WGS 84 geocentric coordinates of:
X = 3 657 660.78 (m)
Y =   255 778.43 (m)
Z = 5 201 387.75 (m)```