# Three & Seven Parameter Similarity Transformations (NZGD1949 to NZGD2000)

This page hosts information on the parameters LINZ provides for three parameter and seven parameter similarity transformations between NZGD1949 and NZGD2000.

Similarity transformations are less accurate than the distortion grid.

Similarity transformations convert the latitude and longitude of a point from one ellipsoid to another which may be centred at a different point and oriented differently. These transformations account for the difference between the ellipsoids underlying the different geodetic datums, but can not account for distortions within the datums themselves. This is why similarity transformations are less accurate than the distortion grid.

Before using a similarity transformation the geographic coordinates of the point must be changed to their Cartesian equivalents. The formulas to do this conversion are described in equations used for datum transformations. While NZGD1949 is a horizontal datum, similarity transformations are three-dimensional. To properly undertake the Cartesian transformation a height of the computation point is required. The normal-orthometric height of the point can be used or a height of zero assumed.

Note: Two different rotation conventions are in common use for defining similarity transformations, Helmert and Bursa-Wolf. The difference between these two conventions is the signs assigned to the rotation components. The transformations described here and the associated conversion formulae follow the Helmert convention. It is critical that transformation software is checked to ensure the correct rotation directions are being applied.

These transformations are formally defined in the LINZ standard LINZS25000: Standard for New Zealand Geodetic Datum 2000, and are summarised in the associated NZGD2000 LINZG25700 Fact sheet.

### Three parameter transformation

The three parameter transformation defines the translation difference TX, TY, TZ) between the origins of the ellipsoids used for each datum. It has a nominal accuracy of ±5 metres.

The three parameter transformation from NZGD1949 to NZGD2000 is:

Translation (metres)
TXTYTZ
54.4-20.1183.1

This transformation can be reversed (ie to convert from NZGD2000 to NZGD1949) by reversing the signs of the above parameters.

These conversions can be implemented using equations used for datum transformations - sample coordinates transformed using the three-parameter conversion are also available in datum transformation examples.

### Seven parameter transformation

The seven parameter transformation defines the translation (TX, TY, TZ), rotation (RX, RY, RZ) and scale change (ΔS) between the origins and axes of the ellipsoids used for each datum. It has a nominal accuracy of ±4 metres.

The seven parameter transformation from NZGD1949 to NZGD2000 is:

Translation (metres)Rotation (seconds of arc)Scale change
(ppm)
TXTYTZRXRYRZΔS
59.47-5.04187.44-0.4700.100-1.024-4.5993

These transformations can be reversed (ie NZGD2000 to NZGD1949) by reversing the signs of the above parameters.