Equations Used for Datum Transformations

The following formulae can be used to transform geographic coordinates between geodetic datums using three and seven parameter similarity transformations.

These formulas are formally defined in the LINZ standard LINZS25000 (Standard for New Zealand Geodetic Datum 2000), and are summarised in the associated NZGD2000 fact sheet (LINZG25700) and LINZG25073 (Fact Sheet - Datum and Projection Transformations).

This conversion is a three-step process:

  • Convert geographic coordinates to their Cartesian equivalents
  • Apply similarity transformation to Cartesian coordinates
  • Convert Cartesian coordinates back to geographic values

Geographic coordinates to Cartesian coordinates

These formulae can be used to convert geographic coordinates, latitude ( Φ ), longitude ( λ ), and height ( h ), into Cartesian coordinates (X, Y, Z ):

Equation to convert geographic coordinates into cartesian coordinates.

Where a and f are obtained from the reference ellipsoid used for the respective geodetic datum and the h is the height of the computation point or approximated as zero

Equation to convert geographic coordinates into cartesian coordinates.

Three parameter transformation

The three parameter transformation is implemented using:

Three parameter transformation equation.

Seven parameter transformation

The Helmert seven parameter similarity transformation is implemented using:

Seven parameter transformation equation.

Note: the rotation parameters ( R ) must be converted from arc-seconds to radians before being used in this equation.

Note: this is a simplified version of the Helmert formulae that applies for small rotation angles.  This is the official formulae to use for the NZGD49-NZGD2000 seven parameter transformation. 

Cartesian coordinates to geographic coordinates

These formulas can be used to convert Cartesian coordinates ( X, Y, Z ) into geographic coordinates latitude ( Φ ), longitude ( λ ), and height ( h ).

Equation to convert cartesian coordinates to geographic coordinates.

Where a and f are obtained from the reference ellipsoid used for the respective geodetic datum:

x.

Note: because NZGD1949 is a horizontal datum the height resulting from the transformation will not be in terms of the output datum, the equation is shown here for completeness.

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