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# New Zealand Map Grid Transformation Formulae

This page explains how to convert NZMG projection coordinates ( N , E ) to their NZGD1949 geographic equivalents and vice versa.

### Projection parameters

a | Semi-major axis of reference ellipsoid |

f | Ellipsoidal flattening |

Origin latitude | |

Origin longitude | |

False Northing | |

False Easting | |

Latitude of computation point | |

Longitude of computation point | |

N | Northing of computation point |

E | Easting of computation point |

The following equations are divided into three sections:

### NZGD1949 geographic to NZMG projection

The conversion of NZGD1949 geographic coordinates ( , ) to NZMG projection coordinates ( N , E ) is achieved in several steps.

The conversion requires the latitude and longitude expressed in decimal degrees (north and east are positive). The constants are provided on the NZMG page and coefficients used in these formulae are listed below.

Note: NZMG applies to coordinates referenced to NZGD1949. Coordinates from other datums, such as NZGD2000 or WGS84, must be converted to NZGD1949 before these formulae can be applied.

Calculate and using:

Calculate a complex polynomial function of the complex number using the formula:

Express the complex number z in terms of its real and imaginary parts and calculate the northing N and easting E from x and y as:

### NZMG projection to NZGD1949 geographic

The conversion from NZMG to latitude and longitude involves an iterative approximation to reverse step 2 above. This starts with the easting E and northing N. The steps to calculate latitude and longitude are as follows. Note that the constants and coefficients used in these formulae are listed below. Derive the complex number z as the following steps.

Note: NZMG applies to coordinates referenced to NZGD1949. Coordinates from other datums, such as NZGD2000 or WGS84, must be converted to NZGD1949 before these formulae can be applied.

Determine the complex number θ as a series of approximations θ_{0} , θ_{1} , θ_{2} and so on. The first approximation is:

Successive approximations are obtained by applying the formula:

Two iterations of this formula will give millimetre transformation accuracy.

Express in terms of its real and imaginary parts and calculate the latitude and longitude as follows

### NZMG coefficients

Coefficient | Real Part | Imaginary Part |
---|---|---|

A_{1} |
0.6399175073 | |

A_{2} |
-0.1358797613 | |

A_{3} |
0.063294409 | |

A_{4} |
-0.02526853 | |

A_{5} |
0.0117879 | |

A_{6} |
-0.0055161 | |

A_{7} |
0.0026906 | |

A_{8} |
-0.001333 | |

A_{9} |
0.00067 | |

A_{10} |
-0.00034 | |

B_{1} |
0.7557853228 | 0 |

B_{2} |
0.249204646 | 0.003371507 |

B_{3} |
-0.001541739 | 0.041058560 |

B_{4} |
-0.10162907 | 0.01727609 |

B_{5} |
-0.26623489 | -0.36249218 |

B_{6} |
-0.6870983 | -1.1651967 |

C_{1} |
1.3231270439 | 0 |

C_{2} |
-0.577245789 | -0.007809598 |

C_{3} |
0.508307513 | -0.112208952 |

C_{4} |
-0.15094762 | 0.18200602 |

C_{5} |
1.01418179 | 1.64497696 |

C_{6} |
1.9660549 | 2.5127645 |

D_{1} |
1.5627014243 | |

D_{2} |
0.5185406398 | |

D_{3} |
-0.03333098 | |

D_{4} |
-0.1052906 | |

D_{5} |
-0.0368594 | |

D_{6} |
0.007317 | |

D_{7} |
0.01220 | |

D_{8} |
0.00394 | |

D_{9} |
-0.0013 |