New Zealand Map Grid: History & Preliminary.

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In 1946 New Zealand introduced a Transverse Mercator mapping system based on the International Spheroid as recommended at the Madrid meeting of the International Geodetic and Geophysical Union in 1924.
The fundamental elements of that spheroid are

major semi-axis, a = 6,378,388 metres
compression, f = 1/297

The projection was base on the Sears(1927) metre-foot relationship
1 metre = 4.97097815656 links.

( 12 inches = 1 foot, 3 feet = 1 yard, 1 link = 7.92 inches
and 100 links = 1 chain)
This all seems very odd today but New Zealand was surveyed in links , chains and acres.
A further complication arose in that New Zealand is a country with a comparatively small extent in longitude and more in latitude. It also consists basically of two large islands. To accommodate this factor the first New Zealand projection was split into two separate Transverse Mercator systems. These were the
North Island National Yards Grid
and South Island National Yards Grid.

Both projections though were Transverse Mercator. However as time evolved it became obvious that there were some serious drawbacks. To overcome the difficulties the New Zealand Map Grid was derived.

The New Zealand Map Grid and GPS

Some more expensive GPS systems with built in maps are generally based on the WGS84 datum. For smaller hand held units however there is often a lot of confusion over which datum or grid to use.

At the present the datum which should be selected is Geodetic Datum 49. This is equivalent to the spheroidal datum used for the New Zealand National Yards Grids. Unless a GPS unit has a built-in conversion it is not possible to output east and north coordinates in terms of the New Zealand Map Grid.

Plans are in train to establish a new datum for mapping in New Zealand.
See the links on my Surveying/Mapping for details. All the information on this page relates solely to Geodetic Datum 1949.

Latitudes and longitudes obtained by using Geodetic Datum 49 are plottable directly on to maps produced in NZ Map Grid but only if the graticule of lats/longs. is shown. Various approximations have been made to enable 'user grids' to be input in a GPS unit. There are all though, just that - approximations. I make no further comment on this.
Further information could probably be found from the popular
news group 'sci.geo.satellite-nav'
My solution has been to write my own moving map software (for PC laptop or desktop) which accepts scanned maps in NZ Map Grid. It converts NMEA information from the GPS to NZMG coordinates. The software though will only work in New Zealand. Anyone interested in the software can Email me. Other software replicates the Garmin 12XL satellites in view/signal strength's screen and a lat/long. to NZMG converter.
It is not my intention to make these available on this page but anyone interested can Email me.

The New Zealand Map Grid

The projection is conformal, but is otherwise unlike any other projection used for mapping. It was derived to give a small range of scale variation over the land area of New Zealand.
New Zealand lies between 166 degrees East and 180 degrees East in longitude and between 34 degrees South and 48 degrees South latitude.
To achieve the small range of scale variation over the land area of New Zealand it has been necessary to abandon the orderly arrangement of scale curves. The result is a unique projection known simply as the New Zealand Map Grid.

Characteristics of the Projection

  1. The true origin of the projection is placed at latitude 41 degrees South, this being the latitude nearest to the middle latitude of the country.
  2. The true origin is placed at longitude 173 degrees East, this being the whole degree nearest to the meridian to which equal perpendiculars can be drawn from the easternmost and westernmost points of the country.
  3. The meridian of longitude 173 degrees East, which is not represented by a straight line, is oriented so that its tangent at the origin is the north-south axis of coordinates.
  4. The true origin is assigned arbitrary coordinates sufficiently large to render all coordinates positive, or east and north of a so-called "false origin".
  5. In a metre coordinate system sufficiently extensive to cover the whole country, the northing must at some stage reach seven integral figures. In the New Zealand Map Grid the coordinates have seven integral figures in all cases.
  6. The Easting is always less than 5,000,000 metres, the Northing always greater than 5,000,000 metres, so that no confusion can arise between Easting and Northing whichever is stated first.
  7. The coordinates assigned to the true origin are sufficiently large for the grid to be extended a considerable distance out to sea without departing from the characteristics stated in (5) and (6).
  8. Thus, the true origin is place at latitude 41 degrees South, longitude 173 degrees East, and the coordinates of this point are
    2,510,000 metres East, 6,023150 metres North.

The land area of New Zealand is thus fitted into a rectangle 1,000,000 metres wide by 1,500,000 metres from north to south. Coordinate values have been alloted to the nominated origin so that the easting ranges from 2,000,000 to 3,000,000 metres, and the Northing ranges from 5,300,000 to 6,800,000 metres. The seemingly strange values assigned to the origin have also been chosen to fit into a scheme of map sheets at a scale of 1:50,000 with boundaries at 10,000 metre values.

Computation.

Lat. & Long. to NZMG

It is important that all latitudes and longitudes used to calculate NZMG coordinates are in terms of Geodetic Datum 1949.
Latitudes and longitudes in other datums (unless already in WGS84) must first be converted to WGS84 datum and then further converted to Geodetic Datum 1949.
Transformation parameters: WGS84 - Geo49
diff X = 84
diff Y = -22
diff Z = 209
Diff.major semi-axis(a) = -251
Diff.flattening * 10^4 =-0.14192702

Formulae for those conversions are not currently provided here.

1.Computation of isometric latitude.

Note: In all of the following
If d.phi is the difference of geodetic latitude in seconds from latitude 41 degrees south, positive northward, then the corresponding difference of isometric latitude (d.u) is given by a Maclaurin series with coefficients (for the International Spheroid) as follows:

	   d.u = 0.6399175073 * (d.phi * 10^-5)
               - 0.1358797613 * (d.phi * 10^-5)^2
                +0.063294409 * (d.phi * 10^-5)^3
                -0.025268530 * (d.phi * 10^-5)^4
                +0.0117879 * (d.phi * 10^-5)^5
                -0.0055161 * (d.phi * 10^-5)^6
                +0.0026906 * (d.phi * 10^-5)^7
                -0.001333 * (d.phi * 10^-5)^8
                +0.00067 * (d.phi * 10^-5)^9
                -0.00034 * (d.phi * 10^-5)^10
These coefficients give 10-figure accuracy for the range of +/-7 degrees from 41 degrees south.

Series defining the projection.

A conformal projection of the spheroid is given by a series in powers of the complex variable zeta, where
zeta = d.u + i(d.lambda)
d.u being the difference in isometric latitude obtained from the above series, and
d.lambda being the difference of longitude in radians from 173 degrees, positive eastward.

That is, coordinates from the true origin are given by

z = a(B1 * zeta + B2 * zeta^2 + B3 * zeta^3 + ...).

For the NZ Map Grid , the series is terminated after the sixth term, and constants are added to the coordinates, which are then given by

N + i(E) = N(origin) + i(E(origin)) + z

For the International Spheroid , the major semi-axis is given by

a = 6,378,388 metres,

and the coefficients of the series defining the NZ Map Grid are as follows, all except B1 being complex numbers. i.e the right hand figures are the imaginery (i) parts.

N(origin) + i(E(origin)) = 6023150.00 + i(2510000.00)
B1 = 0.7557853228
B2 = +0.249204646 and + i(0.003371507)
B3 = -0.001541739 and + i(0.041058560)
B4 = -0.10162907 and + i(0.01727609)
B5 = -0.26623489 and + i(-0.36249218)
B6 = -0.6870983 and + i(-1.1651967)

NZMG to Lat. and Long.

If
z = (N - N(origin) + i(E - E(origin))/a


then
Zeta = c1.z + c2.z^2 + c3.z^3 + ...
Although the direct formula is a six-term polynomial, the inverse formula is an infinite series.
The coefficients of the first six terms are
c1 = 1.3231270439
c2 = -0.577245789 and i(-0.007809598)
c3 = +0.508307513 and i(-0.1152208952)
c4 = -0.150947620 and i(+0.18200602)
c5 = +1.014181790 and i(+1.64497696)
c6 = +1.966054900 and i(+2.5127645)
The value of Zeta given by six terms of the inverse series is a first approximation from which a closer approximation can be obtained by

z + B2.zeta^2 + 2B3.zeta^3 + 3B4.zeta^4 + 4B5.zeta^5 + 5B6.zeta^6
zeta = __________________________________________________________________
B1 + 2B2.zeta + 3b3.zeta^2 + 4B4.zeta^3 + 5B5.zeta^4 + 6B6.zeta^5

A second application of this formula gives sufficient accuracy at any point within the land area of New Zealand.

When the final value of zeta is obtained, the latitude can be derived by from the following inverse series.

      d(phi) * 10^-5 = 1.5627014243 * d.u
                      +0.5185406398 * d.u^2
                      -0.03333098 * d.u^3
                      -0.1052906 * d.u^4
                      -0.0368594 * d.u^5
                      +0.007317 * d.u^6 
                      +0.01220 * d.u^7
                      +0.00394 * d.u^8
                      -0.0013 * d.u^9

Scale coefficient and convergence formulae are not included here.
And, if I ever get around to it, some examples will be here.

Back to last page.
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Or Go to Paul's GPS links page.