NZGeoMap©: A new mapping projection for New Zealand in the twentyfirst century
The following paper has been published in Survey Quarterly 25:2529, March 2001, and is reproduced here by permission of the publisher, the New Zealand Institute of Surveyors. This text has been prepared from the draft manuscript, and may differ in minor details from the published version (in particular, in the Abstract and the Introduction, the published version has an erroneous substitution of the abbreviation NZGD instead of NZMG). NZGeoMap^{©  }A New Mapping Projection for New Zealand in the Twentyfirst Century
W. Ian Reilly, 1/264 Riccarton Road, Christchurch 8004, New Zealand; ireilly_at_southern.co.nz
Abstract
NZGeoMap^{©}^{ }is a new mapping projection that
 is compatible with the New Zealand Geodetic Datum 2000 (NZGD2000);
 offers to map and GIS users a seamless transition from the existing New Zealand Map Grid (NZMG) at intermediate and small scales (1:10 000 or less);
 supports the continuation of mapping based on the existing 1:50 000 topographic series (NZMS260), and the continued utility of rasterscanned topographic data in NZMG;
 offers fewest problems of conversion to NZGD2000compatible projection coordinates from those based on the New Zealand Geodetic Datum 1949 (NZGD49), and avoids unnecessary cost to all those GIS users who would not then need to convert coordinates;
 allows the NZMG readout from lowaccuracy (~10 m) handheld GPS receivers to be treated as NZGeoMap^{©} coordinates;
 offers to all New Zealand map users and producers, particularly those working with largescale mapping, an alternative to proposed projections such as a New Zealandwide Transverse Mercator, which would spell an unnecessarily disruptive and costly break with current practice;
 can be supported by efficient algorithms to transform coordinates between NZGeoMap^{©} and any other projection compatible with NZGD2000.
Introduction
The adoption by Land Information New Zealand (LINZ) of a new geocentric reference system ˜ the New Zealand Geodetic Datum 2000 (NZGD2000) ˜ has sparked on ongoing debate about the need for, and the choice of, mapping projections that are compatible with the new system.
A change in mapping projections is forced by a difference in the size and shape of the reference ellipsoids between the old and the new reference systems adopted for New Zealand. The New Zealand Geodetic Datum 1949 (NZGD49) (Lee, 1978) was a twodimensional system essentially based on the International (Hayford) Ellipsoid 1924, which has an equatorial radius of a = 6 378 388 metres and a flattening f = 1/297. Its successor, NZGD2000, is a threedimensional system, supporting threedimensional geocentric Cartesian coordinates at its most basic, and has associated with it the reference ellipsoid of the Geodetic Reference System 1980 (GRS80), with an equatorial radius of a = 6 378 137 metres and a flattening f = 1/298.257 222 101 (see Jones & Blick, 2000).
The standard projection for topographic mapping in New Zealand for the past quarter century has been the New Zealand Map Grid Projection (NZMG), which was defined in terms of the geometric parameters of the International (Hayford) Ellipsoid 1924, the basis of the geographic coordinates ˜ latitude and longitude ˜ of NZGD49. The NZMG coordinates have also been widely used to store twodimensional position in GIS databases. Because of the change in ellipsoidal parameters, this projection is not compatible with NZGD2000. However, a similar projection can be designed to be compatible with NZGD2000, while giving a good fit to NZMG.
Meanwhile, LINZ has been conducting a campaign of consultation, and has stepped up its advocacy of a New Zealandwide transverse Mercator projection. The arguments in favour are summarised briefly in the LINZ publication Landscan 14 (October 2000), and at greater length in the "Robertson Report". In brief, the points made in these documents are that:
 NZMG (or a similar successor) is "nonstandard" and "complex", and a transverse Mercator projection is "standard", and by implication "simple";
 a few software companies refuse to include "nonstandard" projections in their GIS software;
 "Globalisation" will force us to eschew the "nonstandard";
 there are various (unspecified) "international" and "defence" obligations that require some form of "standard" projection.
I have already commented in print on these points (Reilly 1999, 2000). Point (4) is unsubstantiated; and the "defence" requirements, as far as they can be clearly identified, can be met by overprinting a Universal Transverse Mercator (UTM) grid on any suitable topographic map (such as NZMS260) for training purposes. If a foreign power wishes to invade New Zealand, then perhaps it should be left to sort out our topographic mapping system for itself. Points (1) and (3) are simply juggling with illdefined buzzwords. Point (2) is the only one with any real substance: but some software companies do make provision for NZMG (or other conformal projections similarly expressed), and the coding requirements are technically simple. That other companies don‚t, points to either a fundamental ignorance of their craft on their part, or to a degree of contempt for their clients. This is something that should and could be rectified, and it‚s up to buyers to shop around.
A New Zealandwide transverse Mercator projection (NZTM) was considered and rejected in the 1930s and again in the 1970s. However, it is up to the users of mapping and GIS systems to consider the alternatives, and to decide for themselves what sort of projection they should adopt in order to exploit the benefits of moving to NZGD2000, and how much disruption and expense they are prepared to accept for real ˜ or for illusory ˜ gains. The adoption of an NZTM will impose conversion costs on all users ˜ and predominantly on those in the private and local government sectors rather than on LINZ. The new projection NZGeoMap^{© }is being offered as a concrete alternative to NZTM, to avoid unnecessary costs and to meet the specific needs of users in the New Zealand environment.
The New Projection
The New Zealand GeoMapping Projection 2001^{©}, or NZGeoMap^{©} for short, has been devised in two steps:
 To compute an intermediate projection in exactly the same manner as for NZMG (Reilly 1971, 1973), but substituting the geometric parameters of the GRS80 ellipsoid for those of the International (Hayford) Ellipsoid 1924. The coefficients of the intermediate projection are calculated so as to minimise the mean square scale error over a design set of points at halfdegree intervals, as shown in Fig. 1;
 To transform the intermediate projection to achieve a close fit between the new projection coordinates and the NZMG coordinates over the same design set of points: the result is the NZGeoMap^{©} projection.
Fig. 1. The 228 points used for the minimisation of the mean squared scale error in the design of NZGeoMap^{©}. An identical set was used in the design of the New Zealand Map Grid Projection. (After Reilly, 1971).
The scale error of NZMG is shown in Fig. 2 (which will equally serve for NZGeoMap^{©}).
Fig. 2. Scale error of the New Zealand Map Grid Projection (and, equally, of NZGeoMap^{©}). (Reproduced by kind permission of Land Information New Zealand from its publication Landscan 14 (October 2000). The original was generated by Kim Ollivier at Ollivier and Co.)
A comparison between NZMG and NZGeoMap^{©} is given in Table 1.
Table 1 
NZMG and NZGeoMap^{©} compared over the 228 design points 
Quantity 
NZMG 
NZGeoMap^{©} 
Units 
Rootmeansquare scale error 
1.22 
1.22 
in units of 1.0e4 
Maximum scale error 
3.83 
3.82 
Minimum scale error 
3.16 
3.01 
Rootmeansquare discrepancy 
2.5 metres 
Greatest discrepancy 
5.6 metres at 34° S, 172° E  Notes to Table 1:
 The rootmeansquare scale error is identical to one part in a million.
 The comparison between NZMG and NZGeoMap^{©} coordinates was made at the 228 design points as specified by their geographic coordinates in NZGD49; these were transformed into geographic coordinates in NZGD2000 by using the sevenparameter BuršaWolf model given in Jones & Blick, 1999, p. 21 (Version 1.1 only; version 1.0 contains a serious error). The "discrepancy" is the linear distance between the two positions of the design point as indicated by the NZMG and the NZGeoMap^{©} coordinates respectively.
To discuss the parameters of the NZGeoMap^{©} projection, we need a little mathematics. We begin with a general point P on the reference ellipsoid, with geographic coordinates (j , l ), and define the isometric latitude y of P as
(1) 
y = arctanh( sin j )  e arctanh( e sin j ) 
where the eccentricity e of the ellipsoid is given by
If the projection has an origin P_{0} of (geographic) latitude j_{0} and longitude l_{0}, we further define the complex variable z as the ordered pair of isometric coordinates
(3) 
z = ( y  y_{0}, l  l_{0} ) 
to represent the position of P referred to the origin P_{0}.
We also denote the north x and east y projection coordinates (in metres) by the complex variable
The projection (for NZGeoMap^{©} as for NZMG) is then expressed in terms of z by the complex polynomial
(5) 
z = z_{0} + a å _{n=1..6} B_{n} z ^{n} 
where the B_{n} are the complex coefficients that essentially define the projection.
Notes to Equations (1) to (5)
 Complex variables are inherently associated with conformal projections. Any conformal projection can be expressed in the form of Equation (5), and any transformation between two conformal projections can be similarly expressed.. The technical term "complex" is perhaps an unfortunate one, for it carries the undeserved connotation of "complicated". In fact, the use of complex variables leads to quite simple and compact algorithms.
 A complex variable (a,b) can be defined as an ordered pair of real numbers a and b. The quantities a and b can be viewed as a pair of orthogonal Cartesian coordinates, which is one reason for their usefulness in map projections, and the symbol (a,b) is referred to as the Cartesian form of a complex variable (there is also a polar form which we shan‚t deal with here). The quantity a is often referred to as the "real" part, and b as the "imaginary" part of the Cartesian form, which is also unfortunate, as b is just as real as a.
 The rules of arithmetic for complex variables in the Cartesian form are:
Addition 
(a,b) + (c,d) 
= (a + c, b + d) 
Subtraction 
(a,b)  (c,d) 
= (a  c, b  d) 
Multiplication 
(a,b) ´ (c,d) 
= (a c ˆ b d, b c + a d) 
Division 
(a,b) ¸ (c,d) 
= [(ac + bd) / (c^{2} + d^{2})), ( ˆad + bc)/ (c^{2} + d^{2})] 
Table 2 gives a comparative listing of the parameters defining the ellipsoid and the origins of both NZMG and NZGeoMap^{©}.
Table 2 
Parameters of the NZMG and NZGeoMap^{©} projections 
Parameters 
NZMG 
NZGeoMap^{©} 
Reference ellipsoid 
International (Hayford) 
GRS80 
Equatorial radius a 
6 378 388 m 
6 378 137 m 
Flattening 1/f 
1/297 
1/298.257 222 101 
Geodetic reference system 
NZGD49 
NZGD2000 
Origin of projection j_{0}, l_{0} 
41°S, 173° E 
41°S, 173° E  Table 3 gives a comparative listing of the numerical coefficients of Equation (5) that define both NZMG and NZGeoMap^{©}.
Table 3 
Numerical coefficients of the NZMG and NZGeoMap^{©} projections 
Coefficients 
NZMG 
NZGeoMap^{©} 
z_{0} (m) 
(6023150.000, 2510000.000) 
(6022959.819, 2509988.157) 
B_{1} 
(0.7557853228, 0.0000000000) 
(0.7557882921, 0.0000015469) 
B_{2} 
(0.249204646, 0.003371507) 
(0.249189799, 0.003377608) 
B_{3} 
(0.001541739, 0.041058560) 
(0.001887583, 0.041046085) 
B_{4} 
(0.10162907, 0.01727609) 
(0.10219719, 0.01659788) 
B_{5} 
(0.26623489, 0.36249218) 
(0.25953965, 0.36362466) 
B_{6} 
(0.6870983, 1.1651967) 
(0.6399914, 1.1668540) 
Notes to Table 3:
 The nonrounded z_{0} values of NZGeoMap^{©} result from the adjustment to fit NZMG. Likewise, the nonˆzero second term of B_{1}, indicating nonzero convergence at the origin, is from the same cause: due to the displacement between the centroids of the two ellipsoids involved, the nominal origin of NZGeoMap^{©} has been physically displaced from that of NZMG.
 The coefficients z_{0} and B_{1}…B_{6} inclusive that pertain to NZGeoMap^{©} in Table 3 are declared to be the copyright of the author in terms of the Copyright Act 1994.
 The coefficients z_{0} and B_{1}…B_{6} inclusive that pertain to NZGeoMap^{©} in Table 3 are provisional only. Interested persons are invited to use this coefficient set for evaluation, and in particular to validate any claims made here for the projection; and are invited to send any comments or corrections to the author.
A New Zealand Transverse Mercator Projection
The viewpoint of LINZ, as expressed in the publication Landscan 14 (October 2000) referred to above, suggests the adoption of a single Transverse Mercator projection (NZTM), compatible with NZGD2000, to ultimately replace all the functions of NZMG with respect to NZGD49. It is of some interest, then, to make the sort of comparisons between NZMG and NZTM that we have already done between NZMG and NZGeoMap^{©}.
The suggested origin is the same as that for NZMG, with the scale factor set at 0.9996 (scale error of ˆ4.0e4) on the origin meridian. Using these parameters, one can calculate an approximation to the NZTM as proposed by LINZ. Its scale error is shown in Fig. 3.
Fig. 3. Scale error of the proposed New Zealand Transverse Mercator Projection. (Reproduced by kind permission of Land Information New Zealand from its publication Landscan 14 (October 2000). The original was generated by Kim Ollivier at Ollivier and Co.)
Table 4 gives a comparison between NZTM and NZMG, analogous to that of Table 1.
Table 4 
NZMG and NZTM compared over the 228 design points 
Quantity 
NZMG 
NZTM 
Units 
Rootmeansquare scale error 
1.22 
9.32 
in units of 1.0e4 
Maximum scale error 
3.83 
32.13 
Minimum scale error 
3.16 
4.00 
Rootmeansquare discrepancy 
415 metres 
Greatest discrepancy 
1447 metres at 34° S, 172° E 
Notes to Table 4:
 The rootmeansquare scale error of NZTM is 7.6 times that of NZMG, and of NZGeoMap^{©}.
 To give a fair comparison of the discrepancies, the origin coordinates z_{0} of NZGeoMap^{©} (Table 3) were assigned to NZTM. It should also be noted that otherwise there was, of course, no further attempt the achieve a close fit between NZTM and NZMG.
Recommendations
 The lack of a facility to handle NZMG (or any similar projection) in some commercial software is a technically trivial problem, and easy to fix. The underlying algorithm in Equation (5) is that of the summation of a complex polynomial, a matter of a few lines of code, in addition to a suite of complex arithmetic functions. The use of such an algorithm is not confined to NZMG or similar minimumscaleerror projections ˜ any conformal projection can be expressed in the form of Equation (5) (cf. Lee, 1974). Moreover, a loworder polynomial of the same form can be used for the fast and efficient transformation of coordinates between any two conformal projections ˜ between the TM coordinates of two adjoining meridional circuits, for example, or between meridional circuit coordinates and any national conformal projection. The essential algorithm is just too generally useful to be omitted.
 The introduction of the NZTM proposed by LINZ for "largescale urban, periurban, and related rural mapping" (to quote the Robertson Report), while NZMS260 continues in largely in its present form, will inevitably lead to the simultaneous currency of two incompatible national grid coordinate systems, with all the possibilities that that opens up for confusion in the emergency services. To dispatch a fire engine to a position that is 5 metres in error is no problem: to a position that is 1500 metres in error could be a disaster. The NZGeoMap^{©} projection recommended here is as suitable as NZTM for "largescale urban, periurban, and related rural mapping", while avoiding the confusion of two dissimilar coordinate systems at the level of the emergency services.
 The completion of the 1:50 000 NZMS260 topographic mapping of the whole country over the past quartercentury is undoubtedly the greatest achievement of Lands & Survey/DOSLI/LINZ in recent times. In the 1970s, the stimulus of metrication occasioned a significant advance in topographic mapping, with
 Completely new aerial photography and photogrammetry;
 Increase in map scale from 1:63 360 to 1:50 000;
 Decrease in contour interval from 100 feet to 20 metres;
 Introduction of a single national grid.
 No such improvements are on offer at the present time. The benefits of NZGD2000 can be captured without the forced obsolescence our topographic map series ˜ a change to the NZGeoMap^{©} projection in maps as they are revised will bring about only a shift in the latitudelongitude graticule around the margin.
Conclusions
A change of geodetic reference system from NZGD49 to NZGD2000 can be accomplished simply and at minimum cost. The NZGeoMap^{©}^{ }projection provides a smooth and elegant transition from NZMG coordinates that will capture all the benefits of the change in reference system. The large proportion of lower accuracy coordinates held by many private and local government agencies need not be converted, just relabelled. Nor will it be necessary to supplant the finest topographic map series that this country has ever produced. On the other hand, adoption of an NZTM, as proposed by LINZ, will impose maximum costs on everyone, and no brandishing of buzzwords such as "globalisation" and "standardisation" will conceal the illusory nature of the purported "benefits".
In a letter to the Editor in Survey Quarterly 22 (June 2000), Mr Stan Lusby announced that he had "downloaded nine maps from the web", was particularly pleased that he had "requested a specific place to be at the centre of each map", and was obviously impressed with the discovery that "There are no fixed edges to maps any more". He then draws the somewhat problematic conclusion that "this begins to throw a very different light on what will be needed in the way of map projections in the future".
If one looks at his source in expedia.com, the projection of the maps displayed does not feature at all ˜ it is the technology with which the maps are found, displayed, manipulated. If Mr Lusby looks at swisstopo.ch, he will find examples of what is probably the most advanced interactive cartographic product available ˜ the Atlas of Switzerland. He should know that, while the Swiss have adopted a modern geocentric reference system for some purposes, the maps he will see are on an Oblique Conformal Cylindrical Projection adopted in 1903, which is based on the Bessel Ellipsoid of 1841, itself displaced from the geocentre by about 790 metres (Schneider et al. 1999).
The Swiss example points the way to the evolving technology from which we can all hope to gain. The way forward does not begin by scrapping the existing topographic mapping, but by building on it. A radical change now to our national mapping coordinate system through the introduction of New Zealand Transverse Mercator would be a wasteful diversion of scarce time, money, and effort from the task achieving real technological advance in the presentation and manipulation of spatial data.
Acknowledgments
I thank Dr Hugh Bibby and Dr John Beavan of the Institute of Geological & Nuclear Sciences Ltd for reviewing the manuscript, and for assistance in providing Figure 1.
References
Jones, A., Blick, G., 2000: Where in the world are we? New Zealand Version 1.1, Office of the Surveyor General, PO Box 5501, Wellington. (The earlier Version 1.0 has a serious error in the transformation formula of Insert 7).
Lee, L. P. (1974). The computation of conformal projections. Survey Review 22: 245256.
Lee, L. P. (1978). Firstorder geodetic triangulation of New Zealand 190949 and 197374. Technical Series 1, Department of Lands & Survey, Wellington, 97 pp.
Reilly, W. I. (1971). Leastsquares determination of the coefficients of a conformal mapping transformation. Geophysics Division Report 64, Department of Scientific & Industrial Research, Wellington, 11 pp, 7 figs.
Reilly, W. I. (1973). A conformal mapping projection with minimum scale error. Survey Review 22: 5771.
Reilly, Ian, 1998: Mapping & geodetic coordinate systems: a background to change. Newsletter of the Geological Society of New Zealand 116: 1317 (also New Zealand Geophysical Society Newsletter 50: 1317; reprinted with additions in Survey Quarterly 16:1417, Dec 1998)
Reilly, Ian, 1999: Which way forward? Change and choice in mapping New Zealand. (with an Appendix "The origins of the New Zealand Map Grid Projection"). Survey Quarterly 20:3035.
Reilly, Ian, 2000: Mapping and coordinate systems ˆ an update. Newsletter of the Geological Society of New Zealand 122: 1316.
Schneider, D., Gubler, E., Marti, U., Gurtner, W., 1999: Aufbau der neuen Landesvermessung der Schweiz ŒLV95??? Teil 3: Terrestrische Bezugssysteme und Bezugsrahmen. Bericht 8, Bundesamt für Landestopographie, Wabern.
Biographical Note
W. Ian Reilly, B.A., B.Sc. (N.Z.), D.Sc.(Otago). A.O.S.M., Hon.M.N.Z.I.S., is a retired geophysicist who in 1971 devised the map projection that was later adopted as the basis of the New Zealand Map Grid. He has subsequently maintained an interest in issues of mapping and map projections.
