[This Guidance Note is available in its entirety below, but users are encouraged to download Guidance Note 7, in PDF format or in a MS Word 97-2000 & 6/95 - RTF format (zipped) for improved presentation.]

Guidance Note Number 7

POSC literature pertaining to

Coordinate conversions and coordinate transformations.

1. Map Projections and their Coordinate Conversion Formulas

1.1 Introduction

1.2 Identification of Projected Coordinate Reference Systems

3. Map Projection Parameters
4. Map Projection (Coordinate Conversion) Formulas
1. Lambert Conic Conformal
2. Lambert Conic Near-Conformal
3. Krovak Oblique Conic Conformal
4. Mercator
5. Cassini-Soldner
6. Transverse Mercator
7. Oblique Mercator
8. Stereographic
9. New Zealand Map Grid
10. Tunisia Mining Grid
11. American Polyconic
12. Lambert Azimuthal Equal Area
13. Albers Equal Area
14. Equidistant Cylindrical (Plate Carree)
15. Bonne

2. Formulas for Coordinate Operations other than Map Projections

1. Introduction
2. Coordinate Conversions other than Map Projections
1. Geographic/Geocentric Conversions
3. Coordinate Transformations
1. Offsets
2. Geocentric Translations
3. Molodenski Transformation
4. Helmert Transformation
5. Molodenski-Badedas Transformation
6. Interpolation Methods
4. Coordinate Operation Methods that can be Operations or Transformations
1. Polynomial Transformations
1. General case
2. Polynomial transformations with complex numbers
3. Polynomial transformation for Spain
2. Miscellaneous Linear Coordinate Transformations
1. Affine General Parametric Transformation
2. Affine General Geometric Transformation
3. Affine Orthogonal Geometric Transformation
4. Similarity Transformation

Part 1. Map Projections and their Coordinate Conversion formulas

1.1 Introduction

Setting aside the large number of map projection methods which may be employed for atlas maps, equally small scale illustrative exploration maps, and wall maps of the world or continental areas, EPSG provides reference parameter values for orthomorphic or conformal map projections which are used for medium or large scale topographic or exploration mapping. Here accurate positions are important and sometimes users may wish to scale accurate positions, distances or areas from the maps.

Small scale maps normally assume a spherical earth and the inaccuracies inherent in this assumption are of no consequence at the usual scale of these maps. For medium and large scale sheet maps, or maps and map projection coordinates held digitally to a high accuracy, it is essential that due regard is paid to the actual shape of the earth. Such coordinate reference systems are therefore invariably based on an ellipsoid and its derived map projections. The EPSG data set and this supporting conversion documentation considers only map projections for the ellipsoid.

Though not exhaustive the following list of named map projection methods are those which are most frequently encountered for medium and large scale mapping, some of them much less frequently than others since they are designed to serve only one particular country. They are grouped according to their possession of similar properties, which will be explained later. Except where indicated all are conformal.

Mercator Cylindrical

with one standard parallel

with two standard parallels

Cassini-Soldner (N.B. not conformal) Transverse Cylindrical

Transverse Mercator Group

Transverse Mercator (including south oriented version) Transverse Cylindrical

Universal Transverse Mercator

Gauss-Kruger

Gauss-Boaga

Oblique Mercator Group

Hotine Oblique Mercator Oblique Cylindrical

Oblique Mercator

Laborde Oblique Mercator

New Zealand Map Grid (May also be considered as type of Oblique Stereographic)

Space Oblique Mercator (Not strictly conformal)

Lambert Conical Conformal

with one standard parallel Conical

with two standard parallels

one standard parallel with truncated formulas (n.b. not conformal)

Stereographic Azimuthal

Polar

Oblique and equatorial

1.2. Identification of Map Projection method

If a map or coordinate list is provided for which an EPSG listed coordinate reference system is clearly identifiable, then its name or identifier together with EPSG dataset version number will address the required parameters including the coordinate conversion parameter values. If the coordinate reference system is not listed it will be necessary to create a new coordinate reference system with its own coordinate conversion (parameter) values.

It may often happen that one is presented with a coordinate list or map for which the author or compiler has regrettably failed to provide any indication of parameter values or properties:- no projection name, no grid definition and no statement of ellipsoid or datum. On the map there may be no grid or graticule, or indeed neither.

In order to adequately relate the digital or displayed map data to other data it is necessary to establish the properties of the data or given map from what may be gleaned from their appearance and other information. Geographical coordinates without qualifying information do not allow identification of the coordinate reference system other than that it is a geographic one. Projected or map grid coordinates may, by virtue of the actual and relative magnitudes of the Easting and Northing and knowledge of where in the world they relate to, provide clues as to the map projection. For example eastings between say 150000m and 850000m, allied with 6 or 7 figure northings correlated with latitude may indicate a UTM.

If the map bears neither grid nor graticule it will be useless unless one can identify a number of the point features shown for which one already has coordinate data. One may then be able to superimpose and fit a rectangular grid at appropriate scale from which other coordinate data may be read. If the map carries a grid then the numerical labelling of the grid lines, the assumption that it will be conformal or orthomorphic, and prior knowledge of approximately where in the world it covers may give some indication of the type of projection, but this may not be totally definitive. If the map bears a graticule the nature of the graticule lines will give some indication of the type of projection used in its compilation. For example straight meridians and concentric parallels would suggest a conical projection or, less frequently, a polar azimuthal. If the former, and assuming that it will be orthomorphic, then it will either be with one standard parallel or two and these will have been selected in relation to the latitudinal extent of the area, very possibly those in general use for that state's mapping. If the parallels are equally spaced it will be a simple equidistant conical projection. However for large scale mapping purposes the requirement that it is conformal will dictate that the parallels will not be equally spaced and it is more than likely that it will be some form of Lambert projection with either one or two standard parallels. Unfortunately there is no easy way of detecting which, nor the values of the standard parallels. The country it comes from and its national mapping system, if known, may suggest what these are. The EPSG data set will assist but is not exhaustive.

If both parallels of latitude and meridians are curved the projection has numerous possibilities but a form of Transverse Mercator may well be the most likely. One may attempt to identify the projection by computing the grid values of some of the graticule intersections for several possible projections in turn, plotting these to a rounded value for the estimated scale of the map e.g. 1:50000 or 1:100000, and attempting to fit the overlaid grid plot on the graticule. Repeating this for a number of potential projections for the area may be successful in obtaining a reasonable fit. But bear in mind that paper stretch may slightly distort scale from the nominal scale of the map, and the scale factor used in the graticule to grid conversions is another variable which may take only slightly different values e.g. a Gauss-Kruger takes a central meridian scale factor of unity while a UTM (like Gauss-Kruger, a Transverse Mercator) takes 0.9996.

Digital cartographic techniques make it relatively easy to plot grid and graticule for different projections with different parameters onto transparencies for "trial and error" overlays. The process can be time consuming so it is preferable to make maximum use of the clues which one may infer from the appearance of the map as initially presented, - its origins, its national area, and the conventional projections used for that area.

1.3. Map Projection parameters

A map projection grid is related to the geographical graticule of an ellipsoid through the definition of a coordinate conversion method and a set of parameters appropriate to that method. Differing conversion methods may require different parameters. Any one coordinate conversion method may take several different sets of associated parameter values, each set related to a particular map projection zone applying to a particular country or area of the world. Before setting out the formulas involving these parameters, which enable the coordinate conversions for the projection methods listed above, it is as well to understand the nature of these parameters.

The plane of the map and the ellipsoid surface may be assumed to have one particular point in common. This point is referred to as the natural origin. It is the point from which the values of both the geographical coordinates on the ellipsoid and the grid coordinates on the projection are deemed to increment or decrement for computational purposes. Alternatively it may be considered as the point which in the absence of application of false coordinates has grid coordinates of (0,0). For example, for projected coordinate reference systems using the Cassini-Soldner or Transverse Mercator methods, the natural origin is at the intersection of a chosen parallel and the chosen central meridian (see Figure 2 at end of section). The chosen parallel will frequently but not necessarily be the equator. For the stereographic projection the origin is at the centre of the projection where the plane of the map is imagined to be tangential to the ellipsoid.

Since the natural origin may be at or near the centre of the projection and under normal coordinate circumstances would thus give rise to negative coordinates over parts of the map, this origin is usually given false coordinates which are large enough to avoid this inconvenience. Hence each natural origin will normally have False Easting, FE and False Northing, FN values. For example, the false easting for the origins of all Universal Transverse Mercator zones is 500000m. As the UTM origin lies on the equator, areas north of the equator do not need and are not given a false northing but for mapping southern hemisphere areas the equator origin is given a false northing of 10,000,000m, thus ensuring that no point in the southern hemisphere will take a negative northing coordinate. Figure 4 illustrates the UTM arrangements.

These arrangements suggest that if there are false easting and false northing for the real or natural origin, there is also a Grid Origin which has coordinates (0,0). In general this point is of no consequence though its geographic position may be computed if needed. Sometimes however, rather than base the easting and northing coordinate reference system on the natural origin by giving it FE and FN values, it may be convenient to select a False Origin at a specific meridian/parallel intersection and attribute the false coordinates (0,0) or, more usually, EF and NF to this. The related easting and northing of the natural origin may then be computed if required.

The natural origin will always lie on a meridian of longitude. Longitudes are most commonly expressed relative to the Prime Meridian of Greenwich but some countries, particularly in former times, have preferred to relate their longitudes to a prime meridian through their national astronomic observatory, usually sited in or near their capital city, e.g. Paris for France, Bogota for Colombia. The meridian of the projection zone origin is known as the Longitude of Origin. For certain projection types it is often termed the Central Meridian or abbreviated as CM and provides the direction of the northing axis of the projected coordinate reference system.

Because of the steadily increasing distortion in the scale of the map with increasing distance from the origin, central meridian or other line on which the scale is the nominal scale of the projection, it is usual to limit the extent of a projection to within a few degrees of latitude or longitude of this point or line. Thus, for example, a UTM or other Transverse Mercator projection zone will normally extend only 2 or 3 degrees from the central meridian. For areas beyond this another zone of the projection, with a new origin and central meridian, needs to be used or created. The UTM system has a specified 60 numbered zones, each 6 degrees wide, covering the ellipsoid between the 84 degree North and 80 degree South latitude parallels. Other Transverse Mercator projection zones may be constructed with different central meridians, and different origins chosen to suit the countries or states for which they are used. A number of these are included in the EPSG dataset. Similarly a Lambert Conic Conformal zone distorts most rapidly in the north-south direction and may, as in Texas, be divided into latitudinal bands.

In order to further limit the scale distortion within the coverage of the zone or projection area, some projections introduce a scale factor at the origin (on the central meridian for Transverse Mercator projections), which has the effect of reducing the nominal scale of the map here and making it have the nominal scale some distance away. For example in the case of the UTM and some other Transverse Mercator projections a scale factor of slightly less than unity is introduced on the central meridian thus making it unity on two meridians either side of the central one, and reducing its departure from unity beyond these. The scale factor is a required parameter whether or not it is unity and is usually symbolised as k0.

Thus for projections in the Transverse Mercator group in section 1.1 above, the parameters which are required to completely and unambiguously define the projected coordinate system are:

Latitude of the natural origin

Longitude of the natural origin

Scale factor at the natural origin

False Easting

False Northing

Since the UTM zones obey set rules, it is sufficient to state only the UTM zone number (or central meridian). The remaining parameters from the above list are defined by the rules.

It has been noted that the Transverse Mercator projection is employed for the topographical mapping of longitudinal bands of territories, limiting the amount of scale distortion by limiting the extent of the projection either side of the central meridian. Sometimes the shape, general trend and extent of some countries makes it preferable to apply a single zone of the same kind of projection but with its central line aligned with the trend of the territory concerned rather than with a meridian. So, instead of a meridian forming this true scale central line for one of the various forms of Transverse Mercator, or the equator forming the line for the Mercator, a line with a particular azimuth traversing the territory is chosen, and the same principles of construction are applied to derive what is now an Oblique Mercator. This projection is sometimes referred to as the Hotine Oblique Mercator after the British geodesist who set out its formulas for application to Malaysian Borneo (East Malaysia) and also West Malaysia. Laborde had previously developed the projection system for Madagascar, and Switzerland uses a similar system derived by Rosenmund.

More recently (1974) Lee has derived formulas for a minimum scale factor projection for New Zealand known as the New Zealand Map Grid. The line of minimum scale follows the general alignment of the two main islands. This resembles an Oblique Mercator projection in its effect, but is not strictly an Oblique Mercator. The additional mathematical complexity of the projection enables its derivation via an Oblique Stereographic projection, which is sometimes the way it is classified. Because of its unique formulation inclusion of the New Zealand Map Grid within international mapping software was sporadic; as a consequence New Zealand has reverted to the frequently-encountered Transverse Mercator for its most recent mapping.

The parameters required to define an Oblique Mercator projection are:

Latitude of projection centre (the origin point on the initial line)

Longitude of projection centre

Azimuth of initial line at the projection centre

Scale factor at projection centre

Angle from rectified to skewed grid at the natural origin

and then either

False Easting (easting at the projection natural origin)

False Northing (northing at the projection natural origin)

or

The angle from rectified to skewed grid is normally applied at the natural origin of the projection, that is where the initial line of the projection intersects the aposphere. In some circumstances, for instance in the Alaskan panhandle State Plane zone, this angle is taken to be identical to the azimuth of the initial line at the projection centre. This results in grid and true north coinciding at the projection centre rather than at the natural origin as is more usual.

It is possible to define the azimuth of the initial line through the latitude and longitude of two widely spaced points along that line. This approach is not currently followed by POSC/EPSG.

For Conical map projections, which for the normal aspect may be considered as the projection of the ellipsoid onto an enveloping cone in contact with the ellipsoid along a parallel of latitude, the parallel of contact is known as a standard parallel and the scale is regarded as true along this parallel. Sometimes the cone is imagined to cut the ellipsoid with coincidence of the two surfaces along two standard parallels. All other parallels will be concentric with the chosen standard parallel or parallels but for the Lambert Conical Conformal will have varying separations to preserve the conformal property. All meridians will radiate with equal angular separations from the centre of the parallel circles but will be compressed from the 360 longitude degrees of the ellipsoid to a sector whose angular extent depends on the chosen standard parallel, - or both standard parallels if there are two. Of course the normal longitudinal extent of the projection will depend on the extent of the territory to be projected and will never approach 360 degrees.

As in the case of the Transverse Mercator above it is sometimes desirable to limit the maximum positive scale distortion for the one standard parallel case by distributing it more evenly over the extent of the mapped area. This may be achieved by introducing a scale factor on the standard parallel of slightly less than unity thus making it unity on two parallels either side of it. This achieves the same effect as choosing two specific standard parallels in the first place, on which the nominal scale will be preserved. The projection is then a Lambert Conical Conformal projection with two standard parallels. Although, strictly speaking, the scale on a standard parallel is always the nominal scale of the map and the scale factor on the one or two standard parallels should be unity, it is sometimes convenient to consider a Lambert Conical Conformal projection with one standard parallel yet which has a scale factor on the standard parallel of less than unity. This provision is allowed for by POSC/EPSG, where the single standard parallel is referred to as the latitude of the natural origin. For an ellipsoidal projection the natural origin will fall slightly poleward of the mean of the latitudes of the two standard parallels.

A longitude of origin or central meridian will again be chosen to bisect the area of the map or, more usually, the total national map area for the country or state concerned. Where this cuts the one standard parallel will be the natural origin of the projected coordinate system and, as for the Transverse Mercator, it will be given a False Easting and False Northing to ensure that there are no negative coordinates within the projected area (see Figure 5). Where two standard parallels are specified a false origin may be chosen at the intersection of a specific parallel with the central meridian outside the mapped area. This point will be given easting at false origin and northing at false origin to ensure that no negative coordinates will result. Figure 6 illustrates these arrangements.

It is clear that any number of Lambert projection zones may be formed according to which standard parallel or standard parallels are chosen and this is clearly exemplified by those which are used for many of the United States State Plane coordinate zones. They are normally chosen either, for one standard parallel, to approximately bisect the latitudinal extent of the country or area or, for two standard parallels, to embrace most of the latitudinal extent of the area. In the latter case the aim is to minimise the maximum scale distortion which will affect the mapped area and various formulas have been developed by different mathematicians to select the appropriate standard parallels to achieve this. Kavraisky was one mathematician who derived a recipe for choosing the standard parallels to achieve minimal scale distortion. But however the selection of the standard parallels is made the same projection formulas apply. Thus the parameters needed to specify a projection in the Lambert projected coordinate system group will be as follows:

For a Lambert Conical Conformal with one standard parallel (1SP),

Latitude of the natural origin (the Standard Parallel)

Longitude of the natural origin (the Central Meridian)

Scale factor at the natural origin (on the Standard Parallel)

False Easting

False Northing

For a Lambert Conical Conformal with two standard parallels (2SP),

Latitude of the first standard parallel

Latitude of the second standard parallel

Latitude of the false origin

Longitude of the false origin (the Central Meridian)

Easting at the false origin

Northing at the false origin

where the order of the standard parallels is not material if using the formulas which follow.

The limiting case of the Lambert Conic Conformal having the apex of the cone at infinity produces a cylindrical projection, the Mercator. Here, for the single standard parallel case the latitude of natural origin is the equator. For the two standard parallel case the two parallels have equal latitude in the north and south hemispheres. In both one and two standard parallel cases, grid coordinates are for the natural origin at the intersection of the equator and the central meridian (see figure 1). Thus the parameters needed to specify a projection in the Mercator projected coordinate system group will be:

For a Mercator with one standard parallel (1SP),

Latitude of the natural origin (the Equator)

Longitude of the natural origin (the Central Meridian)

Scale factor at the natural origin (on the Equator)

False Easting

False Northing

For a Mercator with two standard parallels (2SP),

Latitude of the first standard parallel

Longitude of the natural origin (the Central Meridian)

False Easting (grid coordinate at the intersection of the CM with the equator)

False Northing

In the formulas that follow the absolute value of the first standard parallel must be used.

For Azimuthal map projections, which are only infrequently used for ellipsoidal topographic mapping purposes, the natural origin will be at the centre of the projection where the map plane is imagined to be tangential to the ellipsoid and which will lie at the centre of the area to be projected. The central meridian will pass through the natural origin. This point will be given a False Easting and False Northing.

The parameters needed to specify the Stereographic map projection are:

Latitude of natural origin

Longitude of natural origin (the central meridian for the oblique case)

Scale Factor at the natural origin

False Easting at the natural origin

False Northing at the natural origin

1.4. Map Projection formulas

Only formulas for computation on the ellipsoid are considered. Projection formulas for the spherical earth are simpler but the spherical figure is inadequate to represent positional data with great accuracy at large map scales for the real earth. Projections of the sphere are only suitable for illustrative maps at scale of 1:1 million or less where precise positional definition is not critical.

The formulas which follow are largely adapted from "Map Projections - A Working Manual" by J.P.Snyder, published by the U.S. Geological Survey as Professional Paper No.1395. As well as providing an extensive overview of most map projections in current general use, and the formulas for their construction for both the spherical and ellipsoidal earth, this excellent publication provides computational hints and details of the accuracies attainable by the formulas. It is strongly recommended that all those who have to deal with map projections for medium and large scale mapping should follow its guidance.

There are a number of different formulas available in the literature for map projections other than those quoted by Snyder. Some are closed formulas; others, for ease of calculation, may depend on series expansions and their precision will generally depend on the number of terms used for computation. Generally those formulas which follow in this chapter will provide results which are accurate to within a decimetre, which is normally adequate for exploration mapping purposes. Coordinate expression and computations for engineering operations are usually consistently performed in grid terms.

The importance of one further variable should be noted. This is the unit of linear measurement used in the definition of projected coordinate systems. For metric map projections the unit of measurement is restricted to this unit. For non-metric map projections the metric ellipsoid semi-major axis needs to be converted to the projected coordinate system linear unit before use in the formulas below. The relevant ellipsoid is obtained through the datum part of the projected coordinate reference system.

In the formulas for map projections which follow, the basic ellipsoidal parameters are represented by symbols and derived as follows:

a is the ellipsoidal semi-major axis

b is the ellipsoidal semi-minor axis

f is the flattening of the ellipsoid where 1/f = a/(a - b)

e is the eccentricity of the ellipsoid where e2 = 2f - f2

e' is the second eccentricity where e'2 = e2 /(1 -e2)

r is the radius of curvature of the meridian at latitude j ,

where r = a(1 - e2)/(1 - e2sin2j )3/2

n is the radius of curvature on the prime vertical (i.e. perpendicular to the meridian) at

latitude j , where n = a /(1 - e2sin2j )1/2

j is the latitude of the point to be converted, positive if North and negative if South of the equator

l is the longitude of the point to be converted, positive if East and negative if West of the prime meridian

j 0 is the latitude of the natural origin

l 0 is the longitude of the natural origin (with respect to the prime meridian)

j F is the latitude of the false origin

l F is the longitude of the false origin (with respect to the prime meridian)

j 1 is the latitude of first standard parallel

j 2 is the latitude of second standard parallel

k0 is the scale factor at the natural origin

E is the Easting measured from the grid origin

N is the Northing measured from the grid origin

FE is the false easting, the Eastings value assigned to the natural origin

FN is the false northing, the Northings value assigned to the natural origin

EF is the Eastings value assigned to the false origin

NF is the Northings value assigned to the false origin

EC is the Eastings value assigned to the projection centre

NC is the Northings value assigned to the projection centre

Different formulas are require for forward and reverse map projection conversions: the forward formula cannot be used for the reverse conversion. However both forward and reverse formulas are explicitly given in the sections below as parts of a single conversion method. As such, map projection methods are described by EPSG as being reversible. Forward and reverse formulas for each conversion method use the projection parameters appropriate to that method with parameter values unchanged.

1.4.1. Lambert Conic Conformal

For territories with limited latitudinal extent but wide longitudinal width it may sometimes be preferred to use a single projection rather than several bands or zones of a Transverse Mercator. The Lambert Conic Conformal may often be adopted in these circumstances. But if the latitudinal extent is also large there may still be a need to use two or more zones if the scale distortion at the extremities of the one zone becomes too large to be tolerable.

Conical projections with one standard parallel are normally considered to maintain the nominal map scale along the parallel of latitude which is the line of contact between the imagined cone and the ellipsoid. For a one standard parallel Lambert the natural origin of the projected coordinate system is the intersection of the standard parallel with the longitude of origin (central meridian). See Figure 5 at end of section 1.3. To maintain the conformal property the spacing of the parallels is variable and increases with increasing distance from the standard parallel, while the meridians are all straight lines radiating from a point on the prolongation of the ellipsoid's minor axis.

Sometimes however, although a one standard parallel Lambert is normally considered to have unity scale factor on the standard parallel, a scale factor of slightly less than unity is introduced on this parallel. This is a regular feature of the mapping of some former French territories and has the effect of making the scale factor unity on two other parallels either side of the standard parallel. The projection thus, strictly speaking, becomes a Lambert Conic Conformal projection with two standard parallels. From the one standard parallel and its scale factor it is possible to derive the equivalent two standard parallels and then treat the projection as a two standard parallel Lambert conical conformal, but this procedure is seldom adopted. Since the two parallels obtained in this way will generally not have integer values of degrees or degrees minutes and seconds it is instead usually preferred to select two specific parallels on which the scale factor is to be unity, as for several State Plane Coordinate systems in the United States.

The choice of the two standard parallels will usually be made according to the latitudinal extent of the area which it is wished to map, the parallels usually being chosen so that they each lie a proportion inboard of the north and south margins of the mapped area. Various schemes and formulas have been developed to make this selection such that the maximum scale distortion within the mapped area is minimised, e.g. Kavraisky in 1934, but whatever two standard parallels are adopted the formulas are the same.

1.4.1.1 Lambert Conic Conformal (2SP) (EPSG coordinate operation method code 9802)

To derive the projected Easting and Northing coordinates of a point with geographical coordinates (j ,l ) the formulas for the Lambert Conic Conformal two standard parallel case (EPSG coordinate operation method code 9802) are:

Easting, E = EF + r sin q

Northing, N = NF + rF – r cos q

where m = cosj /(1 – e2sin2j )1/2 for m1, j 1, and m2, j 2 where j 1 and j 2 are the latitudes of the standard parallels

t = tan(p /4 – j /2)/[(1 – e sinj )/(1 + e sinj )]e/2 for t1, t2, tF and t using j 1,j 2,j F and j respectively

n = (ln m1 – ln m2)/(ln t1 – ln t2)

F = m1/(nt1n)

r = a F tn for rF and r, where rF is the radius of the parallel of latitude of the false origin

q = n(l – l 0)

The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:

j = p /2 – 2arctan{t'[(1 – esinj )/(1 + esinj )]e/2}

l = q '/n +l 0

where

r' = ± {(E – EF)2 + [rF – (N – NF)]2}1/2 , taking the sign of n

t' = (r'/(aF))1/n

q ' = arctan [(E – EF)/(rF – (N – NF))]

and n, F, and rF are derived as for the forward calculation.

Note that the formula for j requires iteration. First calculate t' and then a trial value for j using

j = π/2-2arctan t'. Then use the full equation for j substituting the trial value into the right hand side of the equation. Thus derive a new value for j . Iterate the process until j does not change significantly. The solution should quickly converge, in 3 or 4 iterations.

Example:

For Projected Coordinate Reference System NAD27 / Texas South Central

Parameters:

Ellipsoid Clarke 1866 a = 6378206.400 metres = 20925832.16 US survey feet 1/f = 294.97870

then e = 0.08227185, e2 = 0.00676866

First Standard Parallel j 1 28o23'00"N = 0.49538262 rad

Second Standard Parallel j 2 30o17'00"N = 0.52854388 rad

Latitude False Origin j F 27o50'00"N = 0.48578331 rad

Longitude False Origin l F 99o00'00"W = -1.72787596 rad

Easting at false origin EF 2000000.00 US survey feet

Northing at false origin NF 0.00 US survey feet

Forward calculation for:

Latitude j 28o30'00.00"N = 0.49741884 rad

Longitude l 96o00'00.00"W = -1.67551608 rad

first gives :

m1 = 0.88046050 m2 = 0.86428642

t = 0.59686306 tF = 0.60475101

t1 = 0.59823957 t2 = 0.57602212

n = 0.48991263 F = 2.31154807

r = 37565039.86 rF = 37807441.20

q = 0.02565177

Then Easting E = 2963503.91 US survey feet

Northing N = 254759.80 US survey feet

Reverse calculation for same easting and northing first gives:

q ' = 0.025651765 r' = 37565039.86

t' = 0.59686306

Then Latitude j = 28o30'00.000"N

Longitude l = 96o00'00.000"W

1.4.1.2 Lambert Conic Conformal (1SP) (EPSG coordinate operation method code 9801)

The formulas for the two standard parallel can be used for the Lambert Conic Conformal single standard parallel case (EPSG coordinate operation method code 9801) with minor modifications. Then

E = FE + r sinq

N = FN + r0 - r cosq , using the natural origin rather than the false origin.

where

n = sin j 0

r = aFtn k0 for r0, and r

t is found for t0, j 0 and t, j and m, F, and q are found as for the two standard parallel case.

The reverse formulas for j and l are as for the two standard parallel case above, with n, F and r0 as before and

q ' = arctan{(E – FE)/[r0 – (N – FN)]}

r' = ± {(E – FE)2 + [r0 – (N – FN)]2}1/2

t' = (r'/(ak0F))1/n

Example:

For Projected Coordinate Reference System JAD69 / Jamaica National Grid

Parameters:

Ellipsoid Clarke 1866 a = 6378206.400 m 1/f = 294.97870

then e = 0.08227185, e2 = 0.00676866

Latitude Natural Origin j o 18o 00'00" N = 0.31415927 rad

Longitude Natural Origin l o 77o 00'00" W = -1.34390352 rad

Scale factor at origin ko 1.000000

False Eastings FE 250000.00 m

False Northings FN 150000.00 m

Forward calculation for:

Latitude j = 17o 55' 55.80" N = 0.31297535 rad

Longitude l = 76o 56' 37.26" W = -1.34292061 rad

first gives

m0 = 0.95136402 t0 = 0.72806411

F = 3.39591092 n = 0.30901699

r = 19643955.26 r0 = 19636447.86

q = 0.00030374 t = 0.728965259

Then Easting E = 255966.58 m

Northing N = 142493.51 m

Reverse calculation for the same easting and northing first gives

q ' = 0.000303736

t' = 0.728965259

m0 = 0.95136402

r' = 19643955.26

Then Latitude j = 17o 55' 55.800" N

Longitude l = 76o 56' 37.260" W

1.4.1.3 Lambert Conic Conformal (West Orientated) (EPSG coordinate operation method code 9826)

In older mapping of Denmark and Greenland the Lambert Conic Conformal is used with axes positive north and west. To derive the projected Westing and Northing coordinates of a point with geographical coordinates (j , l ) the formulas are as for the standard Lambert Conic Conformal (1SP) case above (EPSG coordinate operation method code 9801) except for:

W = FE – r.sin(q )

In this formula the term FE retains its definition, i.e. in the Lambert Conic Conformal (West Orientated) method it increases the Westing value at the natural origin. In this method it is effectively false westing (FW).

The reverse formulas to derive the latitude and longitude of a point from its Westing and Northing values are as for the standard Lambert Conic Conformal (1SP) case except for:

q ' = arctan[(FE – W)/{r0 – (N – FN)}]

r' = +/-[(FE – W)² + {r0 – (N – FN)}²]^0.5

1.4.1.4 Lambert Conic Conformal (2 SP Belgium) (EPSG coordinate operation method code 9803)

In 1972, in order to retain approximately the same grid coordinates after a change of geodetic datum, a modified form of the two standard parallel case was introduced in Belgium. The formulas for the regular Lambert Conic Conformal (2SP) case given above are used except for:

Easting, E = EF + r sin (q – a)

Northing, N = NF + rF - r cos (q – a)

and for the reverse formulas

l = [(q ' + a)/n] + l 0

where a = 29.2985 seconds.

In 2000 this modification was replaced through use of the regular Lambert Conic Conformal (2 SP) map projection with appropriately modified parameter values.

Example:

For Projected Coordinate Reference System Belge l972 / Belge Lambert 72

Parameters:

Ellipsoid International 1924 a = 6378388 metres

1/f = 297

then e = 0.08199189, e2 = 0.006722670

First Standard Parallel j 1 49o50'00"N = 0.86975574 rad

Second Standard Parallel j 2 51o10'00"N = 0.89302680 rad

Latitude False Origin j F 90o00'00"N = 1.57079633 rad

Longitude False Origin l F 4o21'24.983"E = 0.07604294 rad

Easting at false origin EF 150000.01 metres

Northing at false origin NF 5400088.44 metres

Forward calculation for:

Latitude j = 50o40'46.4610"N = 0.88452540 rad

Longitude l = 5o48'26.533"E = 0.10135773 rad

first gives :

m1 = 0.64628304 m2 = 0.62834001

t = 0.59686306 tF = 0.00000000

t1 = 0.36750382 t2 = 0.35433583

n = 0.77164219 F = 1.81329763

r = 37565039.86 rF = 0.00

a= 0.00014204 q = 0.01953396

Then Easting E = 251763.20 metres

Northing N = 153034.13 metres

Reverse calculation for same easting and northing first gives:

q ' = 0.01939192 r' = 548041.03

t' = 0.35913403

Then Latitude j = 50o40'46.4610"N

Longitude l = 5o48'26.533"E

1.4.2. Lambert Conic Near-Conformal (EPSG coordinate operation method code 9817)

The Lambert Conformal Conic with one standard parallel formulas, as published by the Army Map Service, are still in use in several countries. The AMS uses series expansion formulas for ease of computation, as was normal before the electronic computer made such approximate methods unnecessary. Where the expansion series have been carried to enough terms the results are the same as the above formulas to the centimetre level. However in some countries the expansion formulas were truncated to the third order and the map projection is not fully conformal. The full formulas are used in Libya but from 1915 for France, Morocco, Algeria, Tunisia and Syria the truncated formulas were used. In 1943 in Algeria and Tunisia, from 1948 in France, from 1953 in Morocco and from 1973 in Syria the truncated formulas were replaced with the full formulas.

To compute the Lambert Conic Near-Conformal the following formulas are used:

E = FE + r sinq

N = FN + M + r sinq tan (q / 2) using the natural origin rather than the false origin.

Compute constants for the ellipse:

n = (a – b)/(a+b)

A' = a [ 1– n + 5 (n² – n³ ) / 4 + 81 ( n⁴ – n⁵ ) / 64]*p /180

B' = 3 a [ n – n² + 7 ( n³ – n⁴ ) / 8 + 55 n⁵ / 64] / 2

C' = 15 a [ n² – n³ + 3 ( n⁴ – n⁵ ) / 4 ] / 16

D' = 35 a [ n³ – n⁴ + 11 n⁵ / 16 ] / 48

E' = 315 a [ n⁴ – n⁵ ] / 512

Then compute the meridional arc from the equator to the parallel.

so = A' j o – B' sin 2j o + C' sin 4j o – D' sin 6j o + E' sin 8j o, where j o in the first term is in degrees

s = A' j – B' sin 2j + C' sin 4j – D' sin 6j + E' sin 8j , where j in the first term is in degrees

m = s – so

A = 1 / (6 r o n o)

M = ko ( m + Am³ + Bm⁴+ Cm⁵+ Dm⁶ ) This is the term that is truncated to the third order.

M_S = M per second of arc = M / ((j – j o) * 3600)

q = (l – l 0) sin j o

ro = ko n o / tan j o

r = ro – M

The reverse formulas for j and l from E and N with ro and M_S as above:

j = M’/ (M_S *3600) + j o where j o and j are in degrees

l = l 0 + q ‘ / sin j o where l o and l are in radians

where

X = E – FE Y = N – FN

q ' = arctan [X / (ro – Y)]

r' = X / sin q '

M' = r' – ro

Example:

For Projected Coordinate Reference System: Deir ez Zor / Levant Zone

Parameters:

Ellipsoid Clarke 1880 (IGN) a = 6378249.2 m 1/f = 293.46602

then b = 6356515.000 n = 0.001706682563

Latitude Natural Origin j o 34o 39'00" N = 0.604756586 rad

Longitude Natural Origin l o 37o 21'00" E = 0.651880476 rad

Scale factor at origin ko 0.99962560

False Eastings FE 300000.00 m

False Northings FN 300000.00 m

Forward calculation for:

Latitude j 37o 31' 17.625" N = 0.654874806 rad

Longitude l 34o 08' 11.291" E = 0.595793792 rad

first gives

A = 4.1067494 * 10^-15 A' =111131.8633

B' = 16300.64407 C' = 17.38751 D' = 0.02308 E' = 0.000033

so = 3835482.233 s = 4154101.458 m = 318619.225

M = 318632.72 M_S = 30.82262319

q = -0.03188875 ro = 9235264.405 r = 8916631.685

Then Easting E = 15707.96 m (c.f. E = 15708.00 using full formulas)

Northing N = 623165.96 m (c.f. N = 623167.20 using full formulas)

Reverse calculation for the same easting and northing first gives

q ' = -0.03188875

r' = 8916631.685

M' = 318632.72

Latitude j = 0.654874806 rad = 37o 31' 17.625" N

Longitude l = 0.595793792 rad = 34o 08' 11.291" E

1.4.3. Krovak Oblique Conformal Conic (EPSG coordinate operation method code 9819)

The normal case of the Lambert Conformal conic is for the axis of the cone to be coincident with the minor axis of the ellipsoid, that is the axis of the cone is normal to the ellipsoid at a geographic pole. For the Oblique Conformal Conic the axis of the cone is normal to the ellipsoid at a defined location and its extension cuts the minor axis at a defined angle. This map projection is used in the Czech Republic and Slovakia under the name ‘Krovak’ projection. The map projection method is similar in principle to the Oblique Mercator (see section 1.4.5). The geographic coordinates on the ellipsoid are first reduced to conformal coordinates on the conformal (Gaussian) sphere. These spherical coordinates are then projected onto the oblique cone and converted to grid coordinates. The pseudo standard parallel is defined on the conformal sphere after its rotation, to obtain the oblique aspect of the projection. It is then the parallel on this sphere at which the map projection is true to scale; on the ellipsoid it maps as a complex curve. A scale factor may be applied to the map projection to increase the useful area of coverage.

The defining parameters for the Krovak oblique conformal conic map projection are:

j _c = latitude of centre of the projection

l _c = longitude of centre of the projection

j ₁ = latitude of pseudo standard parallel

k_c = scale factor on the pseudo standard parallel

E_c = False Easting of the centre of the projection at the apex of the cone

N_c = False Northing of the centre of the projection at the apex of the cone

From these the following constants for the projection may be calculated :

B = ((1 + e² cos⁴(j _c)) / (1 – e² ))^0.5

A = a (1 – e² )^0.5 / ( 1 – e² sin² (j _c))

g ₀ = asin(sin (j _c) / B)

t₀ = tan(p / 4 + g ₀/ 2) . ((1 + e sin (j _c)) / (1 – e sin (j _c)))^e.B/2 / tan(p / 4 + j _c/ 2) ^B

n = sin j ₁

r0 = k_c A / tan (j ₁)

To derive the projected Easting and Northing coordinates of a point with geographical coordinates (j ,l ) the formulas for the oblique conic conformal are:

Easting: E = Ec + r cos q

Northing: N = Nc + r sinq

where

U = 2 (atan ( k tan^B(j / 2 + p / 4 ) ((1 + e sin (j )) / (1 – e sin (j )))^e.B/2 ) – p / 4)

V = B (l _c – _l )

S = asin ( cos (a _c) sin ( U ) + sin (a _c) cos (U) cos (V) )

D = asin ( cos ( U ) sin ( V ) / cos ( S ) )

q = n D

r = r0 tan(p / 4 + g ₀/ 2) / tan ⁿ ( S/2 + p / 4 )

Note that the terms Easting and Northing here refer to the two map grid ordinates. Their actual geographic direction depends upon the azimuth of the centre line.

The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:

j _j = 2*(atan(k^{-1/ B} tan ^{1/ B} ( U’/2 + p / 4 ) ((1 + e sin ( j _j-1) / ( 1 – e sin ( j _j-1)) ^e/2 ) – p / 4)

where j = 1,2 and the latitude is found by iteration.

l = l c– V’ / B

where

r' = [(E – EF)2 + (N – NF)2]1/2

q ' = arctan [(E – Ec)/(N – Nc)]

D' = q ' / sin ( j ₁)

S' = 2*(atan((r0 / r )^1/n tan(p / 4 + j ₀/ 2)) - p / 4)

U' = asin ( cos (a _c) sin (S') – sin (a _c) cos (S') cos (D') )

V' = asin ( cos (S') sin (D') / cos (U'))

Example:

For Projected Coordinate Reference System: S-JTSK (Ferro) / Krovak

N.B. Krovak projection uses Ferro as the prime meridian. This has a longitude with reference to Greenwich of 17 deg 40 min West. To apply the formulas the defining longitudes must be corrected to the Greenwich meridian.

Parameters:

Ellipsoid Bessel 1841 a = 6377397.155m 1/f = 299.15281

then e = 0.081696831 e2 = 0.006674372

Latitude of projection centre (j _c) 49o 30'00" N = 0.604756586 rad

Longitude of Origin 42o 30'00" East of Ferro

Longitude of Ferro is 17o 40'00" West of Greenwich

Longitude of Origin (l _c) 24o 50'00" East of Greenwich = 0.433423431 rad

Latitude of pseudo standard parallel (j ₁) 78o 30'00" N

Azimuth of centre line (a _c) 30o 17' 17.303"

Scale factor on pseudo standard parallel (ko) 0.99990

Easting at projection centre (E_c) 0.00 m

Northing at projection centre (N_c) 0.00 m

Projection constants:

B = 1.000597498

A = 6380703.61

g ₀ = 0.863239103

t₀ = 1.003419164

n = 0.979924705

r0 = 1298039.005

Forward calculation for:

Latitude j 50o 12' 32.4416" N = 0.876312566 rad

Longitude l 16o 50' 59.1790" E = 0.294083999 rad

Gives

U = 0.875596949

V = 0.139422687

S = 1.386275049

D = 0.506554623

q = 0.496385389

r = 1194731.014

Then "Easting" E = 1050538.643 m

"Northing" N = 568990.997 m

where "Easting" increases southwards and "Northing" increases westwards.

Reverse calculation for the same "Easting" and "Northing" gives

r' = 1194731.014

q ' = 0.496385389

D' = 0.506554623

S' = 1.386275049

U' = 0.875596949

V' = 0.139422687

j 1 = 0.876310601

j 2 = 0.876312560

j 3 = 0.876312566

Latitude j = 0.876312566 rad = 50o 12' 32.4416" N

Longitude l = 0.595793792 rad = 16o 50' 59.1790" E

1.4.4. Mercator (EPSG coordinate operation method codes 9804 and 9805)

The Mercator map projection is a special limiting case of the Lambert Conic Conformal map projection with the equator as the single standard parallel. All other parallels of latitude are straight lines and the meridians are also straight lines at right angles to the equator, equally spaced. It is the basis for the transverse and oblique forms of the projection. It is little used for land mapping purposes but is in almost universal use for navigation charts. As well as being conformal, it has the particular property that straight lines drawn on it are lines of constant bearing. Thus navigators may derive their course from the angle the straight course line makes with the meridians.

In the few cases in which the Mercator projection is used for terrestrial applications or land mapping, such as in Indonesia prior to the introduction of the Universal Transverse Mercator, a scale factor may be applied to the projection. This has the same effect as choosing two standard parallels on which the true scale is maintained at equal north and south latitudes either side of the equator.

The formulas to derive projected Easting and Northing coordinates are:

For the two standard parallel case, k0 is first calculated from

k0 = cosj 1/(1 – e2sin2j 1)1/2

where j 1 is the absolute value of the first standard parallel (i.e. positive).

Then, for both one and two standard parallel cases,

E = FE + ak0(l – l 0)

N = FN + ak0 ln{tan(p /4 + j /2)[(1 – esinj )/(1 + esinj )]e/2 }

where symbols are as listed above and logarithms are natural.

The reverse formulas to derive latitude and longitude from E and N values are:

j = c + (e2/2 + 5e4/24 + e6/12 + 13e8/360) sin(2c )

+ (7e4/48 + 29e6/240 + 811e8/11520) sin(4c )

+ (7e6/120 + 81e8/1120) sin(6c ) + (4279e8/161280) sin(8c )

where c = p /2 – 2 arctan t

t = B (FN-N)/(ak0⁾

B = base of the natural logarithm, 2.7182818...

and for the 2 SP Case, k0 is calculated as for the forward transformation above.

l = [(E – FE)/ak0] + l 0

Examples

1. Mercator (with two standard parallels) (EPSG coordinate operation method code 9805)

For Projected Coordinate Reference System Pulkovo 1942 / Mercator Caspian Sea

Parameters:

Ellipsoid Krassowski 1940 a = 6378245.00m 1/f = 298.300

then e = 0.08181333, e2 = 0.00669342

Latitude first SP j 0 42o00'00"N = 0.73303829 rad

Longitude Natural Origin l 0 51o00'00"E = 0.89011792 rad

False Eastings FE 0.00 m

False Northings (at equator) FN 0.00 m

then natural origin at latitude of 0oN has scale factor k0= 0.74426089

Forward calculation for:

Latitude j 53o00'00.00"N = 0.9250245 rad

Longitude l 53o00'00.00"E = 0.9250245 rad

gives Easting E = 165704.29 m

Northing N = 5171848.07 m

Reverse calculation for same easting and northing first gives :

t = 0.33639129

c = 0.92179596

Then Latitude j = 53o00'00.000"N

Longitude l = 53o00'00.000"E

2. Mercator (with the equator as standard parallel) (EPSG coordinate operation method code 9804)

For Projected Coordinate Reference System Makassar / NEIEZ

Parameters:

Ellipsoid Bessel 1841 a = 6377397.155 m 1/f = 299.15281

then e = 0.08169683

Latitude Natural Origin j o 00o00'00"N = 0.0000000 rad

Longitude Natural Origin l o 110o00'00"E = 1.91986218 rad

Scale factor ko 0.997

False Eastings FE 3900000.00 m

False Northings FE 900000.00 m

Forward calculation for:

Latitude j = 3o00'00.00"S = -0.05235988 rad

Longitude l = 120o00'00.00"E = 2.09439510 rad

gives

Easting E = 5009726.58 m

Northing N = 569150.82 m

Reverse calculation for same easting and northing first gives :

t = 1.0534121

c = -0.0520110

Then Latitude j = 3o00'00.000"S

Longitude l = 120o00'00.000"E

1.4.5. Cassini-Soldner (EPSG coordinate operation method code 9806)

The Cassini-Soldner projection is the ellipsoidal version of the Cassini projection for the sphere. It is not conformal but as it is relatively simple to construct it was extensively used in the last century and is still useful for mapping areas with limited longitudinal extent. It has now largely been replaced by the conformal Transverse Mercator which it resembles. Like this, it has a straight central meridian along which the scale is true, all other meridians and parallels are curved, and the scale distortion increases rapidly with increasing distance from the central meridian.

The formulas to derive projected Easting and Northing coordinates are:

Easting, E = FE + n [A – TA3/6 – (8 – T + 8C)TA5/120]

Northing, N = FN + M – M0 + n tanj [A2/2 + (5 – T + 6C)A4/24]

where A = (l – l 0)cosj

T = tan2j

C = e2 cos2j /(1 - e2)

n = a /(1 – e2sin2j )1/2

and M, the distance along the meridian from equator to latitude j , is given by

M = a[1 – e2/4 – 3e4/64 – 5e6/256 –....)j – (3e2/8 + 3e4/32 + 45e6/1024 +....)sin2j

+ (15e4/256 + 45e6/1024 +.....)sin4j – (35e6/3072 + ....)sin6j + .....]

with j in radians.

M0 is the value of M calculated for the latitude of the chosen origin. This may not necessarily be chosen as the equator.

To compute latitude and longitude from Easting and Northing the reverse formulas are:

j = j 1 – (n 1tanj 1/r 1)[D2/2 – (1 + 3T1)D4/24]

l = l 0 + [D – T1D3/3 + (1 + 3T1)T1D5/15]/cosj 1

where

n 1 = a /(1 – e2sin2j 1)1/2

r 1 = a(1 – e2)/(1 – e2sin2j 1)3/2

j 1 is the latitude of the point on the central meridian which has the same Northing as the point whose coordinates are sought, and is found from:

j 1 = m 1 + (3e1/2 – 27e13/32 +.....)sin2m 1 + (21e12/16 – 55e14/32 + ....)sin4m 1

+ (151e13/96 +.....)sin6m 1 + (1097e14/512 – ....)sin8m 1 + ......

where

e1 = [1 – (1 – e2)1/2]/[1 + (1 – e2)1/2]

m 1 = M1/[a(1 – e2/4 – 3e4/64 – 5e6/256 – ....)]

M1 = M0 + (N – FN)

T1 = tan2j 1

D = (E – FE)/n 1

For Projected Coordinate Reference System Trinidad 1903 / Trinidad Grid

Parameters:

Ellipsoid Clarke 1858 a = 20926348 ft = 31706587.88 links

b = 20855233 ft

then 1/f = 294.97870, e2 = 0.00676866

Latitude of natural origin j 0 10o26'30"N = 0.182241463 rad

Longitude of natural origin l 0 61o20'00"W = -1.07046861 rad

False Eastings FE 430000.00 links

False Northings FN 325000.00 links

Forward calculation for:

Latitude j 10o00'00.00" N = 0.17453293 rad

Longitude l 62o00'00.00"W = – 1.08210414 rad

A = -0.01145876 C = 0.00662550

T = 0.03109120 M = 5496860.24

n = 31709831.92 M0 = 5739691.12

Then Easting E = 66644.94 links

Northing N = 82536.22 links

Reverse calculation for same easting and northing first gives :

e1 = 0.00170207 D = -0.01145875

T1 = 0.03109544 M1 = 5497227.34

n 1 = 31709832.34 m 1 = 0.17367306

j 1 = 0.17454458 r 1 = 31501122.40

Then Latitude j = 10o00'00.000"N

Longitude l = 62o00'00.000"W

1.4.6. Transverse Mercator

1.4.6.1 General Case (EPSG coordinate operation method code 9807)

The Transverse Mercator projection in its various forms is the most widely used projected coordinate system for world topographical and offshore mapping. All versions have the same basic characteristics and formulas. The differences which distinguish the different forms of the projection which are applied in different countries arise from variations in the choice of values for the the coordinate conversion parameters, namely the latitude of the natural origin, the longitude of the natural origin (central meridian), the scale factor at the natural origin (on the central meridian), and the values of False Easting and False Northing, which embody the units of measurement, given to the origin. Additionally there are variations in the width of the longitudinal zones for the projections used in different territories.

The following table indicates the variations in the coordinate conversion parameters which distinguish the different forms of the Transverse Mercator projection and are used in the EPSG Transverse Mercator map projection operations:

The most familiar and commonly used Transverse Mercator in the oil industry is the Universal Transverse Mercator (UTM) whose natural origin lies on the equator. However, some territories use a Transverse Mercator with a natural origin at a latitude of natural origin closer to that territory.

In EPSG the coordinate conversion method is considered to be the same for all forms of the Transverse Mercator projection. The formulas to derive the projected Easting and Northing coordinates for the normal case (EPSG coordinate operation method code 9807) are in the form of a series as follows:

Northing, N = FN + k0{M – M0 + n tanj [A2/2 + (5 – T + 9C + 4C2)A4/24 +

(61 – 58T + T2 + 600C – 330e'2)A6/720]}

where T = tan2j

C = e2 cos2j /(1 – e2)

A = (l – l 0)cosj , with l and l 0 in radians

n = a /(1 – e2sin2j )1/2

M = a[(1 – e2/4 – 3e4/64 – 5e6/256 –....)j – (3e2/8 + 3e4/32 + 45e6/1024+....)sin2j

+ (15e4/256 + 45e6/1024 +.....)sin4j – (35e6/3072 + ....)sin6j + .....]

with j in radians and M0 for j 0, the latitude of the origin, derived in the same way.

The reverse formulas to convert Easting and Northing projected coordinates to latitude and longitude are:

j = j 1 – (n 1tanj 1/r 1)[D2/2 – (5 + 3T1 + 10C1 – 4C12 – 9e'2)D4/24

+ (61 + 90T1 + 298C1 + 45T12 – 252e'2 – 3C12)D6/720]

l = l 0 + [D – (1 + 2T1 + C1)D3/6 + (5 – 2C1 + 28T1 – 3C12 + 8e'2 + 24T12)D5/120] / cosj 1

where

n 1 = a /(1 – e2sin2j 1)1/2

r 1 = a(1 – e2)/(1 – e2sin2j 1)3/2

j 1 may be found as for the Cassini projection from:

j 1 = m 1 + (3e1/2 – 27e13/32 +.....)sin2m 1 + (21e12/16 – 55e14/32 + ....)sin4m 1

+ (151e13/96 +.....)sin6m 1 + (1097e14/512 – ....)sin8m 1 + ......

and where

e1 = [1 – (1 – e2)1/2]/[1 + (1 – e2)1/2]

m 1 = M1/[a(1 – e2/4 – 3e4/64 – 5e6/256 – ....)]

M1 = M0 + (N – FN)/k0

T1 = tan2j 1

C1 = e'2cos2j 1

e'2 = e2 /(1 – e2)

D = (E – FE)/(n 1k0)

For areas south of the equator the value of latitude j will be negative and the formulas above, to compute the E and N, will automatically result in the correct values. Note that the false northings of the origin, if the equator, will need to be large to avoid negative northings and for the UTM projection is in fact 10,000,000m. Alternatively, as in the case of Argentina's Transverse Mercator (Gauss-Kruger) zones, the origin is at the south pole with a northings of zero. However each zone central meridian takes a false easting of 500000m prefixed by an identifying zone number. This ensures that instead of points in different zones having the same eastings, every point in the country, irrespective of its projection zone, will have a unique set of projected system coordinates. Strict application of the above formulas, with south latitudes negative, will result in the derivation of the correct Eastings and Northings.

Similarly, in applying the reverse formulas to determine a latitude south of the equator, a negative sign for j results from a negative j 1 which in turn results from a negative M1.

For Projected Coordinate Reference System OSGB 1936 / British National Grid

Parameters:

Ellipsoid Airy 1830 a = 6377563.396 m 1/f = 299.32496

then e'2 = 0.00671534 e2 = 0.00667054

Latitude Natural Origin j o 49o00'00"N = 0.85521133 rad

Longitude Natural Origin l o 2o00'00"W = -0.03490659 rad

Scale factor ko 0.9996013

False Eastings FE 400000.00 m

False Northings FN – 100000.00 m

Forward calculation for:

Latitude j 50o30'00.00"N = 0.88139127 rad

Longitude l 00o30'00.00"E = 0.00872665 rad

A = 0.02775415 C = 0.00271699

T = 1.47160434 M = 5596050.46

M0 = 5429228.60 n = 6390266.03

Then Easting E = 577274.99 m

Northing N = 69740.50 m

Reverse calculations for same easting and northing first gives :

e1 = 0.00167322 m 1 = 0.87939562

M1 = 5599036.80 n 1 = 6390275.88

j 1 = 0.88185987 D = 0.02775243

r 1 = 6372980.21 C1 = 0.00271391

T1 = 1.47441726

Then Latitude j = 50o30'00.000"N

Longitude l = 00o30'00.000"E

1.4.6.2 Transverse Mercator Zoned Grid System (EPSG coordinate operation method code 9824)

When the growth in distortion away from the projection origin is of concern, a projected coordinate reference system cannot be used far from its origin. A means of creating a grid system over a large area but also limiting distortion is to have several grid zones with most defining parameters being made common. Coordinates throughout the system are repeated in each zone. To make coordinates unambiguous the easting is prefixed by the relevant zone number. This procedure was adopted by German mapping in the 1930’s through the Gauss-Kruger systems and later by American military mapping through the Universal Transverse Mercator (or UTM) grid system. (Note: subsequent civilian adoption of the systems usually ignores the zone prefix to easting. Where this is the case the formulas below do not apply: use the standard TM formula separately for each zone).

The parameter Longitude of Natural Origin (l 0) is changed from being a defining parameter to a derived parameter, replaced by two other defining parameters, the Initial Longitude (the western limit of zone 1) (l I) and the Zone Width (W). Each of the remaining four Transverse Mercator defining parameters – Latitude of Natural Origin, Scale factor at natural origin, False easting and False Northing – have the same parameter values in every zone.

The standard Transverse Mercator formulas above are modified as follows:

Zone number, Z, = int((Long + LongI + W) / W) with Long, LongI and W in degrees.

where (LongI) is the Initial Longitude of the zoned grid system

and W is the width of each zone of the zoned grid system.

If Long < 0, Long = (Long + 360) degrees.

Then,

Long0 = [Z * W] – [LongI + (W/2)]

For the forward calculation,

Easting, E = Z*10^6 + FE + k0.nu[A + (1 – T + C)A^3/6 + (5 – 18T + T^2 + 72C – 58e'^2)A^5/120]