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3.4.3 Projected coordinate system parameters

A projection grid is related to the geographical graticule of an ellipsoid through the definition of a projection transformation method and a set of parameters appropriate to that method. Differing methods may require different parameters. Any one coordinate transformation 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 transformations for the projections 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 (which in Epicentre is a vertex) is called 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 Cassini-Soldner or Transverse Mercator projected coordinate systems the natural origin is at the intersection of a chosen parallel and the chosen central meridian (Figures 2 and 3 below). The chosen parallel will frequently but not necessarily be the equator. For the stereographic projection the origin is at the center 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 center 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 500,000m. 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 position may be computed if needed. Sometimes however, rather than base the easting and northing coordinate 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 observatory, usually sited in 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 in Epicentre. 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 system. These two terms will be used interchangeably.

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 and 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 Epicentre. 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 symbolized as k0.

Thus for projections of the Transverse Mercator method, the parameters which are required to completely and unambiguously define the projected coordinate system are:

Longitude of the natural origin (Central Meridian)

Latitude of the natural origin

Scale factor at the natural origin (on the Central Meridian)

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. See Table 3-3 for more details.

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. 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.

The parameters required to define an Oblique Mercator projection are:

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

Longitude of projection center

Azimuth of initial line at the projection center

Scale factor at projection center

Angle from rectified to skewed grid at the natural origin

False Easting (easting at the projection natural origin)

False Northing (northing at the projection natural origin)

Easting at the projection center

Northing at the projection center

The angle from skewed to rectified 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 center. This results in grid and true north coinciding at the projection center rather than at the natural origin as is more natural.

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.

The Conical projections: When a cone is placed on an ellispoid, eith the conical axis corresponding to the polar axis of the ellipsoid, the cone will contact 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 Conic Conformal will have varying separations to preserve the conformal property. All meridians will radiate with equal angular separations from the center 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, 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 in Epicentre, 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 below). 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. See Figure 6 below for these arangements.

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 minimize the maximum scale distortion which will affect the mapped area. 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 One standard parallel Lambert Conical Conformal,

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 Two standard parallel Lambert Conical Conformal,

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, reprinted below.). Thus the parameters needed to specify a projection in the Mercator projected coordinate system group will be:

For a One standard parallel Mercator,

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 Two standard parallel Mercator,

Latitude of the first standard parallel

Longitude of the natural origin (the Central Meridian)

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

False Northing

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

For Azimuthal projections, which are only infrequently used for ellipsoidal topographic mapping purposes, the natural origin will be at the center of the projection where the map plane is imagined to be tangential to the ellipsoid and which will lie at the center 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 projected coordinate system 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

Table 3-2: Summary of parameters required to specify Projected Coordinate Systems

x

 * or two points defining the central line

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