Transverse Mercator projection


The transverse Mercator map projection is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the UTM. When paired with a suitable geodetic datum, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.

Standard and transverse aspects

The transverse Mercator projection is the transverse aspect of the standard Mercator projection. They share the same underlying mathematical construction and consequently the transverse Mercator inherits many traits from the normal Mercator:
Since the central meridian of the transverse Mercator can be chosen at will, it may be used to construct highly accurate maps anywhere on the globe. The secant, ellipsoidal form of the transverse Mercator is the most widely applied of all projections for accurate large-scale maps.

Spherical transverse Mercator

In constructing a map on any projection, a sphere is normally chosen to model the Earth when the extent of the mapped region exceeds a few hundred kilometers in length in both dimensions. For maps of smaller regions, an ellipsoidal model must be chosen if greater accuracy is required; see next section. The spherical form of the transverse Mercator projection was one of the seven new projections presented, in 1772, by Johann Heinrich Lambert. Lambert did not name his projections; the name transverse Mercator dates from the second half of the nineteenth century. The principal properties of the transverse projection are here presented in comparison with the properties of the normal projection.

Normal and transverse spherical projections

Ellipsoidal transverse Mercator

The ellipsoidal form of the transverse Mercator projection was developed by Carl Friedrich Gauss in 1825 and further analysed by Johann Heinrich Louis Krüger in 1912. The projection is known by several names: Gauss Conformal or Gauss-Krüger in Europe; the transverse Mercator in the US; or Gauss–Krüger transverse Mercator generally. The projection is conformal with a constant scale on the central meridian. Throughout the twentieth century the Gauss–Krüger transverse Mercator was adopted, in one form or another, by many nations ; in addition it provides the basis for the Universal Transverse Mercator series of projections. The Gauss–Krüger projection is now the most widely used projection in accurate large-scale mapping.
The projection, as developed by Gauss and Krüger, was expressed in terms of low order power series which were assumed to diverge in the east-west direction, exactly as in the spherical version. This was proved to be untrue by British cartographer E. H. Thompson, whose unpublished exact version of the projection, reported by L. P. Lee in 1976, showed that the ellipsoidal projection is finite. This is the most striking difference between the spherical and ellipsoidal versions of the transverse Mercator projection: Gauss–Krüger gives a reasonable projection of the whole ellipsoid to the plane, although its principal application is to accurate large-scale mapping "close" to the central meridian.

Features

In most applications the Gauss–Krüger coordinate system is applied to a narrow strip near the central meridians where the differences between the spherical and ellipsoidal versions are small, but nevertheless important in accurate mapping. Direct series for scale, convergence and distortion are functions of eccentricity and both latitude and longitude on the ellipsoid: inverse series are functions of eccentricity and both x and y on the projection. In the secant version the lines of true scale on the projection are no longer parallel to central meridian; they curve slightly. The convergence angle between projected meridians and the x constant grid lines is no longer zero so that a grid bearing must be corrected to obtain an azimuth from true north. The difference is small, but not negligible, particularly at high latitudes.

Implementations of the Gauss–Krüger projection

In his 1912 paper, Krüger presented two distinct solutions, distinguished here by the expansion parameter:
The Krüger–λ series were the first to be implemented, possibly because they were much easier to evaluate on the hand calculators of the mid twentieth century.
The Krüger–n series have been implemented by the following nations.
Higher order versions of the Krüger–n series have been implemented to seventh order by Ensager and Poder and to tenth order by Kawase. Apart from a series expansion for the transformation between latitude and conformal latitude, Karney has implemented the series to thirtieth order.

Exact Gauss–Krüger and accuracy of the truncated series

An exact solution by E. H. Thompson is described by L. P. Lee. It is constructed in terms of elliptic functions which can be calculated to arbitrary accuracy using algebraic computing systems such as Maxima. Such an implementation of the exact solution is described by Karney.
The exact solution is a valuable tool in assessing the accuracy of the truncated n and λ series. For example, the original 1912 Krüger–n series compares very favourably with the exact values: they differ by less than 0.31 μm within 1000 km of the central meridian and by less than 1 mm out to 6000 km. On the other hand, the difference of the Redfearn series used by Geotrans and the exact solution is less than 1 mm out to a longitude difference of 3 degrees, corresponding to a distance of 334 km from the central meridian at the equator but a mere 35 km at the northern limit of an UTM zone. Thus the Krüger–n series are very much better than the Redfearn λ series.
The Redfearn series becomes much worse as the zone widens. Karney discusses Greenland as an instructive example. The long thin landmass is centred on 42W and, at its broadest point, is no more than 750 km from that meridian while the span in longitude reaches almost 50 degrees. Krüger–n is accurate to within 1 mm but the Redfearn version of the Krüger–λ series has a maximum error of 1 kilometre.
Karney's own 8th-order series is accurate to 5 nm within 3900 km of the central meridian.

Formulae for the spherical transverse Mercator

Spherical normal Mercator revisited

The normal cylindrical projections are described in relation to a cylinder tangential at the equator with axis along the polar axis of the sphere. The cylindrical projections are constructed so that all points on a meridian are projected to points with x = aλ and y a prescribed function of φ. For a tangent Normal Mercator projection the formulae which guarantee conformality are:
Conformality implies that the point scale, k, is independent of direction: it is a function of latitude only:
For the secant version of the projection there is a factor of k on the right hand side of all these equations: this ensures that the scale is equal to k on the equator.

Normal and transverse graticules

The figure on the left shows how a transverse cylinder is related to the conventional graticule on the sphere. It is tangential to some arbitrarily chosen meridian and its axis is perpendicular to that of the sphere. The x- and y-axes defined on the figure are related to the equator and central meridian exactly as they are for the normal projection. In the figure on the right a rotated graticule is related to the transverse cylinder in the same way that the normal cylinder is related to the standard graticule. The 'equator', 'poles' and 'meridians' of the rotated graticule are identified with the chosen central meridian, points on the equator 90 degrees east and west of the central meridian, and great circles through those points.
The position of an arbitrary point on the standard graticule can also be identified in terms of angles on the rotated graticule: φ′ is an effective latitude and −λ′ becomes an effective longitude. are related to the rotated graticule in the same way that. The Cartesian axes are related to the rotated graticule in the same way that the axes axes are related to the standard graticule.
The tangent transverse Mercator projection defines the coordinates in terms of −λ′ and φ′ by the transformation formulae of the tangent Normal Mercator projection:
This transformation projects the central meridian to a straight line of finite length and at the same time projects the great circles through E and W to infinite straight lines perpendicular to the central meridian. The true parallels and meridians have no simple relation to the rotated graticule and they project to complicated curves.

The relation between the graticules

The angles of the two graticules are related by using spherical trigonometry on the spherical triangle NM′P defined by the true meridian through the origin, OM′N, the true meridian through an arbitrary point, MPN, and the great circle WM′PE. The results are:

Direct transformation formulae

The direct formulae giving the Cartesian coordinates follow immediately from the above. Setting x = y′ and y = −x′
The above expressions are given in Lambert and also in Snyder, Maling and Osborne.

Inverse transformation formulae

Inverting the above equations gives

Point scale

In terms of the coordinates with respect to the rotated graticule the point scale factor is given by k = sec φ′: this may be expressed either in terms of the geographical coordinates or in terms of the projection coordinates:
The second expression shows that the scale factor is simply a function of the distance from the central meridian of the projection. A typical value of the scale factor is k = 0.9996 so that k = 1 when x is approximately 180 km. When x is approximately 255 km and k = 1.0004: the scale factor is within 0.04% of unity over a strip of about 510 km wide.

Convergence

The convergence angle γ at a point on the projection is defined by the angle measured from the projected meridian, which defines true north, to a grid line of constant x, defining grid north. Therefore, γ is positive in the quadrant north of the equator and east of the central meridian and also in the quadrant south of the equator and west of the central meridian. The convergence must be added to a grid bearing to obtain a bearing from true north. For the secant transverse Mercator the convergence may be expressed either in terms of the geographical coordinates or in terms of the projection coordinates:

Formulae for the ellipsoidal transverse Mercator

Details of actual implementations
The projection coordinates resulting from the various developments of the ellipsoidal transverse Mercator are Cartesian coordinates such that the central meridian corresponds to the x axis and the equator corresponds to the y axis. Both x and y are defined for all values of λ and ϕ. The projection does not define a grid: the grid is an independent construct which could be defined arbitrarily. In practice the national implementations, and UTM, do use grids aligned with the Cartesian axes of the projection, but they are of finite extent, with origins which need not coincide with the intersection of the central meridian with the equator.
The true grid origin is always taken on the central meridian so that grid coordinates will be negative west of the central meridian. To avoid such negative grid coordinates, standard practice defines a false origin to the west of the grid origin: the coordinates relative to the false origin define eastings and northings which will always be positive. The false easting, E0, is the distance of the true grid origin east of the false origin. The false northing, N0, is the distance of the true grid origin north of the false origin. If the true origin of the grid is at latitude φ0 on the central meridian and the scale factor the central meridian is k0 then these definitions give eastings and northings by:
The terms "eastings" and "northings" do not mean strict east and north directions. Grid lines of the transverse projection, other than the x and y axes, do not run north-south or east-west as defined by parallels and meridians. This is evident from the global projections shown above. Near the central meridian the differences are small but measurable. The difference between the north-south grid lines and the true meridians is the angle of convergence.