Transverse Mercator projection

When paired with a suitable geodetic datum, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.

The secant, ellipsoidal form of the transverse Mercator is the most widely applied of all projections for accurate large-scale maps.

For maps of smaller regions, an ellipsoidal model must be chosen if greater accuracy is required; see next section.

The ellipsoidal form of the transverse Mercator projection was developed by Carl Friedrich Gauss in 1822[5] and further analysed by Johann Heinrich Louis Krüger in 1912.

(There are other conformal generalisations of the transverse Mercator from the sphere to the ellipsoid but only Gauss-Krüger has a constant scale on the central meridian.)

[citation needed] 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.

[citation needed] 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.

In the secant version the lines of true scale on the projection are no longer parallel to central meridian; they curve slightly.

In his 1912[6] 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.

[8] It is constructed in terms of elliptic functions (defined in chapters 19 and 22 of the NIST[24] handbook) which can be calculated to arbitrary accuracy using algebraic computing systems such as Maxima.

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.

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.

For a tangent Normal Mercator projection the (unique) formulae which guarantee conformality are:[27] 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 k0 on the right hand side of all these equations: this ensures that the scale is equal to k0 on the equator.

The figure on the left shows how a transverse cylinder is related to the conventional graticule on 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.

Setting x = y′ and y = −x′ (and restoring factors of k0 to accommodate secant versions) The above expressions are given in Lambert[1] and also (without derivations) in Snyder,[13] Maling[28] and Osborne[27] (with full details).

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.

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.