Transverse Mercator projection

• Near the central meridian (Greenwich in the above example) the projection has low distortion and the shapes of Africa, western Europe, the British Isles, Greenland, and Antarctica compare favourably with a globe.
• The central regions of the transverse projections on sphere and ellipsoid are indistinguishable on the small-scale projections shown here.
• The meridians at 90° east and west of the chosen central meridian project to horizontal lines through the poles. The more distant hemisphere is projected above the north pole and below the south pole.
• The equator bisects Africa, crosses South America and then continues onto the complete outer boundary of the projection; the top and bottom edges and the right and left edges must be identified (i.e. they represent the same lines on the globe). (Indonesia is bisected.)
• Distortion increases towards the right and left boundaries of the projection but it does not increase to infinity. Note the Galapagos Islands where the 90° west meridian meets the equator at bottom left.
• The map is conformal. Lines intersecting at any specified angle on the ellipsoid project into lines intersecting at the same angle on the projection. In particular parallels and meridians intersect at 90°.
• The point scale factor is independent of direction at any point so that the shape of a small region is reasonably well preserved. The necessary condition is that the magnitude of scale factor must not vary too much over the region concerned. Note that while South America is distorted greatly the island of Ceylon is small enough to be reasonably shaped although it is far from the central meridian.
• The choice of central meridian greatly affects the appearance of the projection. If 90°W is chosen then the whole of the Americas is reasonable. If 145°E is chosen the Far East is good and Australia is oriented with north up.

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 (except on the equator) 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.

In his 1912[6] paper, Krüger presented two distinct solutions, distinguished here by the expansion parameter:

• Krüger–n (paragraphs 5 to 8): Formulae for the direct projection, giving the coordinates x and y, are fourth order expansions in terms of the third flattening, n (the ratio of the difference and sum of the major and minor axes of the ellipsoid). The coefficients are expressed in terms of latitude (φ), longitude (λ), major axis (a) and eccentricity (e). The inverse formulae for φ and λ are also fourth order expansions in n but with coefficients expressed in terms of x, y, a and e.
• Krüger–λ (paragraphs 13 and 14): Formulae giving the projection coordinates x and y are expansions (of orders 5 and 4 respectively) in terms of the longitude λ, expressed in radians: the coefficients are expressed in terms of φ, a and e. The inverse projection for φ and λ are sixth order expansions in terms of the ratio x/a, with coefficients expressed in terms of y, a and e. (See Transverse Mercator: Redfearn series.)

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.

• Lee–Redfearn–OSGB: In 1945, L. P. Lee[9] confirmed the λ expansions of Krüger and proposed their adoption by the OSGB[10] but Redfearn (1948)[11] pointed out that they were not accurate because of (a) the relatively high latitudes of Great Britain and (b) the great width of the area mapped, over 10 degrees of longitude. Redfearn extended the series to eighth order and examined which terms were necessary to attain an accuracy of 1 mm (ground measurement). The Redfearn series are still the basis of the OSGB map projections.[10]
• Thomas–UTM: The λ expansions of Krüger were also confirmed by Paul Thomas in 1952:[12] they are readily available in Snyder.[13] His projection formulae, completely equivalent to those presented by Redfearn, were adopted by the United States Defence Mapping Agency as the basis for the UTM.[14] They are also incorporated into the Geotrans[15] coordinate converter made available by the United States National Geospatial-Intelligence Agency .
• Other countries: The Redfearn series are the basis for geodetic mapping in many countries: Australia, Germany, Canada, South Africa to name but a few. (A list is given in Appendix A.1 of Stuifbergen 2009.)[16]
• Many variants of the Redfearn series have been proposed but only those adopted by national cartographic agencies are of importance. For an example of modifications which do not have this status see Transverse Mercator: Bowring series). All such modifications have been eclipsed by the power of modern computers and the development of high order n-series outlined below. The precise Redfearn series, although of low order, cannot be disregarded as they are still enshrined in the quasi-legal definitions of OSGB and UTM etc.

The Krüger–n series have been implemented (to fourth order in n) by the following nations.

• France[17]
• Finland[18]
• Sweden[19]
• Japan[20]

Higher order versions of the Krüger–n series have been implemented to seventh order by Engsager and Poder[21] and to tenth order by Kawase.[22] Apart from a series expansion for the transformation between latitude and conformal latitude, Karney has implemented the series to thirtieth order.[23]

An exact solution by E. H. Thompson is described by L. P. Lee.[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.[25] Such an implementation of the exact solution is described by Karney (2011).[23]

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 (in n) series is accurate to 5 nm within 3900 km of the central meridian.

The normal aspect of a tangent cylindrical projection of the sphere

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 ${\displaystyle x=a\lambda }$ (where ${\displaystyle a}$ is the Earth radius) and ${\displaystyle y}$ is a prescribed function of ${\displaystyle \phi }$. For a tangent Normal Mercator projection the (unique) formulae which guarantee conformality are:[26]

${\displaystyle x=a\lambda \,,\qquad y=a\ln \left[\tan \left({\frac {\pi }{4}}+{\frac {\varphi }{2}}\right)\right]={\frac {a}{2}}\ln \left[{\frac {1+\sin \varphi }{1-\sin \varphi }}\right].}$

Conformality implies that the point scale, k, is independent of direction: it is a function of latitude only:

${\displaystyle k(\varphi )=\sec \varphi .\,}$

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.

Transverse mercator 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' (E and W) 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.

Transverse mercator geometry

The position of an arbitrary point (φ,λ) on the standard graticule can also be identified in terms of angles on the rotated graticule: φ′ (angle M′CP) is an effective latitude and −λ′ (angle M′CO) becomes an effective longitude. (The minus sign is necessary so that (φ′,λ′) are related to the rotated graticule in the same way that (φ,λ) are related to the standard graticule). The Cartesian (x′,y′) axes are related to the rotated graticule in the same way that the axes (x,y) axes are related to the standard graticule.

The tangent transverse Mercator projection defines the coordinates (x′,y′) in terms of −λ′ and φ′ by the transformation formulae of the tangent Normal Mercator projection:

${\displaystyle x'=-a\lambda '\,\qquad y'={\frac {a}{2}}\ln \left[{\frac {1+\sin \varphi '}{1-\sin \varphi '}}\right].}$

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 (which include the equator) to infinite straight lines perpendicular to the central meridian. The true parallels and meridians (other than equator and central meridian) have no simple relation to the rotated graticule and they project to complicated curves.

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:[26]

${\displaystyle {\begin{aligned}\sin \varphi '&=\sin \lambda \cos \varphi ,\\\tan \lambda '&=\sec \lambda \tan \varphi .\end{aligned}}}$

The direct formulae giving the Cartesian coordinates (x,y) follow immediately from the above. Setting x = y′ and y = −x′ (and restoring factors of k₀ to accommodate secant versions)

${\displaystyle {\begin{aligned}x(\lambda ,\varphi )&={\frac {1}{2}}k_{0}a\ln \left[{\frac {1+\sin \lambda \cos \varphi }{1-\sin \lambda \cos \varphi }}\right],\\[5px]y(\lambda ,\varphi )&=k_{0}a\arctan \left[\sec \lambda \tan \varphi \right],\end{aligned}}}$

The above expressions are given in Lambert[1] and also (without derivations) in Snyder,[13] Maling[27] and Osborne[26] (with full details).

Inverting the above equations gives

${\displaystyle {\begin{aligned}\lambda (x,y)&=\arctan \left[\sinh {\frac {x}{k_{0}a}}\sec {\frac {y}{k_{0}a}}\right],\\[5px]\varphi (x,y)&=\arcsin \left[{\mbox{sech}}\;{\frac {x}{k_{0}a}}\sin {\frac {y}{k_{0}a}}\right].\end{aligned}}}$

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:

${\displaystyle {\begin{aligned}k(\lambda ,\varphi )&={\frac {k_{0}}{\sqrt {1-\sin ^{2}\lambda \cos ^{2}\varphi }}},\\[5px]k(x,y)&=k_{0}\cosh \left({\frac {x}{k_{0}a}}\right).\end{aligned}}}$

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 k₀ = 0.9996 so that k = 1 when x is approximately 180 km. When x is approximately 255 km and k₀ = 1.0004: the scale factor is within 0.04% of unity over a strip of about 510 km wide.

The angle of 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[26] either in terms of the geographical coordinates or in terms of the projection coordinates:

${\displaystyle {\begin{aligned}\gamma (\lambda ,\varphi )&=\arctan(\tan \lambda \sin \varphi ),\\[5px]\gamma (x,y)&=\arctan \left(\tanh {\frac {x}{k_{0}a}}\tan {\frac {y}{k_{0}a}}\right).\end{aligned}}}$