The transverse Mercator map projection (TM, TMP) 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 Universal Transverse Mercator. 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 transverse Mercator projection is the transverse aspect of the standard (or Normal) 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 (of narrow width) 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.
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.^{[1]}^{[2]} (The text is also available in a modern English translation.^{[3]}) Lambert did not name his projections; the name transverse Mercator dates from the second half of the nineteenth century.^{[4]} The principal properties of the transverse projection are here presented in comparison with the properties of the normal projection.
Normal Mercator | Transverse Mercator | |||
---|---|---|---|---|
• | The central meridian projects to the straight line x = 0. Other meridians project to straight lines with x constant. | • | The central meridian projects to the straight line x = 0. Meridians 90 degrees east and west of the central meridian project to lines of constant y through the projected poles. All other meridians project to complicated curves. | |
• | The equator projects to the straight line y = 0 and parallel circles project to straight lines of constant y. | • | The equator projects to the straight line y = 0 but all other parallels are complicated closed curves. | |
• | Projected meridians and parallels intersect at right angles. | • | Projected meridians and parallels intersect at right angles. | |
• | The projection is unbounded in the y direction. The poles lie at infinity. | • | The projection is unbounded in the x direction. The points on the equator at ninety degrees from the central meridian are projected to infinity. | |
• | The projection is conformal. The shapes of small elements are well preserved. | • | The projection is conformal. The shapes of small elements are well preserved. | |
• | Distortion increases with y. The projection is not suited for world maps. Distortion is small near the equator and the projection (particularly in its ellipsoidal form) is suitable for accurate mapping of equatorial regions. | • | Distortion increases with x. The projection is not suited for world maps. Distortion is small near the central meridian and the projection (particularly in its ellipsoidal form) is suitable for accurate mapping of narrow regions. | |
• | Greenland is almost as large as Africa; the actual area is about one fourteenth that of Africa. | • | When Greenland and Africa are both near the central meridian, their shapes are good and the ratio of the areas is a good approximation to actual values. | |
• | The point scale factor is independent of direction. It is a function of y on the projection. (On the sphere it depends on latitude only.) The scale is true on the equator. | • | The point scale factor is independent of direction. It is a function of x on the projection. (On the sphere it depends on both latitude and longitude.) The scale is true on the central meridian. | |
• | The projection is reasonably accurate near the equator. Scale at an angular distance of 5° (in latitude) away from the equator is less than 0.4% greater than scale at the equator, and is about 1.54% greater at an angular distance of 10°. | • | The projection is reasonably accurate near the central meridian. Scale at an angular distance of 5° (in longitude) away from the central meridian is less than 0.4% greater than scale at the central meridian, and is about 1.54% at an angular distance of 10°. | |
• | In the secant version the scale is reduced on the equator and it is true on two lines parallel to the projected equator (and corresponding to two parallel circles on the sphere). | • | In the secant version the scale is reduced on the central meridian and it is true on two lines parallel to the projected central meridian. (The two lines are not meridians.) | |
• | Convergence (the angle between projected meridians and grid lines with x constant) is identically zero. Grid north and true north coincide. | • | Convergence is zero on the equator and non-zero everywhere else. It increases as the poles are approached. Grid north and true north do not coincide. | |
• | Rhumb lines (of constant azimuth on the sphere) project to straight lines. |
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.^{[6]}
The projection is known by several names: the (ellipsoidal) transverse Mercator in the US; Gauss conformal or Gauss–Krüger in Europe; or Gauss–Krüger transverse Mercator more generally. Other than just a synonym for the ellipsoidal transverse Mercator map projection, the term Gauss–Krüger may be used in other slightly different ways:
The projection is conformal with a constant scale on the central meridian. (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.) Throughout the twentieth century the Gauss–Krüger transverse Mercator was adopted, in one form or another, by many nations (and international bodies);^{[7]} 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.^{[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. This was proved to be untrue by British cartographer E. H. Thompson, whose unpublished exact (closed form) version of the projection, reported by Laurence Patrick Lee in 1976,^{[8]} showed that the ellipsoidal projection is finite (below). 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.^{[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. 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:
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 (to fourth order in n) by the following nations.
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]}^{[26]}
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 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 (where is the Earth radius) and is a prescribed function of . 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 k_{0} on the right hand side of all these equations: this ensures that the scale is equal to k_{0} on the equator.
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.
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:
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:^{[27]}
The direct formulae giving the Cartesian coordinates (x,y) follow immediately from the above. Setting x = y′ and y = −x′ (and restoring factors of k_{0} 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).
Inverting the above equations gives
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} = 0.9996 so that k = 1 when x is approximately 180 km. When x is approximately 255 km and k_{0} = 1.0004: the scale factor is within 0.04% of unity over a strip of about 510 km wide.
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^{[27]} either in terms of the geographical coordinates or in terms of the projection coordinates:
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 (and possibly north or south) of the grid origin: the coordinates relative to the false origin define eastings and northings which will always be positive. The false easting, E_{0}, is the distance of the true grid origin east of the false origin. The false northing, N_{0}, is the distance of the true grid origin north of the false origin. If the true origin of the grid is at latitude φ_{0} on the central meridian and the scale factor the central meridian is k_{0} 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.