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.
Standard and transverse aspects
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: Both projections are cylindrical: for the normal Mercator, the axis of the cylinder coincides with the polar axis and the line of tangency with the equator. For the transverse Mercator, the axis of the cylinder lies in the equatorial plane, and the line of tangency is any chosen meridian, thereby designated the central meridian. Both projections may be modified to secant forms, which means the scale has been reduced so that the cylinder slices through the model globe. Both exist in spherical and ellipsoidal versions. Both projections are conformal, so that the point scale is independent of direction and local shapes are well preserved; Both projections have constant scale on the line of tangency (the equator for the normal Mercator and the central meridian for the transverse).
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.
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, Johann Heinrich. 1772. Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten. In Beyträge zum Gebrauche der Mathematik und deren Anwendung, part 3, section 6)Albert Wangerin (Editor), 1894. Ostwalds Klassiker der exakten Wissenschaften (54). Published by Wilhelm Engelmann. This is Lambert's paper with additional comments by the editor. Available at the University of Michigan Historical Math Library. (The text is also available in a modern English translation.) Lambert did not name his projections; the name transverse Mercator dates from the second half of the nineteenth century. This is an excellent survey of virtually all known projections from antiquity to 1993. The principal properties of the transverse projection are here presented in comparison with the properties of the normal projection.
Normal and transverse spherical projections
Normal MercatorTransverse Mercator •The central meridian projects to the straight line . Other meridians project to straight lines with constant. •The central meridian projects to the straight line . Meridians 90 degrees east and west of the central meridian project to lines of constant through the projected poles. All other meridians project to complicated curves. • The equator projects to the straight line and parallel circles project to straight lines of constant . •The equator projects to the straight line 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 direction. The poles lie at infinity. •The projection is unbounded in the 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 . 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 . 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 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 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 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.
Tissot's Indicatrix in transverse projections
The Mercator projection is a conformal transformation, independently of whether it is normal, oblique or transverse. This property is shown with an animation passing smoothly from the normal aspect (equatorial projection) to the transverse aspect of a spherical Mercator projection. The indicatrices remain circular everywhere on the map, though of different sizes as a result of the local scale changes.
During the transformation the central meridian is rotated from the -axis in the polar projection to the -axis in the transverse projection, as though the Earth were rotated inside the projection cylinder. In both cases cropping is applied symmetrically on the -axis, the cylinder axis, such that the map’s outline does not change. The equator of the Earth is highlighted as a thick green line. It extends to infinity in the transverse projection.
Ellipsoidal transverse Mercator The ellipsoidal form of the transverse Mercator projection was developed by Carl Friedrich Gauss in 1822Gauss, Karl Friedrich, 1825. "Allgemeine Auflösung der Aufgabe: die Theile einer gegebnen Fläche auf einer andern gegebnen Fläche so abzubilden, daß die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird" Preisarbeit der Kopenhagener Akademie 1822. Schumacher Astronomische Abhandlungen, Altona, no. 3, p. 5–30. [Reprinted, 1894, Ostwald's Klassiker der Exakten Wissenschaften, no. 55: Leipzig, Wilhelm Engelmann, p. 57–81, with editing by Albert Wangerin, pp. 97–101. Also in Herausgegeben von der Gesellschaft der Wissenschaften zu Göttingen in Kommission bei Julius Springer in Berlin, 1929, v. 12, pp. 1–9.] and further analysed by Johann Heinrich Louis Krüger in 1912.Krüger, L. (1912). Konforme Abbildung des Erdellipsoids in der Ebene. Royal Prussian Geodetic Institute, New Series 52.
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: Sometimes, the term is used for a particular computational method for transverse Mercator: that is, how to convert between latitude/longitude and projected coordinates. There is no simple closed formula to do so when the earth is modelled as an ellipsoid. But the Gauss–Krüger method gives the same results as other methods, at least if you are sufficiently near the central meridian: less than 100 degrees of longitude, say. Further away, some methods become inaccurate. The term is also used for a particular set of transverse Mercator projections used in narrow zones in Europe and South America, at least in Germany, Turkey, Austria, Slovenia, Croatia, Bosnia-Herzegovina, Serbia, Montenegro, North Macedonia, Finland and Argentina. This Gauss–Krüger system is similar to the universal transverse Mercator system, but the central meridians of the Gauss–Krüger zones are only 3° apart, as opposed to 6° in UTM.
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); The EEA recommends the transverse Mercator for conformal pan-European mapping at scales larger than 1:500,000. 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 (closed form) version of the projection, reported by Laurence Patrick Lee in 1976, Supplement No. 1 to The Canadian Cartographer 13. pp. 1–14, 92–101, and 107–114. 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.
Features 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.)