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 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.(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. 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.||•||Greenland and Africa are both near to 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 1825 [ citation needed ]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. (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); 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 L. P. Lee in 1976, [ citation needed ]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.
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 1912paper, 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 Ensager and Poderand 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.
An exact solution by E. H. Thompson is described by L. P. Lee.It is constructed in terms of elliptic functions (defined in chapters 19 and 22 of the NIST handbook) 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 (2011).
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 x = aλ and y a prescribed function of φ. For a tangent Normal Mercator projection the (unique) 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 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. 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:
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)
The above expressions are given in Lambertand also (without derivations) in Snyder, Maling and Osborne (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 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 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 expressedeither 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, 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.
In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the geodetic latitude as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six auxiliary latitudes that are used in special applications.
The Mercator projection is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because it is unique in representing north as up and south as down everywhere while preserving local directions and shapes. The map is thereby conformal. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator but accelerates with increasing latitude to become infinite at the poles. So, for example, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.
A geographic coordinate system (GCS) is a coordinate system associated with positions on Earth. A GCS can give positions:
In navigation, a rhumb line, rhumb, or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north.
The use of orthographic projection in cartography dates back to antiquity. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective projection, in which the sphere is projected onto a tangent plane or secant plane. The point of perspective for the orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, where the horizon is a great circle. The shapes and areas are distorted, particularly near the edges.
The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.
The equirectangular projection (also called the equidistant cylindrical projection or la carte parallélogrammatique projection, and which includes the special case of the plate carrée projection, is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. The projection maps meridians to vertical straight lines of constant spacing, and circles of latitude to horizontal straight lines of constant spacing. The projection is neither equal area nor conformal. Because of the distortions introduced by this projection, it has little use in navigation or cadastral mapping and finds its main use in thematic mapping. In particular, the plate carrée has become a standard for global raster datasets, such as Celestia and NASA World Wind, because of the particularly simple relationship between the position of an image pixel on the map and its corresponding geographic location on Earth.
The sinusoidal projection is a pseudocylindrical equal-area map projection, sometimes called the Sanson–Flamsteed or the Mercator equal-area projection. Jean Cossin of Dieppe was one of the first mapmakers to use the sinusoidal, appearing in a world map of 1570.
The Universal Transverse Mercator (UTM) is a system for assigning coordinates to locations on the surface of the Earth. Like the traditional method of latitude and longitude, it is a horizontal position representation, which means it ignores altitude and treats the earth as a perfect ellipsoid. However, it differs from global latitude/longitude in that it divides earth into 60 zones and projects each to the plane as a basis for its coordinates. Specifying a location means specifying the zone and the x, y coordinate in that plane. The projection from spheroid to a UTM zone is some parameterization of the transverse Mercator projection. The parameters vary by nation or region or mapping system.
In cartography, a Tissot's indicatrix is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize local distortions due to map projection. It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the resulting diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map.
The Miller cylindrical projection is a modified Mercator projection, proposed by Osborn Maitland Miller in 1942. The latitude is scaled by a factor of 4⁄5, projected according to Mercator, and then the result is multiplied by 5⁄4 to retain scale along the equator. Hence:
The van der Grinten projection is a compromise map projection, which means that it is neither equal-area nor conformal. Unlike perspective projections, the van der Grinten projection is an arbitrary geometric construction on the plane. Van der Grinten projects the entire Earth into a circle. It largely preserves the familiar shapes of the Mercator projection while modestly reducing Mercator's distortion. Polar regions are subject to extreme distortion.
The Cassini projection is a map projection described by César-François Cassini de Thury in 1745. It is the transverse aspect of the equirectangular projection, in that the globe is first rotated so the central meridian becomes the "equator", and then the normal equirectangular projection is applied. Considering the earth as a sphere, the projection is composed of the operations:
In cartography, the cylindrical equal-area projection is a family of cylindrical, equal-area map projections.
The equidistant conic projection is a conic map projection commonly used for maps of small countries as well as for larger regions such as the continental United States that are elongated east-to-west.
The article Transverse Mercator projection restricts itself to general features of the projection. This article describes in detail one of the (two) implementations developed by Louis Krüger in 1912; that expressed as a power series in the longitude difference from the central meridian. These series were recalculated by Lee in 1946, by Redfearn in 1948, and by Thomas in 1952. They are often referred to as the Redfearn series, or the Thomas series. This implementation is of great importance since it is widely used in the U.S. State Plane Coordinate System, in national and also international mapping systems, including the Universal Transverse Mercator coordinate system (UTM). They are also incorporated into the Geotrans coordinate converter made available by the United States National Geospatial-Intelligence Agency. When paired with a suitable geodetic datum, the series deliver high accuracy in zones less than a few degrees in east-west extent.
This article is about the Bowring series of the transverse mercator. In 1989 Bernard Russel Bowring gave formulas for the Transverse Mercator that are simpler to program but retain millimeter accuracy.
The Eckert II projection is an equal-area pseudocylindrical map projection. In the equatorial aspect the network of longitude and latitude lines consists solely of straight lines, and the outer boundary has the distinctive shape of an elongated hexagon. It was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections. Within each pair, the meridians have the same shape, and the odd-numbered projection has equally spaced parallels, whereas the even-numbered projection has parallels spaced to preserve area. The pair to Eckert II is the Eckert I projection.
The rectangular polyconic projection is a map projection was first mentioned in 1853 by the U.S. Coast Survey, where it was developed and used for portions of the U.S. exceeding about one square degree. It belongs to the polyconic projection class, which consists of map projections whose parallels are non-concentric circular arcs except for the equator, which is straight. Sometimes the rectangular polyconic is called the War Office projection due to its use by the British War Office for topographic maps. It is not used much these days, with practically all military grid systems having moved onto conformal projection systems, typically modeled on the transverse Mercator projection.
The Nicolosi globular projection is a polyconic map projection invented about the year 1,000 by the Iranian polymath al-Biruni. As a circular representation of a hemisphere, it is called globular because it evokes a globe. It can only display one hemisphere at a time and so normally appears as a "double hemispheric" presentation in world maps. The projection came into use in the Western world starting in 1660, reaching its most common use in the 19th century. As a "compromise" projection, it preserves no particular properties, instead giving a balance of distortions.
|Wikimedia Commons has media related to Mercator projections .|