Transverse Mercator
The transverse Mercator projection in its various forms is the most widely used projected coordinate system for world topographical and offshore mapping. It is a conformal projection in which a chosen meridian projects to a straight line at constant scale.
Classification |
Transverse and oblique cylindrical |
Available forms |
Forward and inverse, spherical and ellipsoidal |
Defined area |
Global, with full accuracy within 3900 km of the central meridian |
Alias |
tmerc |
Domain |
2D |
Input type |
Geodetic coordinates |
Output type |
Projected coordinates |
Usage
Prior to the development of the Universal Transverse Mercator coordinate system, several European nations demonstrated the utility of grid-based conformal maps by mapping their territory during the interwar period. Calculating the distance between two points on these maps could be performed more easily in the field (using the Pythagorean theorem) than was possible using the trigonometric formulas required under the graticule-based system of latitude and longitude. In the post-war years, these concepts were extended into the Universal Transverse Mercator/Universal Polar Stereographic (UTM/UPS) coordinate system, which is a global (or universal) system of grid-based maps.
The following table gives special cases of the Transverse Mercator projection.
Projection Name |
Areas |
Central meridian |
Zone width |
Scale Factor |
---|---|---|---|---|
Transverse Mercator |
World wide |
Various |
less than 1000 km |
Various |
Transverse Mercator south oriented |
Southern Africa |
2° intervals E of 11°E |
2° |
1.000 |
UTM North hemisphere |
World wide equator to 84°N |
6° intervals E & W of 3° E & W |
Usually 6°, wider for Norway and Svalbard |
0.9996 |
UTM South hemisphere |
World wide north of 80°S to equator |
6° intervals E & W of 3° E & W |
Always 6° |
0.9996 |
Gauss-Kruger |
Former USSR, Yugoslavia, Germany, S. America, China |
Various, according to area |
Usually less than 6°, often less than 4° |
1.0000 |
Gauss Boaga |
Italy |
Various, according to area |
6° |
0.9996 |
Example using Gauss-Kruger on Germany area (aka EPSG:31467)
$ echo 9 51 | proj +proj=tmerc +lon_0=9 +x_0=3500000 +ellps=bessel
3500000.00 5651505.56
Example using Gauss Boaga on Italy area (EPSG:3004)
$ echo 15 42 | proj +proj=tmerc +lon_0=15 +k_0=0.9996 +x_0=2520000 +ellps=intl
2520000.00 4649858.60
Parameters
Note
All parameters for the projection are optional.
- +approx
Added in version 6.0.0.
Use the Evenden-Snyder algorithm described below under "Legacy ellipsoidal form". It is faster than the default algorithm, but is less accurate and diverges beyond 3° from the central meridian.
- +algo=auto/evenden_snyder/poder_engsager
Added in version 7.1.
Selects the algorithm to use. The hardcoded value and the one defined in proj.ini default to
poder_engsager
; that is the most precise one.When using auto, a heuristics based on the input coordinate to transform is used to determine if the faster Evenden-Snyder method can be used, for faster computation, without causing an error greater than 0.1 mm (for an ellipsoid of the size of Earth)
- +lon_0=<value>
Central meridian/longitude of natural origin, longitude of origin or longitude of false origin (naming and meaning depend on the projection method).
Defaults to 0.0.
Note
The default convention is to interpret this value as decimal degrees. To specify radians instead, follow the value with the "r" character.
Example: +lon_0=1.570796r
See Projection Units for more information.
- +lat_0=<value>
Latitude of natural origin, latitude of false origin or latitude of projection centre (naming and meaning depend on the projection method).
Defaults to 0.0.
Note
The default convention is to interpret this value as decimal degrees. To specify radians instead, follow the value with the "r" character.
Example: +lat_0=1.570796r
See Projection Units for more information.
- +ellps=<value>
The name of a built-in ellipsoid definition.
See Ellipsoids for more information, or execute
proj -le
for a list of built-in ellipsoid names.Defaults to "GRS80".
- +R=<value>
Radius of the sphere, given in meters. If used in conjunction with
+ellps
,+R
takes precedence.See Ellipsoid size parameters for more information.
- +k_0=<value>
Scale factor. Determines scale factor used in the projection.
Defaults to 1.0.
- +x_0=<value>
False easting, easting at false origin or easting at projection centre (naming and meaning depend on the projection method). Always in meters.
Defaults to 0.0.
- +y_0=<value>
False northing, northing at false origin or northing at projection centre (naming and meaning depend on the projection method). Always in meters.
Defaults to 0.0.
Mathematical definition
The formulation given here for the Transverse Mercator projection is due to Krüger [Krueger1912] who gave the series expansions accurate to \(n^4\), where \(n = (a-b)/(a+b)\) is the third flattening. These series were extended to sixth order by Engsager and Poder in [Poder1998] and [Engsager2007]. This gives full double-precision accuracy within 3900 km of the central meridian (about 57% of the surface of the earth) [Karney2011tm]. The error is less than 0.1 mm within 7000 km of the central meridian (about 89% of the surface of the earth).
This formulation consists of three steps: a conformal projection from the ellipsoid to a sphere, the spherical transverse Mercator projection, rectifying this projection to give constant scale on the central meridian.
The scale on the central meridian is \(k_0\) and is set by +k_0
.
Option +lon_0
sets the central meridian; in the formulation
below \(\lambda\) is the longitude relative to the central meridian.
Options +lat_0
, +x_0
, and +y_0
serve to
translate the projected coordinates so that at \((\phi, \lambda) =
(\phi_0, \lambda_0)\), the projected coordinates are \((x,y) =
(x_0,y_0)\). To simplify the formulas below, these options are set to
zero (their default values).
Because the projection is conformal, the formulation is most conveniently given in terms of complex numbers. In particular, the unscaled projected coordinates \(\eta\) (proportional to the easting, \(x\)) and \(\xi\) (proportional to the northing, \(y\)) are combined into the single complex quantity \(\zeta = \xi + i\eta\), where \(i=\sqrt{-1}\). Then any analytic function \(f(\zeta)\) defines a conformal mapping (this follows from the Cauchy-Riemann conditions).
Spherical form
Because the full (ellipsoidal) projection includes the spherical projection as one of the components, we present the spherical form first with the coordinates tagged with primes, \(\phi'\), \(\lambda'\), \(\zeta' = \xi' + i\eta'\), \(x'\), \(y'\), so that they can be distinguished from the corresponding ellipsoidal coordinates (without the primes). The projected coordinates for the sphere are given by
Forward projection
Inverse projection
Ellipsoidal form
The projected coordinates are given by
The series for conversion between ellipsoidal and spherical geographic coordinates and ellipsoidal and spherical projected coordinates are given in matrix notation where \(\mathbf S(\theta)\) and \(\mathbf N\) are the row and column vectors of length 6
and \(\mathsf C_{\alpha,\beta}\) are upper triangular \(6\times6\) matrices.
Relation between geographic coordinates
Instead of using this analytical formula for \(\phi'\), the conversions between \(\phi\) and \(\phi'\) use the series approximations:
Here \(\phi'\) is the conformal latitude (sometimes denoted by \(\chi\)) and \(\mathsf C_{\chi,\phi}\) and \(\mathsf C_{\phi,\chi}\) are the coefficients in the trigonometric series for converting between \(\phi\) and \(\chi\).
Relation between projected coordinates
On the central meridian (\(\lambda = \lambda' = 0\)), \(\zeta' = \phi'\) is the conformal latitude \(\chi\) and \(\zeta\) plays the role of the rectifying latitude (sometimes denoted by \(\mu\)). \(\mathsf C_{\mu,\chi}\) and \(\mathsf C_{\chi,\mu}\) are the coefficients in the trigonometric series for converting between \(\chi\) and \(\mu\).
Legacy ellipsoidal form
The formulas below describe the algorithm used when giving the
+approx
option. They are originally from [Snyder1987],
but here quoted from [Evenden1995] and [Evenden2005]. These
are less accurate that the formulation above and are only valid within
about 5 degrees of the central meridian. Here \(M(\phi)\) is the
meridional distance.