Transverse Mercator

The transverse Mercator projection in its various forms is the most widely used projected coordinate system for world topographical and offshore mapping.


Transverse and oblique cylindrical

Available forms

Forward and inverse, spherical and ellipsoidal

Defined area

Global, but reasonably accurate only within 15 degrees of the central meridian





Input type

Geodetic coordinates

Output type

Projected coordinates

Transverse Mercator

proj-string: +proj=tmerc


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


Central meridian

Zone width

Scale Factor

Transverse Mercator

World wide


less than 6°


Transverse Mercator south oriented

Southern Africa

2° intervals E of 11°E


UTM North hemisphere

World wide equator to 84°N

6° intervals E & W of 3° E & W

Always 6°


UTM South hemisphere

World wide north of 80°S to equator

6° intervals E & W of 3° E & W

Always 6°



Former USSR, Yugoslavia, Germany, S. America, China

Various, according to area

Usually less than 6°, often less than 4°


Gauss Boaga


Various, according to area


Example using Gauss-Kruger on Germany area (aka EPSG:31467)

$ echo 9 51 | proj +proj=tmerc +lat_0=0 +lon_0=9 +k_0=1 +x_0=3500000 +y_0=0 +ellps=bessel +units=m
3500000.00  5651505.56

Example using Gauss Boaga on Italy area (EPSG:3004)

$ echo 15 42 | proj +proj=tmerc +lat_0=0 +lon_0=15 +k_0=0.9996 +x_0=2520000 +y_0=0 +ellps=intl +units=m
2520000.00  4649858.60



All parameters for the projection are optional.


New in version 6.0.0.

Use the algorithm described in section “Ellipsoidal Form” below. It is faster than the default algorithm, but also diverges faster as the distance from the central meridian increases.


New 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)

Note that +approx and +algo are mutually exclusive.


Longitude of projection center.

Defaults to 0.0.


Latitude of projection center.

Defaults to 0.0.


See proj -le for a list of available ellipsoids.

Defaults to “GRS80”.


Radius of the sphere given in meters. If used in conjunction with +ellps +R takes precedence.


Scale factor. Determines scale factor used in the projection.

Defaults to 1.0.


False easting.

Defaults to 0.0.


False northing.

Defaults to 0.0.

Mathematical definition

The formulas describing the Transverse Mercator below are quoted from Evenden’s [Evenden2005].

\(\phi_0\) is the latitude of origin that match the center of the map. It can be set with +lat_0.

\(k_0\) is the scale factor at the natural origin (on the central meridian). It can be set with +k_0.

\(M(\phi)\) is the meridional distance.

Spherical form

Forward projection

\[B = \cos \phi \sin \lambda\]
\[x = \frac{k_0}{2} \ln(\frac{1+B}{1-B})\]
\[y = k_0 ( \arctan(\frac{\tan(\phi)}{\cos \lambda}) - \phi_0)\]

Inverse projection

\[D = \frac{y}{k_0} + \phi_0\]
\[x' = \frac{x}{k_0}\]
\[\phi = \arcsin(\frac{\sin D}{\cosh x'})\]
\[\lambda = \arctan(\frac{\sinh x'}{\cos D})\]

Ellipsoidal form

The formulas below describe the algorithm used when giving the +approx option. They are originally from [Snyder1987], but here quoted from [Evenden1995]. The default algorithm is given by Poder and Engsager in [Poder1998]

Forward projection

\[N = \frac{k_0}{(1 - e^2 \sin^2\phi)^{1/2}}\]
\[R = \frac{k_0(1-e^2)}{(1-e^2 \sin^2\phi)^{3/2}}\]
\[t = \tan(\phi)\]
\[\eta = \frac{e^2}{1-e^2}cos^2\phi\]
\[\begin{split}x &= k_0 \lambda \cos \phi \\ &+ \frac{k_0 \lambda^3 \cos^3\phi}{3!}(1-t^2+\eta^2) \\ &+ \frac{k_0 \lambda^5 \cos^5\phi}{5!}(5-18t^2+t^4+14\eta^2-58t^2\eta^2) \\ &+\frac{k_0 \lambda^7 \cos^7\phi}{7!}(61-479t^2+179t^4-t^6)\end{split}\]
\[\begin{split}y &= M(\phi) \\ &+ \frac{k_0 \lambda^2 \sin(\phi) \cos \phi}{2!} \\ &+ \frac{k_0 \lambda^4 \sin(\phi) \cos^3\phi}{4!}(5-t^2+9\eta^2+4\eta^4) \\ &+ \frac{k_0 \lambda^6 \sin(\phi) \cos^5\phi}{6!}(61-58t^2+t^4+270\eta^2-330t^2\eta^2) \\ &+ \frac{k_0 \lambda^8 \sin(\phi) \cos^7\phi}{8!}(1385-3111t^2+543t^4-t^6)\end{split}\]

Inverse projection

\[\phi_1 = M^-1(y)\]
\[N_1 = \frac{k_0}{1 - e^2 \sin^2\phi_1)^{1/2}}\]
\[R_1 = \frac{k_0(1-e^2)}{(1-e^2 \sin^2\phi_1)^{3/2}}\]
\[t_1 = \tan(\phi_1)\]
\[\eta_1 = \frac{e^2}{1-e^2}cos^2\phi_1\]
\[\begin{split}\phi &= \phi_1 \\ &- \frac{t_1 x^2}{2! R_1 N_1} \\ &+ \frac{t_1 x^4}{4! R_1 N_1^3}(5+3t_1^2+\eta_1^2-4\eta_1^4-9\eta_1^2t_1^2) \\ &- \frac{t_1 x^6}{6! R_1 N_1^5}(61+90t_1^2+46\eta_1^2+45t_1^4-252t_1^2\eta_1^2) \\ &+ \frac{t_1 x^8}{8! R_1 N_1^7}(1385+3633t_1^2+4095t_1^4+1575t_1^6)\end{split}\]
\[\begin{split}\lambda &= \frac{x}{\cos \phi N_1} \\ &- \frac{x^3}{3! \cos \phi N_1^3}(1+2t_1^2+\eta_1^2) \\ &+ \frac{x^5}{5! \cos \phi N_1^5}(5+6\eta_1^2+28t_1^2-3\eta_1^2+8t_1^2\eta_1^2) \\ &- \frac{x^7}{7! \cos \phi N_1^7}(61+662t_1^2+1320t_1^4+720t_1^6)\end{split}\]