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

Transverse Mercator

proj-string: +proj=tmerc

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

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

0.9996

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

Parameters

Note

All parameters for the projection are optional.

+approx

New 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

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.

+lon_0=<value>

Longitude of projection center.

Defaults to 0.0.

+lat_0=<value>

Latitude of projection center.

Defaults to 0.0.

+ellps=<value>

See proj -le for a list of available ellipsoids.

Defaults to “GRS80”.

+R=<value>

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

+k_0=<value>

Scale factor. Determines scale factor used in the projection.

Defaults to 1.0.

+x_0=<value>

False easting.

Defaults to 0.0.

+y_0=<value>

False northing.

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 spherial 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

\[x' = k_0 R \eta';\qquad y' = k_0 R \xi'\]

Forward projection

\[\xi' = \tan^{-1}\biggl(\frac{\tan\phi'}{\cos\lambda'}\biggr)\]
\[\eta' = \sinh^{-1}\biggl(\frac{\sin\lambda'} {\sqrt{\tan^2\phi' + \cos^2\lambda'}}\biggr)\]

Inverse projection

\[\phi' = \tan^{-1}\biggl(\frac{\sin\xi'} {\sqrt{\sinh^2\eta' + \cos^2\xi'}}\biggr)\]
\[\lambda' = \tan^{-1}\biggl(\frac{\sinh\eta'}{\cos\xi'}\biggr)\]

Ellipsoidal form

The projected coordinates are given by

\[\zeta = \xi + i\eta;\qquad x = k_0 A \eta;\qquad y = k_0 A \xi\]
\[A = \frac a{1+n}\biggl(1 + \frac14 n^2 + \frac1{64} n^4 + \frac1{256}n^6\biggr)\]

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

\[\mathbf S(\theta) = \begin{pmatrix} \sin 2\theta & \sin 4\theta & \sin 6\theta & \sin 8\theta & \sin 10\theta & \sin 12\theta \end{pmatrix}\]
\[\begin{split}\mathbf N = \begin{pmatrix} n \\ n^2 \\ n^3\\ n^4 \\ n^5 \\ n^6 \end{pmatrix}\end{split}\]

and \(\mathsf C_{\alpha,\beta}\) are upper triangular \(6\times6\) matrices.

Relation between geographic coordinates

\[\lambda' = \lambda\]
\[\phi' = \tan^{-1}\sinh\bigl(\sinh^{-1}\tan\phi - e \tanh^{-1}(e\sin\phi)\bigr)\]

Instead of using this analytical formula for \(\phi'\), the conversions between \(\phi\) and \(\phi'\) use the series approximations:

\[\phi' = \phi + \mathbf S(\phi) \cdot \mathsf C_{\chi,\phi} \cdot \mathbf N\]
\[\phi = \phi' + \mathbf S(\phi') \cdot \mathsf C_{\phi,\chi} \cdot \mathbf N\]
\[\begin{split}\mathsf C_{\chi,\phi} = \begin{pmatrix} -2& \frac{2}{3}& \frac{4}{3}& -\frac{82}{45}& \frac{32}{45}& \frac{4642}{4725} \\ & \frac{5}{3}& -\frac{16}{15}& -\frac{13}{9}& \frac{904}{315}& -\frac{1522}{945} \\ & & -\frac{26}{15}& \frac{34}{21}& \frac{8}{5}& -\frac{12686}{2835} \\ & & & \frac{1237}{630}& -\frac{12}{5}& -\frac{24832}{14175} \\ & & & & -\frac{734}{315}& \frac{109598}{31185} \\ & & & & & \frac{444337}{155925} \\ \end{pmatrix}\end{split}\]
\[\begin{split}\mathsf C_{\phi,\chi} = \begin{pmatrix} 2& -\frac{2}{3}& -2& \frac{116}{45}& \frac{26}{45}& -\frac{2854}{675} \\ & \frac{7}{3}& -\frac{8}{5}& -\frac{227}{45}& \frac{2704}{315}& \frac{2323}{945} \\ & & \frac{56}{15}& -\frac{136}{35}& -\frac{1262}{105}& \frac{73814}{2835} \\ & & & \frac{4279}{630}& -\frac{332}{35}& -\frac{399572}{14175} \\ & & & & \frac{4174}{315}& -\frac{144838}{6237} \\ & & & & & \frac{601676}{22275} \\ \end{pmatrix}\end{split}\]

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

\[\zeta = \zeta' + \mathbf S(\zeta') \cdot \mathsf C_{\mu,\chi} \cdot \mathbf N\]
\[\zeta' = \zeta + \mathbf S(\zeta) \cdot \mathsf C_{\chi,\mu} \cdot \mathbf N\]
\[\begin{split}\mathsf C_{\mu,\chi} = \begin{pmatrix} \frac{1}{2}& -\frac{2}{3}& \frac{5}{16}& \frac{41}{180}& -\frac{127}{288}& \frac{7891}{37800} \\ & \frac{13}{48}& -\frac{3}{5}& \frac{557}{1440}& \frac{281}{630}& -\frac{1983433}{1935360} \\ & & \frac{61}{240}& -\frac{103}{140}& \frac{15061}{26880}& \frac{167603}{181440} \\ & & & \frac{49561}{161280}& -\frac{179}{168}& \frac{6601661}{7257600} \\ & & & & \frac{34729}{80640}& -\frac{3418889}{1995840} \\ & & & & & \frac{212378941}{319334400} \\ \end{pmatrix}\end{split}\]
\[\begin{split}\mathsf C_{\chi,\mu} = \begin{pmatrix} -\frac{1}{2}& \frac{2}{3}& -\frac{37}{96}& \frac{1}{360}& \frac{81}{512}& -\frac{96199}{604800} \\ & -\frac{1}{48}& -\frac{1}{15}& \frac{437}{1440}& -\frac{46}{105}& \frac{1118711}{3870720} \\ & & -\frac{17}{480}& \frac{37}{840}& \frac{209}{4480}& -\frac{5569}{90720} \\ & & & -\frac{4397}{161280}& \frac{11}{504}& \frac{830251}{7257600} \\ & & & & -\frac{4583}{161280}& \frac{108847}{3991680} \\ & & & & & -\frac{20648693}{638668800} \\ \end{pmatrix}\end{split}\]

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.

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

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

Further reading

  1. Wikipedia