Geographic coordinate conversion

The length PQ, called the prime vertical radius, is ${\displaystyle N(\phi )}$. The length IQ is equal to ${\displaystyle \,e^{2}N(\phi )}$. ${\displaystyle R=(X,\,Y,\,Z)}$.

Geodetic coordinates (latitude ${\displaystyle \ \phi }$, longitude ${\displaystyle \ \lambda }$, height ${\displaystyle h}$) can be converted into ECEF coordinates using the following equation:[3]

${\displaystyle {\begin{aligned}X&=\left(N(\phi )+h\right)\cos {\phi }\cos {\lambda }\\Y&=\left(N(\phi )+h\right)\cos {\phi }\sin {\lambda }\\Z&=\left({\frac {b^{2}}{a^{2}}}N(\phi )+h\right)\sin {\phi }\end{aligned}}}$

where

${\displaystyle N(\phi )={\frac {a^{2}}{\sqrt {a^{2}\cos ^{2}\phi +b^{2}\sin ^{2}\phi }}}={\frac {a}{\sqrt {1-e^{2}\sin ^{2}\phi }}},}$

and ${\displaystyle a}$ and ${\displaystyle b}$ are the equatorial radius (semi-major axis) and the polar radius (semi-minor axis), respectively. ${\displaystyle e^{2}=1-{\frac {b^{2}}{a^{2}}}}$ is the square of the first numerical eccentricity of the ellipsoid. The prime vertical radius of curvature ${\displaystyle \,N(\phi )}$ is the distance from the surface to the Z-axis along the ellipsoid normal (see "Radius of curvature on the Earth").

The following equation holds for the longitude in the same way as in the geocentric coordinates system:

${\displaystyle {\frac {X}{\cos \lambda }}-{\frac {Y}{\sin \lambda }}=0.}$

And the following equation holds for the latitude:

${\displaystyle {\frac {p}{\cos \phi }}-{\frac {Z}{\sin \phi }}-e^{2}N(\phi )=0,}$

where ${\displaystyle p={\sqrt {X^{2}+Y^{2}}}}$, as the parameter ${\displaystyle h}$ is eliminated by subtracting

${\displaystyle {\frac {p}{\cos \phi }}=N+h}$

and

${\displaystyle {\frac {Z}{\sin \phi }}={\frac {b^{2}}{a^{2}}}N+h.}$

The orthogonality of the coordinates is confirmed via differentiation:

${\displaystyle {\begin{aligned}{\begin{pmatrix}dX\\dY\\dZ\end{pmatrix}}&={\begin{pmatrix}-\sin \lambda &-\sin \phi \cos \lambda &\cos \phi \cos \lambda \\\cos \lambda &-\sin \phi \sin \lambda &\cos \phi \sin \lambda \\0&\cos \phi &\sin \phi \\\end{pmatrix}}{\begin{pmatrix}dE\\dN\\dU\end{pmatrix}},\\[3pt]{\begin{pmatrix}dE\\dN\\dU\end{pmatrix}}&={\begin{pmatrix}\left(N(\phi )+h\right)\cos \phi &0&0\\0&M(\phi )+h&0\\0&0&1\\\end{pmatrix}}{\begin{pmatrix}d\lambda \\d\phi \\dh\end{pmatrix}},\end{aligned}}}$

where

${\displaystyle M(\phi )={\frac {a\left(1-e^{2}\right)}{\left(1-e^{2}\sin ^{2}\phi \right)^{\frac {3}{2}}}}}$

(see also "Meridian arc on the ellipsoid").

The conversion of ECEF coordinates to geodetic coordinates (such WGS84) is the same as that of the geocentric one for the longitude:

${\displaystyle \lambda =\arctan {\frac {Y}{X}}}$.

The conversion for the latitude involves a bit complicated calculation and is known to be solved by using several methods shown as below. It is, however, sensitive to small accuracy due to ${\displaystyle Rn}$ and ${\displaystyle h}$ being maybe 10⁶ apart.[4][5]

The following Bowring's irrational geodetic-latitude equation[6] is efficient to be solved by Newton–Raphson iteration method:[7][8]

${\displaystyle \kappa -1-{\frac {e^{2}a\kappa }{\sqrt {p^{2}+\left(1-e^{2}\right)Z^{2}\kappa ^{2}}}}=0,}$

where ${\displaystyle \kappa ={\frac {p}{Z}}\tan \phi .}$ The height is calculated as:

${\displaystyle {\begin{aligned}h&=e^{-2}\left(\kappa ^{-1}-{\kappa _{0}}^{-1}\right){\sqrt {p^{2}+Z^{2}\kappa ^{2}}},\\\kappa _{0}&=\left(1-e^{2}\right)^{-1}.\end{aligned}}}$

The iteration can be transformed into the following calculation:

${\displaystyle \kappa _{i+1}={\frac {c_{i}+\left(1-e^{2}\right)Z^{2}\kappa _{i}^{3}}{c_{i}-p^{2}}}=1+{\frac {p^{2}+\left(1-e^{2}\right)Z^{2}\kappa _{i}^{3}}{c_{i}-p^{2}}},}$

where ${\displaystyle c_{i}={\frac {\left(p^{2}+\left(1-e^{2}\right)Z^{2}\kappa _{i}^{2}\right)^{\frac {3}{2}}}{ae^{2}}}.}$

The constant ${\displaystyle \,\kappa _{0}}$ is a good starter value for the iteration when ${\displaystyle h\approx 0}$. Bowring showed that the single iteration produces a sufficiently accurate solution. He used extra trigonometric functions in his original formulation.

The quartic equation of ${\displaystyle \kappa }$, derived from the above, can be solved by Ferrari's solution[9][10] to yield:

${\displaystyle {\begin{aligned}\zeta &=\left(1-e^{2}\right){\frac {z^{2}}{a^{2}}},\\[4pt]\rho &={\frac {1}{6}}\left({\frac {p^{2}}{a^{2}}}+\zeta -e^{4}\right),\\[4pt]s&={\frac {e^{4}\zeta p^{2}}{4\rho ^{3}a^{2}}},\\[4pt]t&={\sqrt[{3}]{1+s+{\sqrt {s(s+2)}}}},\\[4pt]u&=\rho \left(t+1+{\frac {1}{t}}\right),\\[4pt]v&={\sqrt {u^{2}+e^{4}\zeta }},\\[4pt]w&=e^{2}{\frac {u+v-\zeta }{2v}},\\[4pt]\kappa &=1+e^{2}{\frac {{\sqrt {u+v+w^{2}}}+w}{u+v}}.\end{aligned}}}$

A number of techniques and algorithms are available but the most accurate, according to Zhu,[11] is the following procedure established by Heikkinen,[12] as cited by Zhu. It is assumed that geodetic parameters ${\displaystyle \{a,\,b,\,e\}}$ are known

${\displaystyle {\begin{aligned}r&={\sqrt {X^{2}+Y^{2}}}\\[3pt]e'^{2}&={\frac {a^{2}-b^{2}}{b^{2}}}\\[3pt]F&=54b^{2}Z^{2}\\[3pt]G&=r^{2}+\left(1-e^{2}\right)Z^{2}-e^{2}\left(a^{2}-b^{2}\right)\\[3pt]c&={\frac {e^{4}Fr^{2}}{G^{3}}}\\[3pt]s&={\sqrt[{3}]{1+c+{\sqrt {c^{2}+2c}}}}\\[3pt]P&={\frac {F}{3\left(s+1+{\frac {1}{s}}\right)^{2}G^{2}}}\\[3pt]Q&={\sqrt {1+2e^{4}P}}\\[3pt]r_{0}&={\frac {-Pe^{2}r}{1+Q}}+{\sqrt {{\frac {1}{2}}a^{2}\left(1+{\frac {1}{Q}}\right)-{\frac {P\left(1-e^{2}\right)Z^{2}}{Q(1+Q)}}-{\frac {1}{2}}Pr^{2}}}\\[3pt]U&={\sqrt {\left(r-e^{2}r_{0}\right)^{2}+Z^{2}}}\\[3pt]V&={\sqrt {\left(r-e^{2}r_{0}\right)^{2}+\left(1-e^{2}\right)Z^{2}}}\\[3pt]z_{0}&={\frac {b^{2}Z}{aV}}\\[3pt]h&=U\left(1-{\frac {b^{2}}{aV}}\right)\\[3pt]\phi &=\arctan \left[{\frac {Z+e'^{2}z_{0}}{r}}\right]\\[3pt]\lambda &=\operatorname {arctan2} [Y,\,X]\end{aligned}}}$

Note: arctan2[Y, X] is the four-quadrant inverse tangent function.

For small e² the power series

${\displaystyle \kappa =\sum _{i\geq 0}\alpha _{i}e^{2i}}$

starts with

${\displaystyle {\begin{aligned}\alpha _{0}&=1;\\\alpha _{1}&={\frac {a}{\sqrt {Z^{2}+p^{2}}}};\\\alpha _{2}&={\frac {aZ^{2}{\sqrt {Z^{2}+p^{2}}}+2a^{2}p^{2}}{2\left(Z^{2}+p^{2}\right)^{2}}}.\end{aligned}}}$

To convert from geodetic coordinates to local ENU coordinates is a two-stage process:

1. Convert geodetic coordinates to ECEF coordinates
2. Convert ECEF coordinates to local ENU coordinates

To transform from ECEF coordinates to the local coordinates we need a local reference point, typically this might be the location of a radar. If a radar is located at ${\displaystyle \left\{X_{r},\,Y_{r},\,Z_{r}\right\}}$ and an aircraft at ${\displaystyle \left\{X_{p},\,Y_{p},\,Z_{p}\right\}}$ then the vector pointing from the radar to the aircraft in the ENU frame is

${\displaystyle {\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}-\sin \lambda _{r}&\cos \lambda _{r}&0\\-\sin \phi _{r}\cos \lambda _{r}&-\sin \phi _{r}\sin \lambda _{r}&\cos \phi _{r}\\\cos \phi _{r}\cos \lambda _{r}&\cos \phi _{r}\sin \lambda _{r}&\sin \phi _{r}\end{bmatrix}}{\begin{bmatrix}X_{p}-X_{r}\\Y_{p}-Y_{r}\\Z_{p}-Z_{r}\end{bmatrix}}}$

Note: ${\displaystyle \ \phi }$ is the geodetic latitude. A prior version of this page showed use of the geocentric latitude (${\displaystyle \ \phi ^{\prime }}$). The geocentric latitude is not the appropriate up direction for the local tangent plane. If the original geodetic latitude is available it should be used, otherwise, the relationship between geodetic and geocentric latitude has an altitude dependency, and is captured by:

${\displaystyle \tan \phi ^{\prime }={\frac {Z_{r}}{\sqrt {X_{r}^{2}+Y_{r}^{2}}}}={\frac {N(\phi )(1-f)^{2}+h}{N(\phi )+h}}\tan \phi }$

Obtaining geodetic latitude from geocentric coordinates from this relationship requires an iterative solution approach, otherwise the geodetic coordinates may be computed via the approach in the section above labeled "From ECEF to geodetic coordinates."

The geocentric and geodetic longitude have the same value. This is true for the Earth and other similar shaped planets because their latitude lines (parallels) can be considered in much more degree perfect circles when compared to their longitude lines (meridians).

${\displaystyle \tan \lambda ={\frac {Y_{r}}{X_{r}}}}$

Note: Unambiguous determination of ${\displaystyle \ \phi }$ and ${\displaystyle \ \lambda }$ requires knowledge of which quadrant the coordinates lie in.

This is just the inversion of the ECEF to ENU transformation so

${\displaystyle {\begin{bmatrix}X\\Y\\Z\end{bmatrix}}={\begin{bmatrix}-\sin \lambda &-\sin \phi \cos \lambda &\cos \phi \cos \lambda \\\cos \lambda &-\sin \phi \sin \lambda &\cos \phi \sin \lambda \\0&\cos \phi &\sin \phi \end{bmatrix}}{\begin{bmatrix}x\\y\\z\end{bmatrix}}+{\begin{bmatrix}X_{r}\\Y_{r}\\Z_{r}\end{bmatrix}}}$

Conversion of coordinates and map positions among different map projections reference to the same datum may be accomplished either through direct translation formulas from one projection to another, or by first converting from a projection ${\displaystyle A}$ to an intermediate coordinate system, such as ECEF, then converting from ECEF to projection ${\displaystyle B}$. The formulas involved can be complex and in some cases, such as in the ECEF to geodetic conversion above, the conversion has no closed-form solution and approximate methods must be used. References such as the DMA Technical Manual 8358.1[13] and the USGS paper Map Projections: A Working Manual[14] contain formulas for conversion of map projections. It is common to use computer programs to perform coordinate conversion tasks, such as with the DoD and NGA supported GEOTRANS program.[15]

Use of the Helmert transform in the transformation from geodetic coordinates of datum ${\displaystyle A}$ to geodetic coordinates of datum ${\displaystyle B}$ occurs in the context of a three-step process:[16]

1. Convert from geodetic coordinates to ECEF coordinates for datum ${\displaystyle A}$
2. Apply the Helmert transform, with the appropriate ${\displaystyle A\to B}$ transform parameters, to transform from datum ${\displaystyle A}$ ECEF coordinates to datum ${\displaystyle B}$ ECEF coordinates
3. Convert from ECEF coordinates to geodetic coordinates for datum ${\displaystyle B}$

In terms of ECEF XYZ vectors, the Helmert transform has the form[16]

${\displaystyle {\begin{bmatrix}X_{B}\\Y_{B}\\Z_{B}\end{bmatrix}}={\begin{bmatrix}c_{x}\\c_{y}\\c_{z}\end{bmatrix}}+\left(1+s\times 10^{-6}\right){\begin{bmatrix}1&-r_{z}&r_{y}\\r_{z}&1&-r_{x}\\-r_{y}&r_{x}&1\end{bmatrix}}{\begin{bmatrix}X_{A}\\Y_{A}\\Z_{A}\end{bmatrix}}.}$

The Helmert transform is a seven-parameter transform with three translation (shift) parameters ${\displaystyle c_{x},\,c_{y},\,c_{z}}$, three rotation parameters ${\displaystyle r_{x},\,r_{y},\,r_{z}}$ and one scaling (dilation) parameter ${\displaystyle s}$. The Helmert transform is an approximate method that is accurate when the transform parameters are small relative to the magnitudes of the ECEF vectors. Under these conditions, the transform is considered reversible.[17]

A fourteen-parameter Helmert transform, with linear time dependence for each parameter,[17]^:131-133 can be used to capture the time evolution of geographic coordinates dues to geomorphic processes, such as continental drift.[18] and earthquakes.[19] This has been incorporated into software, such as the Horizontal Time Dependent Positioning (HTDP) tool from the U.S. NGS.[20]

To eliminate the coupling between the rotations and translations of the Helmert transform, three additional parameters can be introduced to give a new XYZ center of rotation closer to coordinates being transformed. This ten-parameter model is called the Molodensky-Badekas transformation and should not be confused with the more basic Molodensky transform.[17]^:133-134

Like the Helmert transform, using the Molodensky-Badekas transform is a three-step process:

1. Convert from geodetic coordinates to ECEF coordinates for datum ${\displaystyle A}$
2. Apply the Molodensky-Badekas transform, with the appropriate ${\displaystyle A\to B}$ transform parameters, to transform from datum ${\displaystyle A}$ ECEF coordinates to datum ${\displaystyle B}$ ECEF coordinates
3. Convert from ECEF coordinates to geodetic coordinates for datum ${\displaystyle B}$

The transform has the form[21]

${\displaystyle {\begin{bmatrix}X_{B}\\Y_{B}\\Z_{B}\end{bmatrix}}={\begin{bmatrix}X_{A}\\Y_{A}\\Z_{A}\end{bmatrix}}+{\begin{bmatrix}\Delta X_{A}\\\Delta Y_{A}\\\Delta Z_{A}\end{bmatrix}}+{\begin{bmatrix}1&-r_{z}&r_{y}\\r_{z}&1&-r_{x}\\-r_{y}&r_{x}&1\end{bmatrix}}{\begin{bmatrix}X_{A}-X_{A}^{0}\\Y_{A}-Y_{A}^{0}\\Z_{A}-Z_{A}^{0}\end{bmatrix}}+\Delta S{\begin{bmatrix}X_{A}-X_{A}^{0}\\Y_{A}-Y_{A}^{0}\\Z_{A}-Z_{A}^{0}\end{bmatrix}}.}$

where ${\displaystyle \left(X_{A}^{0},\,Y_{A}^{0},\,Z_{A}^{0}\right)}$ is the origin for the rotation and scaling transforms and ${\displaystyle \Delta S}$ is the scaling factor.

The Molodensky-Badekas transform is used to transform local geodetic datums to a global geodetic datum, such as WGS 84. Unlike the Helmert transform, the Molodensky-Badekas transform is not reversible due to the rotational origin being associated with the original datum.[17]^:134

The Molodensky transformation converts directly between geodetic coordinate systems of different datums without the intermediate step of converting to geocentric coordinates (ECEF).[22] It requires the three shifts between the datum centers and the differences between the reference ellipsoid semi-major axes and flattening parameters.

The Molodensky transform is used by the National Geospatial-Intelligence Agency (NGA) in their standard TR8350.2 and the NGA supported GEOTRANS program.[23] The Molodensky method was popular before the advent of modern computers and the method is part of many geodetic programs.

Magnitude of shift in position between NAD27 and NAD83 datum as a function of location.

Grid-based transformations directly convert map coordinates from one (map-projection, geodetic datum) pair to map coordinates of another (map-projection, geodetic datum) pair. An example is the NADCON method for transforming from the North American Datum (NAD) 1927 to the NAD 1983 datum.[24] The High Accuracy Reference Network (HARN), a high accuracy version of the NADCON transforms, have an accuracy of approximately 5 centimeters. The National Transformation version 2 (NTv2) is a Canadian version of NADCON for transforming between NAD 1927 and NAD 1983. HARNs are also known as NAD 83/91 and High Precision Grid Networks (HPGN).[25] Subsequently, Australia and New Zealand adopted the NTv2 format to create grid-based methods for transforming among their own local datums.

Like the multiple regression equation transform, grid-based methods use a low-order interpolation method for converting map coordinates, but in two dimensions instead of three. The NOAA provides a software tool (as part of the NGS Geodetic Toolkit) for performing NADCON transformations.[26][27]

Datum transformations through the use of empirical multiple regression methods were created to achieve higher accuracy results over small geographic regions than the standard Molodensky transformations. MRE transforms are used to transform local datums over continent-sized or smaller regions to global datums, such as WGS 84.[28] The standard NIMA TM 8350.2, Appendix D,[29] lists MRE transforms from several local datums to WGS 84, with accuracies of about 2 meters.[30]

The MREs are a direct transformation of geodetic coordinates with no intermediate ECEF step. Geodetic coordinates ${\displaystyle \phi _{B},\,\lambda _{B},\,h_{B}}$ in the new datum ${\displaystyle B}$ are modeled as polynomials of up to the ninth degree in the geodetic coordinates ${\displaystyle \phi _{A},\,\lambda _{A},\,h_{A}}$ of the original datum ${\displaystyle A}$. For instance, the change in ${\displaystyle \phi _{B}}$ could be parameterized as (with only up to quadratic terms shown)[28]^:9

${\displaystyle \Delta \phi =a_{0}+a_{1}U+a_{2}V+a_{3}U^{2}+a_{4}UV+a_{5}V^{2}+\cdots }$

where

${\displaystyle {\begin{aligned}a_{i}&{\text{, parameters fitted by multiple regression}}\\U&=K(\phi _{A}-\phi _{m})\\V&=K(\lambda _{A}-\lambda _{m})\\K&{\text{, scale factor}}\\\phi _{m},\,\lambda _{m}&{\text{, origin of the datum, }}A\\\end{aligned}}}$

with similar equations for ${\displaystyle \Delta \lambda }$ and ${\displaystyle \Delta h}$. Given a sufficient number of ${\displaystyle (A,\,B)}$ coordinate pairs for landmarks in both datums for good statistics, multiple regression methods are used to fit the parameters of these polynomials. The polynomials, along with the fitted coefficients, form the multiple regression equations.