> For the complete documentation index, see [llms.txt](https://docs.os.uk/more-than-maps/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.os.uk/more-than-maps/a-guide-to-coordinate-systems-in-great-britain/converting-between-3d-cartesian-and-ellipsoidal-latitude-longitude-and-height-coordinates.md).
# Converting between 3D Cartesian and ellipsoidal latitude, longitude and height coordinates
All the formula here are coded into a spreadsheet available from the OS web site
### Converting latitude, longitude and ellipsoid height to 3D Cartesian coordinates
{% hint style="info" %}
*These formulae are used to* **convert** *the format of coordinates between 3D Earth Centred Earth Fixed (ECEF) Cartesian coordinates and latitude, longitude and ellipsoidal height. This is **NOT** a datum* transformation*.*
{% endhint %}
Values are required for the following ellipsoid constants: the semi-major axis length *a* and eccentricity squared .png?alt=media). The latter can be calculated from *a* and *b* or the flattening *f* by
$$
e^2 = \frac{a^2 - b^2}{a^2}= 2f -f^2
$$
The Cartesian coordinates *x y* and *z* of a point are obtained from the latitude *φ*, longitude *λ* and ellipsoid height *H* by
$$
\nu = \frac{a}{\sqrt{1-e^2sin^2\phi}}\\
x = (v+H)cos\phi cos\lambda\\
y = (v+H)cos\phi sin\lambda \\
z = ((1-e^2)v+H)\sin\phi
$$
$$
x = (v+H)cos\phi\lambda\\
$$
$$
y = (v+H)cos\phi\lambda\\
$$
$$
z = ((1-e^2)v+H\sin\phi\\
$$
Here’s a worked example using the GRS80 ellipsoid. Intermediate values are shown here to 10 decimal places. Compute all values using double-precision arithmetic.
Latitude, *φ* 53° 36′ 43.1653″ N\
Longitude, *λ* 001° 39′ 51.9920″ W\
Ellipsoidal height, *H* 299.800 m
*e²* 6.6943800355E-03\
*ν* 6.3920173768E+06
*x* 3790644.900 m\
*y* -110149.210 m\
*z* 5111482.970 m
### Converting 3D Cartesian coordinates to latitude, longitude and ellipsoid height
Again, we need the defining constants of the ellipsoid. Longitude *λ* is easily computed from Cartesian coordinates, remembering to be careful about the quadrant of the resulting angle:
$$
\lambda = arctan (\frac{y}{x})
$$
The latitude *φ* is obtained by an iterative procedure. The initial value of latitude is given by
\lambda = arctan (\frac{z}{p(1-e^2})
where
$$
p = \sqrt{x^2 + Y^2}
$$
*φ* is iteratively improved by repeatedly computing *ν* from equation:
$$
\nu = \frac{a}{\sqrt{1-e^2sin^2\phi}}\\
$$
(using the latest value of *φ*) and then a new value for *φ* by
$$
\phi = arctan\[\frac{z+e^2vsin\phi}{p}]
$$
until the change between two successive values of *φ* is smaller than the precision to which you want to calculate the latitude. Ellipsoid height *H* is then given by:
$$
H = \frac{p}{cos\phi} - v
$$
Here’s a worked example using the GRS80 ellipsoid. Intermediate values are shown here to 10 decimal places. Compute all values using double-precision arithmetic.
*x* 3790644.900 m\
*y* -110149.210 m\
*z* 5111482.970 m
*e²* 6.6943800355E-03\
Initial *φ* 9.3570590125E-01\
Initial *ν* 6.3920173799E+06\
*φ*#2 9.3570575065E-01\
*ν*#2 6.3920173768E+06\
initial *φ* – *φ*#2 1.5059657843E-07\
*φ*#3 9.3570575030E-01\
*ν*#3 6.3920173768E+06\
*φ*#2 – *φ*#3 3.5634395434E-10\
*φ*#4 9.3570575029E-01\
*ν*#4 6.3920173768E+06
*φ*#3 – *φ*#4 8.4310336490E-13\
*φ*#5 9.3570575029E-01\
*ν*#5 6.3920173768E+06\
*φ*#4 – *φ*#5 1.9984014443E-15\
*φ*#6 9.3570575029E-01\
*ν*#6 6.3920173768E+06\
*φ*#5 – *φ*#6 0.0000000000E+00
Latitude, *φ* 53° 36′ 43.1653″ N\
Longitude, *λ* 001° 39′ 51.9920″ W\
Ellipsoidal height, *H* 299.800 m
---
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