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# Ellipsoidal and Cartesian Coordinates Conversion

Fundamentals
Title Ellipsoidal and Cartesian Coordinates Conversion
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Year of Publication 2011

The $(x,y,z)$ ECEF cartesian coordinates can be expressed in the ellipsoidal coordinates $(\varphi, \lambda, h)$, where $\varphi$ and $\lambda$ are, respectively, the latitude and longitude from the ellipsoid, and $h$ the height above it. Figure 1 illustrates the relation between Cartesian and ellipsoidal coordinates.

Figure 1: Cartesian $(x,y,z)$ and ellipsoidal $(\varphi, \lambda, h)$ coordinates

## From Ellipsoidal to Cartesian coordinates

The Cartesian coordinates of a point $(x,y,z)$ can be obtained from the ellipsoidal coordinates $(\varphi, \lambda, h)$ by the next expressions:

$\begin{array}{l} x=(N+h) \cos \varphi\,\cos \lambda\\ y=(N+h)\cos \varphi\,\sin \lambda\\ z=\left ((1-e^2)N+h \right)\sin \varphi\\ \end{array} \qquad \mbox{(1)}$

where $N$ is the radius of curvature in the prime vertical:

$N=\displaystyle \frac{a}{\sqrt{1-e^2\sin^2\varphi}} \qquad \mbox{(2)}$

and where the eccentricity $e$ is related with the semi-major axis $a$, the semi-minor axis $b$ and the flattening factor $f = 1 -\frac{b}{a}$ by:

$e^2=\displaystyle \frac{a^2-b^2}{a^2}=2f-f^2 \qquad \mbox{(3)}$

## From Cartesian to Ellipsoidal coordinates

The ellipsoidal coordinates of a point $(\varphi, \lambda, h)$ can be obtained from the cartesian coordinates $(x,y,z)$ as follows:

The longitude $\lambda$ is given by: $\lambda= \arctan\frac{y}{x} \qquad \mbox{(4)}$

The latitude is computed by an iterative procedure.

1. The initial value is given by:
$\varphi_{(0)}=\arctan\left [ \frac{z}{(1-e^2)\,p} \right ] \qquad \mbox{(5)}$

with $p=\sqrt{x^2+y^2}$.

2. Improved values of $\varphi$, as well as the height $h$, are computed iterating in the equations:

$\begin{array}{l} N_Template:(i)= \frac{a}{\sqrt{1-e^2\sin^2\varphi_{_{(i-1)}}}}\\[0.6cm] h_Template:(i)=\frac{p}{\cos \varphi_{_{(i-1)}}}-N_{(i)}\\[0.5cm] \varphi_Template:(i)= \arctan\left [ \frac{z}{ \left ( 1 - e^2\frac{N_{(i)}}{N_{(i)}+h_{(i)}} \right ) p } \right ] \end{array} \qquad \mbox{(6)}$

The iterations are repeated until the change between two successive values of $\varphi_{(i)}$ are smaller than the precision required.