Reference Ellipsoid
The reference ellipsoid is a mathematically defined surface that approximates
the shape of the Earth (or another planetary body) as an oblate spheroid.
Unlike a perfect sphere, the Earth is slightly flattened at the poles and
bulged at the equator due to its rotation. The reference ellipsoid provides a
simple, standardized model for representing the Earth’s size and shape, serving
as the foundation for geodetic calculations, mapping, and satellite navigation
systems. By specifying parameters such as the equatorial radius \(\Req\)
and flattening \(\flattening\), the reference ellipsoid enables precise
definitions of geographic coordinates, altitudes, and distances on or near the
Earth’s surface.
The reference ellipsoid can be literally thought as an oblate sphere. This
geometric flattening can be mathematically represented by scaling the z-axis of
a sphere of radius \(\Req\) by the factor \(\frac{\Rpo}{\Req}\), where
\(\Rpo\) denotes the polar radius of the ellipsoid. In other words, the
transformation from a sphere to an ellipsoid can be expressed as a linear
mapping, where only the polar (\(z\)-axis) coordinate is compressed
relative to the equatorial plane. The following equation illustrates this
transformation:
\[\begin{split}\PosEll
=
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & \frac{\Rpo}{\Req}
\end{bmatrix}
\PosSph.\end{split}\]
Here, \(\PosSph\) is a position vector on the sphere and \(\PosEll\) is the corresponding point on the ellipsoid. This simple scaling effectively “squashes” the sphere along its axis of rotation, creating the oblate ellipsoidal shape that more accurately models the Earth’s surface.
Parametrization
(1)\[\tan\latRed = \pare{1 - \flattening} \tan\lat\]
Cartesian/Geodetic Coordinates Conversions
\[\begin{split}\Req
\begin{bmatrix}
\cos\lon \cos\latRed \\
\sin\lon \cos\latRed \\
\pare{1 - \flattening}\sin\beta
\end{bmatrix}
+
\frac{
h
}{
\sqrt{
1
-
\pare{2\flattening - \flattening^2}
\cos^2\latRed
}
}
\begin{bmatrix}
\pare{1 - \flattening}\cos\lon \cos\latRed \\
\pare{1 - \flattening}\sin\lon \cos\latRed \\
\sin\beta
\end{bmatrix}
=
\begin{bmatrix}
x\\
y\\
z
\end{bmatrix}\end{split}\]
Which can be rewritten as the following system of coupled equations:
(2)\[\pare{
\Req
+
\frac{
\altitude
\pare{1 - \flattening}
}{
\sqrt{
1
-
\pare{2\flattening - \flattening^2}
\cos^2\latRed
}
}
}
\cos\lon \cos\latRed
=
x\]
(3)\[\pare{
\Req
+
\frac{
\altitude
\pare{1 - \flattening}
}{
\sqrt{
1
-
\pare{2\flattening - \flattening^2}
\cos^2\latRed
}
}
}
\sin\lon \cos\latRed
=
y\]
(4)\[\pare{
\Rpo
+
\frac{
\altitude
}{
\sqrt{
1
-
\pare{2\flattening - \flattening^2}
\cos^2\latRed
}
}
}
\sin\latRed
=
z\]
From (2) and (3), the longitude is determined as
(5)\[\lon = \arctanTWO\pare{y, x}.\]
Moreover, those equations can be even combined to remove the longitude and yield
(6)\[\pare{
\Req
+
\frac{
\altitude \pare{1 - \flattening}
}{
\sqrt{
1
-
\pare{2\flattening - \flattening^2}
\cos^2\latRed
}
}
}
\cos\latRed
=
\sqrt{x^2 + y^2}\]
Equations (4) and (6) will then be used to
retrieve \(\latRed\) and \(alt\). Initially, from (4)
note that:
\[\altitude
=
\frac{z}{\sin\latRed}
-
\Rpo\]
Thus, (6) becomes
\[\pare{
\Req
+
\frac{z}{\sin\latRed} \pare{1 - \flattening}
-
\Rpo \pare{1 - \flattening}
}
\cos\latRed
=
\sqrt{x^2 + y^2}\]
\[\sin\latRed
\cos\latRed
-
A
\sin\latRed
+
B
\cos\latRed
=
0\]
where \(A = \frac{\sqrt{x^2 + y^2}}{\Req - \Rpo\pare{1 - \flattening}}\) and
\(B = \frac{z\pare{1 - \flattening}}{\Req - \Rpo\pare{1 - \flattening}}\).
Cartesian/Geodetic State Conversions
\[\dot{\lon}
=
\frac{
x\dot{y} - y\dot{x}
}{
x^2 + y^2
}\]
(4) and (6) are written as
\[\begin{split}\rHor\pare{\latRed, \alt}\cos\latRed &= \sqrt{x^2 + y^2}
\\[2mm]
\rVer\pare{\latRed, \alt}\sin\latRed &= z\end{split}\]
where:
\[\begin{split}\rHor\pare{\latRed, \alt}
&=
\Req + \pare{1-\flattening}\aux\pare{\latRed} \alt
\\[2mm]
\rVer\pare{\latRed, \alt}
&=
\Rpo + \aux\pare{\latRed} \alt
\\[2mm]
\aux\pare{\latRed}
&=
\frac{
1
}{
\sqrt{
1
-
\pare{2\flattening - \flattening^2}
\cos^2\latRed
}
}\end{split}\]
\[\begin{split}\pare{
\PartialDerivative{\rHor}{\latRed}\pare{\latRed, \alt}\
\dot{\latRed}
+
\PartialDerivative{\rHor}{\alt}\pare{\latRed, \alt}\
\dot{\alt}
}
\cos\latRed
-
\rHor\pare{\latRed, \alt}
\sin\latRed\
\dot{\latRed}
&=
\frac{
x\dot{x} + y\dot{y}
}{
\sqrt{x^2 + y^2}
}
\\[3mm]
\pare{
\PartialDerivative{\rVer}{\latRed}\pare{\latRed, \alt}\
\dot{\latRed}
+
\PartialDerivative{\rVer}{\alt}\pare{\latRed, \alt}\
\dot{\alt}
}
\sin\latRed
+
\rVer\pare{\latRed, \alt}
\cos\latRed\
\dot{\latRed}
&=
\dot{z}\end{split}\]
\[\begin{split}\begin{bmatrix}
\PartialDerivative{\rHor}{\latRed}\pare{\latRed, \alt}
\cos\latRed
-
\rHor\pare{\latRed, \alt}
\sin\latRed\
&
\PartialDerivative{\rHor}{\alt}\pare{\latRed, \alt}
\cos\latRed
\\[3mm]
\PartialDerivative{\rVer}{\latRed}\pare{\latRed, \alt}
\sin\latRed
+
\rVer\pare{\latRed, \alt}
\cos\latRed
&
\PartialDerivative{\rVer}{\alt}\pare{\latRed, \alt}
\sin\latRed
\end{bmatrix}
\begin{bmatrix}
\dot{\latRed}
\\[3mm]
\dot{\alt}
\end{bmatrix}
=
\begin{bmatrix}
\frac{
x\dot{x} + y\dot{y}
}{
\sqrt{x^2 + y^2}
}
\\[3mm]
\dot{z}
\end{bmatrix}\end{split}\]
\[\begin{split}\begin{bmatrix}
\pare{1-\flattening}\aux^\prime\pare{\latRed} \alt
\cos\latRed
-
\rHor\pare{\latRed, \alt}
\sin\latRed\
&
\pare{1-\flattening}\aux\pare{\latRed}
\cos\latRed
\\[3mm]
\aux^\prime\pare{\latRed} \alt
\sin\latRed
+
\rVer\pare{\latRed, \alt}
\cos\latRed
&
\aux\pare{\latRed}
\sin\latRed
\end{bmatrix}
\begin{bmatrix}
\dot{\latRed}
\\[3mm]
\dot{\alt}
\end{bmatrix}
=
\begin{bmatrix}
\frac{
x\dot{x} + y\dot{y}
}{
\sqrt{x^2 + y^2}
}
\\[3mm]
\dot{z}
\end{bmatrix}\end{split}\]
