Convert geocentric latitude to geodetic latitude

Utilities/Axes Transformations

The Geocentric to Geodetic Latitude block converts a geocentric
latitude (*λ*) into geodetic latitude (*μ*).
There are a number of geometric relationships that are used to calculate
the geodetic latitude in this noniterative method. A number of angles
and points are involved in the calculation, which are shown in following
figure.

.

Given geocentric latitude (*λ*) and the
radius (*r*) from the center of the planet (O) to
the center of gravity (P), this noniterative method starts by computing
values for the point of *r* that intercepts the surface
of the planet (S). By rearranging the equation for an ellipse, the
horizontal coordinate, $$\left({x}_{a}\right)$$ is determined.
When equatorial radius (*R*), polar radius $$\left((1-f)R\right)$$ and $${x}_{a}\mathrm{tan}\lambda $$, are substituted for semi-major
axis, semi-minor axis and vertical coordinate $$\left({y}_{a}\right)$$, the resulting equation for $${x}_{a}$$ has the following form:

$${x}_{a}=\frac{(1-f)R}{\sqrt{{\mathrm{tan}}^{2}\lambda +{(1-f)}^{2}}}$$

To determine the geodetic latitude at S$${\mu}_{a}$$, the equation for an ellipse
with equatorial radius (*R*), polar radius $$\left((1-f)R\right)$$is used again. This time it is
used to define $${y}_{a}$$ in terms of $${x}_{a}$$.

$${y}_{a}=\sqrt{{R}^{2}-{x}_{a}^{2}}(1-f)$$

Additionally, the relationship between geocentric latitude at the planet's surface and geodetic latitude is used.

$${\mu}_{a}=\text{atan}\left(\frac{\mathrm{tan}\lambda}{{(1-f)}^{2}}\right)$$

Using the relationship $$\mathrm{tan}\lambda ={y}_{a}/{x}_{a}$$ and the two equations above, the resulting equation for $${\mu}_{a}$$ is obtained.

$${\mu}_{a}=\text{atan}\left(\frac{\sqrt{{R}^{2}-{x}_{a}{}^{2}}}{(1-f){x}_{a}}\right)$$

The correct sign of $${\mu}_{a}$$ is
determined by testing *λ* and if *λ* is
less than zero $${\mu}_{a}$$ changes sign accordingly.

In order to calculate the geodetic latitude of P, a number of geometric relationships are required to be calculated. These calculations follow.

The radius $$\left({r}_{a}\right)$$ from the center of the planet (O) to the surface of the planet (S) is calculated by using trigonometric relationship.

$${r}_{a}=\frac{{x}_{a}}{\mathrm{cos}\lambda}$$

The distance from S to P is defined by:

$$l=r-{r}_{a}$$

The angular difference between geocentric latitude and geodetic
latitude at S(*δλ*) is defined by:

$$\delta \lambda ={\mu}_{a}-\lambda $$

Using $$l$$ and *δλ*,
the segment TP or the mean sea-level altitude (*h*)
is estimated.

$$h=l\mathrm{cos}\delta \lambda $$

The equation for the radius of curvature in the Meridian $$\left({\rho}_{a}\right)$$ at $${\mu}_{a}$$ is

$${\rho}_{a}=\frac{R{(1-f)}^{2}}{{\left(1-(2f-{f}^{2}){\mathrm{sin}}^{2}{\mu}_{a}\right)}^{3/2}}$$

Using $$l$$, *δλ*, *h*,
and $${\rho}_{a}$$, the angular difference between
geodetic latitude at S $$\left(\mu \right)$$and geodetic latitude
at P $$\left({\mu}_{a}\right)$$is defined as:

$$\delta \mu =\text{atan}\left(\frac{l\mathrm{sin}\delta \lambda}{{\rho}_{a}+h}\right)$$

Subtracting *δμ* from $${\mu}_{a}$$ then gives *μ*.

$$\mu ={\mu}_{a}-\delta \mu $$

**Units**Specifies the parameter and output units:

Units

Radius from CG to Center of Planet

Equatorial Radius

`Metric (MKS)`

Meters

Meters

`English`

Feet

Feet

This option is only available when

**Planet model**is set to`Earth (WGS84)`

.**Planet model**Specifies the planet model to use:

`Custom`

or`Earth (WGS84)`

.**Flattening**Specifies the flattening of the planet. This option is only available with

**Planet model**set to`Custom`

.

**Equatorial radius of planet**Specifies the radius of the planet at its equator. The units of the equatorial radius parameter should be the same as the units for radius. This option is only available with

**Planet model**set to`Custom`

.

Input | Dimension Type | Description |
---|---|---|

First | Scalar | Contains the geocentric latitude, in degrees. Latitude values can be any value. However, values of +90 and -90 may return unexpected values because of singularity at the poles. |

Second | Scalar | Contains the radius from center of the planet to the center of gravity. |

Output | Dimension Type | Description |
---|---|---|

First | Scalar | Contains the geodetic latitude, in degrees. |

This implementation generates a geodetic latitude that lies between ±90 degrees.

Jackson, E. B., *Manual for a Workstation-based Generic
Flight Simulation Program (LaRCsim) Version 1.4*, NASA
TM 110164, April, 1995.

Hedgley, D. R., Jr., “An Exact Transformation from Geocentric to Geodetic Coordinates for Nonzero Altitudes,” NASA TR R-458, March, 1976.

Clynch, J. R., “Radius of the Earth - Radii Used in Geodesy,” Naval Postgraduate School, 2002, https://core.ac.uk/download/pdf/36732690.pdf.

Stevens, B. L., and F. L. Lewis, *Aircraft Control
and Simulation*, John Wiley & Sons, New York, 1992.

Edwards, C. H., and D. E. Penny, *Calculus and Analytical
Geometry 2nd Edition*, Prentice-Hall, Englewood Cliffs,
New Jersey, 1986.