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# Geocentric to Geodetic Latitude

Convert geocentric latitude to geodetic latitude

## Library

Utilities/Axes Transformations

## Description

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.

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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 $\left(\left(1-f\right)R\right)$ 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:

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

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

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

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

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

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

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

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

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

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.

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

## Dialog Box

Units

Specifies the parameter and output units:

Units

Radius from CG to Center of Planet

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.

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.

## Inputs and Outputs

InputDimension TypeDescription

First

ScalarContains 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

ScalarContains the radius from center of the planet to the center of gravity.

OutputDimension TypeDescription

First

ScalarContains the geodetic latitude, in degrees.

## Assumptions and Limitations

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

## References

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.