Documentation

# lla2flat

Convert from geodetic latitude, longitude, and altitude to flat Earth position

## Syntax

```flatearth_pos = lla2flat(lla, llo, psio, href) flatearth_pos = lla2flat(lla, llo, psio, href, ellipsoidModel) flatearth_pos = lla2flat(lla, llo, psio, href, flattening, equatorialRadius) ```

## Description

```flatearth_pos = lla2flat(lla, llo, psio, href)``` estimates an array of flat Earth coordinates, `flatearth_pos`, from an array of geodetic coordinates, `lla`. This function estimates the `flatearth_pos` value with respect to a reference location that `llo`, `psio`, and `href` define.

```flatearth_pos = lla2flat(lla, llo, psio, href, ellipsoidModel)``` estimates the coordinates for a specific ellipsoid planet.

```flatearth_pos = lla2flat(lla, llo, psio, href, flattening, equatorialRadius)``` estimates the coordinates for a custom ellipsoid planet defined by `flattening` and `equatorialRadius`.

## Input Arguments

 `lla` `m`-by-3 array of geodetic coordinates (latitude, longitude, and altitude), in [degrees, degrees, meters]. Latitude and longitude values can be any value. However, latitude values of +90 and -90 may return unexpected values because of singularity at the poles. `llo` Reference location, in degrees, of latitude and longitude, for the origin of the estimation and the origin of the flat Earth coordinate system. `psio` Angular direction of flat Earth x-axis (degrees clockwise from north), which is the angle in degrees used for converting flat Earth x and y coordinates to the North and East coordinates. `href` Reference height from the surface of the Earth to the flat Earth frame with regard to the flat Earth frame, in meters. `ellipsoidModel` Specific ellipsoid planet model. This function supports only `'WGS84'`. Default: WGS84 `flattening` Custom ellipsoid planet defined by flattening. `equatorialRadius` Planetary equatorial radius, in meters.

## Output Arguments

 `flatearth_pos` Flat Earth position coordinates, in meters.

## Examples

Estimate coordinates at latitude, longitude, and altitude:

```p = lla2flat( [ 0.1 44.95 1000 ], [0 45], 5, -100 ) p = 1.0e+004 * 1.0530 -0.6509 -0.0900```

Estimate coordinates at multiple latitudes, longitudes, and altitudes, specifying the WGS84 ellipsoid model:

```p = lla2flat( [ 0.1 44.95 1000; -0.05 45.3 2000 ], [0 45], 5, -100, 'WGS84' ) p = 1.0e+004 * 1.0530 -0.6509 -0.0900 -0.2597 3.3751 -0.1900```

Estimate coordinates at multiple latitudes, longitudes, and altitudes, specifying a custom ellipsoid model:

```f = 1/196.877360; Re = 3397000; p = lla2flat( [ 0.1 44.95 1000; -0.05 45.3 2000 ], [0 45], 5, -100, f, Re ) p = 1.0e+004 * 0.5588 -0.3465 -0.0900 -0.1373 1.7975 -0.1900```

## Tips

• This function assumes that the flight path and bank angle are zero.

• This function assumes that the flat Earth `z`-axis is normal to the Earth only at the initial geodetic latitude and longitude. This function has higher accuracy over small distances from the initial geodetic latitude and longitude. It also has higher accuracy at distances closer to the equator. The function calculates a longitude with higher accuracy when the variations in latitude are smaller. Additionally, longitude is singular at the poles.

## Algorithms

The estimation begins by finding the small changes in latitude and longitude from the output latitude and longitude minus the initial latitude and longitude.

`$\begin{array}{l}d\mu =\mu -{\mu }_{0}\\ {d}_{\iota }=\iota -{\iota }_{0}\end{array}$`

To convert geodetic latitude and longitude to the North and East coordinates, the estimation uses the radius of curvature in the prime vertical (RN) and the radius of curvature in the meridian (RM). RN and RM are defined by the following relationships:

`$\begin{array}{l}{R}_{N}=\frac{R}{\sqrt{1-\left(2f-{f}^{2}\right){\mathrm{sin}}^{2}{\mu }_{0}}}\\ {R}_{M}={R}_{N}\frac{1-\left(2f-{f}^{2}\right)}{1-\left(2f-{f}^{2}\right){\mathrm{sin}}^{2}{\mu }_{0}}\end{array}$`

where (R) is the equatorial radius of the planet and $f$ is the flattening of the planet.

Small changes in the North (dN) and East (dE) positions are approximated from small changes in the North and East positions by

`$\begin{array}{l}dN=\frac{d\mu }{\text{atan}\left(\frac{1}{{R}_{M}}\right)}\\ dE=\frac{d\iota }{\text{atan}\left(\frac{1}{{R}_{N}\mathrm{cos}{\mu }_{0}}\right)}\end{array}$`

With the conversion of the North and East coordinates to the flat Earth x and y coordinates, the transformation has the form of

`$\left[\begin{array}{c}{p}_{x}\\ {p}_{y}\end{array}\right]=\left[\begin{array}{cc}\mathrm{cos}\psi & \mathrm{sin}\psi \\ -\mathrm{sin}\psi & \mathrm{cos}\psi \end{array}\right]\left[\begin{array}{c}N\\ E\end{array}\right]$`

where

`$\left(\psi \right)$`

is the angle in degrees clockwise between the x-axis and north.

The flat Earth z-axis value is the negative altitude minus the reference height (href).

`${p}_{z}=-h-{h}_{ref}$`

## References

Etkin, B., Dynamics of Atmospheric Flight. New York: John Wiley & Sons, 1972.

Stevens, B. L., and F. L. Lewis, Aircraft Control and Simulation, 2nd ed. New York: John Wiley & Sons, 2003.