# Azimuthal equidistant projection for small distances (<10km)

I know that the azimuthal equidistant projection preserves the distance and direction of each point with respect to the reference point. However, it does not preserve the distance nor the direction between other points.

Currently, I would like to use this projection to project some points(latitude and longitude) to local XY coordinates and all the points including the reference point are within 10km distance.

What would be the error in the distance and direction between two points (other than the reference) in this range? Is there a way to calculate such an error?

I am using this to localize a robot using a GPS sensor.

The scaling of distances measured along circles of equal distance from the reference point can be calculated as `d/(R sin (d/R))` where `d` is the distance from the reference point and `R` is Earth's radius; this gives you an error of appx. `0.000041 %` at 10 km distance.
The equation can be derived by relating the radius of the circle in the projection `Rf` to the real radius of Earth's circle of latitude `R sin(f)`, where `f = d/R` is the angular distance of the projected point from the reference point. This gives `f/sin(f) = d/(R sin (d/R))`.