I have the position of a RADAR station and some aitrcrafts' azimuth, elevation and range data. How can I calculate the latitude longitude of those aircrafts?
Hapgood, M. A. (1992). Space physics coordinate transformations: A user guide. Planetary and Space Science, 40(5), 711-717.
You have two spherical coordinates. In essence. You must add them. What I am going to do here is a generalization that the radius of the earth is the same everywhere. I know that it is not, but because you have a geoid, you are better off making the assumption in this case. You have two spherical coordinates. To add them, we have to put them in Cartesian coordinates and then convert backward to spherical (and then convert to lat long because of the declination variable phi).
Since they do not start at the same point, you have to convert them.
Coordinate 1:
R = 3959 miles + elevation of the radar site
Theta = Longitude
Phi = 90 - Latitude (comes down from true north, rather than up from the equator)
Coordinate 2:
R = Range
Theta = Azimuth
Phi = 90 - Elevation
The conversion rules are:
x = R sin (phi) cos (theta)
y = R sin (phi) sin (theta)
z = R cos (phi)
Because we have cartesian coordinates after that point, we can add them across.
Final position = (x1 + x2 , y1 + y2, z1 + z2)
To get your spherical coordinates, you need to now convert backward:
R(final) = sqrt(x^2+y^2+z^2)
Theta(final) = arctan(y/x)
Phi(final) = arctan(sqrt(x^2+y^2)/z^2)
The latitude and longitude are then:
Lat = 90 - Phi(final)
Long = Theta(final)
Hopefully this helps. This is a vector problem. Any book on vector algebra will give you enough information to carry this task out (you just have to apply it to the scenario).
Here is a calculator that can help with that.
https://www.fcc.gov/media/radio/find-terminal-coordinates
To use it, you will have to use range and elevation to arrive at the distance. That is simple trigonometry as both of them are straight-line distances.
This may also help (it refers to spacecraft, but that shouldn't make a difference as you just have lower elevation):
http://www.mssl.ucl.ac.uk/grid/iau/extra/local_copy/SP_coords/local.htm
Thanks matthew but do you have reference or equation working behind the calculator?
Hapgood, M. A. (1992). Space physics coordinate transformations: A user guide. Planetary and Space Science, 40(5), 711-717.
You have two spherical coordinates. In essence. You must add them. What I am going to do here is a generalization that the radius of the earth is the same everywhere. I know that it is not, but because you have a geoid, you are better off making the assumption in this case. You have two spherical coordinates. To add them, we have to put them in Cartesian coordinates and then convert backward to spherical (and then convert to lat long because of the declination variable phi).
Since they do not start at the same point, you have to convert them.
Coordinate 1:
R = 3959 miles + elevation of the radar site
Theta = Longitude
Phi = 90 - Latitude (comes down from true north, rather than up from the equator)
Coordinate 2:
R = Range
Theta = Azimuth
Phi = 90 - Elevation
The conversion rules are:
x = R sin (phi) cos (theta)
y = R sin (phi) sin (theta)
z = R cos (phi)
Because we have cartesian coordinates after that point, we can add them across.
Final position = (x1 + x2 , y1 + y2, z1 + z2)
To get your spherical coordinates, you need to now convert backward:
R(final) = sqrt(x^2+y^2+z^2)
Theta(final) = arctan(y/x)
Phi(final) = arctan(sqrt(x^2+y^2)/z^2)
The latitude and longitude are then:
Lat = 90 - Phi(final)
Long = Theta(final)
Hopefully this helps. This is a vector problem. Any book on vector algebra will give you enough information to carry this task out (you just have to apply it to the scenario).
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