I'm trying to find a way to calculate look angles to not just GEO satellites, but LEO, MEO, and HEO sats. The only calculator I've found myself is here, which was referenced for a similar problem on Stack Exchange in regards to calculating the longitude of GEO satellites GIVEN look angles here: Computing GEO satellite's longitude from elevation/azimuth from a given latitude/longitude?

I'm interested in two functions/formulas, both of which are calculated on that first link I posted.

  1. $Az, \ El = f(earth_{lat}, \ earth_{lon}, \ sat_{lat}, \ sat_{lon}, \ altitude)$. I.e. look angle from earth station and satellite locations
  2. $sat_{lat}, \ sat_{lon} = f(earth_{lat}, \ earth_{lon}, \ az, \ el, \ alt)$

So far I've started by subtracting the ECEF (geocentric) vector to the station from that of the satellite:

$v_{1} = [r*cos(earth_{lat})*cos(earth_{lon}), \ r*cos(earth_{lat})*sin(earth_{lon}), \ r*(1-f)*asin(earth_{lat})]⋅T$

$v_{2} = [R*cos(sat_{lat})*cos(sat_{lon}), R*cos(sat_{lat})*sin(sat_{lon}), R*(1-f)*sin(sat_{lat})]⋅T$

$V = v_{2}-v_{1}$

Where $r$ is the radius of Earth, $f$ is the flattening of Earth, and $R = sat_{alt} + r$.

If I'm not mistaken this should yield the vector pointing from the station to the satellite. So now I just have to transform these coordinates from ECEF to ENU (topocentric):

$t_{1} = [-sin(earth_{lon}), cos(earth_{lon}), 0]$

$t_{2} = [-sin(earth_{lat})*cos(earth_{lon}), -sin(earth_{lat})*sin(earth_{lat}), cos(earth_{lat})]$

$t_{3} = [cos(earth_{lat})*cos(earth_{lon}), cos(earth_{lat})*sin(earth_{lon}), sin(earth_{lat})]$

$T = [t_{1}, t_{2}, t_{3}]$ (ECEF to ENU transformation matrix)

$V_{ENU} = T*V$

Lastly, I calculate the azimuth and elevation:

$Az = arctan2(V_{ENU(y)}] / V_{ENU(x)}] )$ $El = arcsin(V_{ENU(z)}/ ||V_{ENU}||] )$

This keeps giving me the wrong answer, at least according to that calculator I referenced. What am I missing?

  • $\begingroup$ You might want to include the write/wrong answers you're working with, and code if that's what you're using. The logic sounds correct. $\endgroup$ May 10, 2022 at 21:02

1 Answer 1


In your equation for $t_2$, the middle term ought to be $$-\sin(earth_{LAT})*\sin(earth_{LON})$$ not $$-\sin(earth_{LAT})*\sin(earth_{LAT})$$

Check to see if your code has the same error. You should also try using the transpose of your T matrix, and see if that fixes it. Beyond that, I'd ask where are you getting your satellite positions? Might they be in ECI, rather than ECF? Expressing what a HEO orbit looks like in ECF is a bit weird.


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