This project lets you determine the azimuth and elevation of a dish antenna you see from some location. You need to know your location (we call this the observer location in the following) and the location of the antenna:

• The height of both locations
• The distance
• The direction the observer is located from the antenna (for azimuth correction)

There is currently much code to originally derive the equations needed to compute the dish azimuth and elevation. The antenna is seen from the observer location as an ellipse (e.g. when you take a photo). The ellipse major axis is called l1, the minor axis l2.

Our coordinate system: Z points up, Y points away from the observer to the antenna, X points right as seen from the observer. This is a right handed coordinate system. The right-hand rule holds, to determine which is the positive rotation angle around an axis, point your right-hand thumb in the direction of the positive axis, then the fingers point in the direction of the positive rotation angle.

To sum up:

• Measure l1, l2 of ellipse and compute beta = arccos (l2 / l1)
• Measure alpha: angle between l1 and vertical Z-axis, positive angles turn the ellipse right, we measure by how much ellipsis needs to be turned to be upright
• In the real world determine psi, our angle to the antenna. If we're higher than the antenna, the angle is positive, if we're lower, negative. If we're the same height psi = 0.

• Compute theta, phi, gamma:

  theta=-asin(sin(alpha)*sin(beta))
  phi=-asin((cos(alpha)*sin(beta))/cos(theta))
  gamma=asin((cos(phi)*sin(theta))/sin(beta))

• Compute theta_r, phi_r, xi:

  theta_r=asin(cos(psi)*sin(theta)+cos(phi)*sin(psi)*cos(theta))
  phi_r=asin((sin(phi)*cos(theta))/cos(theta_r)) 
  xi=-asin((sin(phi)*sin(psi))/cos(theta_r))

Now theta_r is the real elevation of the antenna and phi_r is the real (well uncorrected, we're still asuming we're in the north of the antenna) azimuth.

Azimuth correction (correcting the asumption that we're north of the antenna) is left as an exercise.

This project contains an OpenScad file to simulate the rotation. The antenna without rotation (azimuth = 0, elevation = 0) points to the observer. You can rotate the antenna:

• By specifying a non-zero elevation
• By specifying a non-zero azimuth
• By specifying a non-zero phi-angle between observer and antenna, a possitive angle means the observer is higher than the antenna

Rotations in openscad get an object to rotate as parameter, so the following code:

rotate ([0, 0, -51.066])
rotate ([0, 62.966, 0])
schuessel (ele = 60, azi = 45, psi = 20)

Specifies a dish (Schüssel in German) with 60° elevation and 45° azimuth where the observer is located 20° above the antenna. This antenna is then rotated by 62.966° around the Y-axis, then around the Z-axis by -51.066°.

After starting openscad with ellipse.scad, you want to select Front from the View menu. An un-rotated antenna is visible as a circle (so it points at the observer). The inside of the antenna is yellow, the backside is green.