# How do you calculate the deviation from an RNAV waypoint radial?

As a clarification at the beginning: I'm talking about non-GPS, pure VOR-DME based RNAV devices here.

On RNAV devices that use just the selected VOR-DME, not GPS, the phantom radio station is set up by a desired bearing from the VOR-DME station and a distance. Then you can select a desired radial from that phantom station (instead of the VOR-DME station), and the deviation from that radial is then indicated by the CDI.

Given

• the aircraft position relative to the station as radial $$R_a$$ in degrees and distance $$D_a$$ in nautical miles
• the desired phantom station position relative to the VOR-DME station as radial $$R_s$$ in degrees and distance $$D_s$$ in nautical miles
• the selected radial of that phantom station $$R_{cdi}$$ in degrees

how would you calculate the deviation $$D_{cdi}$$ from the selected radial $$R_{cdi}$$ of the phantom station ?

### EDIT:

This is what I have tried so far:

• $$x_a = D_a * cos(rad(R_a))$$, $$x_a$$ being the X position of the aircraft relative to the VOR-DME
• $$y_a = D_a * sin(rad(R_a))$$, $$y_a$$ being the Y position of the aircraft relative to the VOR-DME
• $$x_s = D_s * cos(rad(R_s))$$, $$x_s$$ being the X position of the selected RNAV phantom station relative to the VOR-DME
• $$y_s = D_s * sin(rad(R_s))$$, $$y_s$$ being the Y position of the selected RNAV phantom station relative to the VOR-DME
• $$D_{as} = \sqrt {(x_a - x_s) ^ 2 + (y_a - y_s) ^ 2}$$, $$D_{as}$$ being the distance between the selected phantom station and the aircraft in nautical miles
• $$B = deg(atan2(y_a - y_s, x_a - x_s))$$, $$B$$ being the bearing of the selected phantom station to the aircraft in degrees
• $$D_r = sin(rad(R_s - B)) * D_{as}$$, $$D_r$$ being the deviation of the aircraft from the selected radial of the selected phantom station

I've tried using these formulas, but I ended up with nonsense values for $$D_{as}$$ - 0 if $$D_s$$ was 90, close to zero for smaller values and 90 for $$D_s = 0$$. I must say that I was never good in trigonometry … so please, can you tell me what is wrong with theses formulas ?

• It’s just triangles and trig. What have you tried. How far have you got?
– Jim
Feb 2, 2022 at 3:43
• @Jim I edited my question. Feb 2, 2022 at 14:23
• Maybe you could get an answer on Math SE? Seems like a better fit… Feb 2, 2022 at 15:50
• @MichaelHall maybe you are right … but anyways it looks like I just made a mistake somewhere as it's giving plausible values now. Feb 2, 2022 at 16:36

To keep it simple and hide the trig here we can do it with vectors.

Define Va as the vector from the VOR to the aircraft. (The VOR/DME gives this directly in polar coordinates (Radial, distance).

Define Vs as the vector from the VOR to the phantom station. This is a given when setting up the phantom station.

Define Vsa as the vector from the phantom station to the aircraft.

Then: Va = Vs + Vsa

And Vsa = Va - Vs

The CDI shows angular deviation from the selected radial. Define a unit vector vcdi for the selected CDI radial.

The dot product can be used to get the angle:

vcdiVsa = ‖vcdi‖ ‖Vsacos α

Where α is now the angular deviation of the aircraft to the selected CDI radial.

But ‖vcdi‖ ( a unit vector) is 1.

So vcdiVsa = ‖Vsacos α

• Could you please add an example calculation, say, for $V_a = (135°, ~14.14 nm)$ and $V_s = (90°, 10 nm)$ (so $V_{sa} = (45°, ~4.14 nm)$) and $V_{cdi} = (180°, 1)$ - because I don't get how to calculate the CDI needle deflection from $V_{sa}$ and $V_{cdi}$. Mar 27, 2022 at 11:15