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

Question

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 ?

Answer 1

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

Using your subscript nomenclature:

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

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

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

Then: V_a = V_s + V_sa

And V_sa = V_a - V_s

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

The dot product can be used to get the angle:

v_cdi ⋅ V_sa = ‖v_cdi‖ ‖V_sa‖ cos α

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

But ‖v_cdi‖ ( a unit vector) is 1.

So v_cdi ⋅ V_sa = ‖V_sa‖ cos α