DFC asks -
"how would the FMS decide where to locate the "abeam B" waypoint?"
It does it like this -
(1) Calculate the Track from A to C,
(2) Calculate the Track and Distance from A to B (Remembering that the Distance is in Degrees subtended to the Earth's centre, not ground miles),
(3) Find the absolute value of the Difference in Track from A-C and A-B.
Use the following equation to find the distance (in degrees) from the Abeam B position to B -
Distance Abeam = Arc Sin (Cos(90-Difference in Track) X Cos(90-Distance from A to B))..... The answer is in Degrees.
Now use the following equation to find the distance (in degrees) from A to the Abeam B position on the A to C Track -
Distance to Abeam Point = Arc Sin (Tan (Distance Abeam found in first step) X Tan (90-Difference in Track).
The FMC then applies the distance to Abeam Point found above to the Spherical Triangle formed by the Pole, Position A, and the Abeam point to calculate it's Lat & Long and Instantaneous Track.
That's how the FMC does it!
To give an example, Track from A-C is 280 degrees, Track and Distance from A-B is 285 degrees and 10 degrees (600 nm). Difference in Track is 285-280=5 degrees.
Distance Abeam = Arc Sin (Cos(90-5) X Cos(90-10)) = .867172409 Degrees (52.03 nm)
Distance to Abeam Point = Arc Sin (Tan (.867172409) X Tan (90-5)) = 9.962710787 Degrees (597.76 nm).
That angular distance is then applied to the A-C Great Circle Track using Spherical Trigonometry to find Position and other relevant data.
DFC, you quoted
99% of the time we navigate using great circle tracks i.e. we follow either straight lines on something like a conformal lambert's conic projection chart . That is
CATEGORICALLY NOT a Great Circle. It's close, with accuracy dependant upon proximity of the Standard Parallels and your Track's displacement from them, but Great Circles they aint. Only straight lines on Gnomonic Charts are Great Circles, but these are impractical for navigation due to extreme scale varaition over large distances.
Does
ANYONE know
ANYTHING about Spherical Trigonometry any more?
ANYONE? It is, after all, at the heart of Great Circle Tracking that our FMC/FMSs provide us every day!!!
Regards,
Old Smokey