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Great Circle Track - FMC/FMGS

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Great Circle Track - FMC/FMGS

Old 19th Oct 2009, 07:02
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Great Circle Track - FMC/FMGS

Hi Guys,

Here's something that has been bothering me for the past week. As we all know, our FMC/FMGS calculates the IMT and dist from point A to point B based on great circle track.

My question is, if we go direct from point A to point C, bypassing point B, as compared to point A to Point C with abeam point of point B inserted into the FMC, will the distance travelled by the a/c be different?

My logic being the FMC will calculate the great circle track from point A to ABEAM point B then another great circle track from ABEAM point B to point C. Will the recalculation differs from a direct great circle track from point A to C ?

My guess is there will be difference in distance travelled in both cases.
Any comments ?

Hope the above explaination is clear enough

Thanks for your help,
Cheers,
lion-g
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Old 19th Oct 2009, 12:18
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I promise you, the Track from A to C, abeam point B, will be an exact Great Circle. It would be identical to Track from A to C without any intermediate points.

What may be confusing you is that on the LEGS Page, Initial Magnetic Track (IMT) is shown, and, as the Magnetic Track changes constantly throughout any Great Circle Track, the IMT abeam of point B will without doubt be different to the IMT at point A. Put in 10 Abeam waypoints on the same Great Circle Track, and you will see 10 IMT's All of them are components of ONE GC Track from A to C.

Another point which might possibly cause confusion is when you take a 'Direct To' with Abeam Waypoints, when the first Abeam Waypoint is quite close. The FMC computes Turn Radius for the first segment (that is to Abeam of the first waypoint) but not the entire Track. Thus, it is possible to see a kink in the Track abeam the first waypoint - The Fix? - Let the aircraft settle down on the new 'kinked' track, and re-insert Direct To the distant point with Abeam Points selected again, et voila!, the kink is gone

See you on line

Old Smokey
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Old 20th Oct 2009, 21:21
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There are an infinite number of points that are "abeam point B". If we understand your question, you're proposing to travel via the one point that's "abeam point B" and is also on the great circle from A to C. So you're always on the great circle, if you (or the machine) calculated the lat-lon of that point correctly. Are you asking whether you can trust the machine to do that?

If that's what you're wondering, there are calculators on the web you can use to check the machine.
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Old 20th Oct 2009, 21:34
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I'm with Old Smokey on this one! The great circle line between A and C remains ...... a great circle! You can have a multitude of abeams wherever you like - it isn't going to change the track of A to C.

(An' anyway, Ol' Smokey knows his onions like you would not believe! That is deep dark water, arguing the toss with Ol' Smokey! He will done wup yo' ass!)
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Old 21st Oct 2009, 20:48
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The Great Circle track between any two given points of equal latitude will produce a true heading of either East or West at the mid longditude. If you were to travel between N60.00.00 W10.00.00 and N60.00.00 W50.00.00, you would cross W30.00.00 on a true heading of 270 degrees True (the apogee of the great circle track). The displacement of track from the N60 Latitude would be proportional to the Sine of the Change of Longditude. The Sine of 60 degrees is 0.5; therefore, the Convergency is 30 degrees. The Initial True Track is half the convergency = 285 degrees. The final True track into N60.00.00 W50.00.00 would be 255 degrees.

If you were to include any other waypoints between those that I've used in the above example (even if they were directly 'on track' of this particular Great Circle example), the Initial True Track (and final True Track) to any other waypont will be different to the example... therefore, the displacement (abeam) will allways be different. If you split a great circle track into segments, the initial True Track between waypoints will allways be different; because they are different Great Circles!

John
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Old 22nd Oct 2009, 06:35
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TheChitterneFlyer has nicely reinforced the truth in different words to mine, but we stand together

Chitterne has stated correctly that in inserting en-route waypoints for a continuous Great Circle Track, the integrity of the Entire GC Track is maintained, but now broken into smaller components, each of which will, by dint of Longitudinal convergence and changeing Magnetic Variation, have differing Initial Magnetic Tracks. This, then, is a series of Great Circles, all of which exactly make up the 'complete' Great Circle. This is akin to what we did in olden days pre-FMC/INS, the Great Circle over a long distance would be broken into smaller 'flyable' Rhumb Lines. Such a case is NOT a continuous GC, but a composite routing, simply because aircraft of earlier eras did not have Great Circle navigational capability.

I will make one very minor concession in the continuous GC discussion, and that is that SOME units calculate the new en-route positions, and round off the results to DMT (Degrees, Minutes, and tenths of Minutes) or DMH (Degrees, Minutes, and hundredths of Minutes). It's then up to the programmer to decide if the intermediate Tracks are from the 'rounded off' positions, or the much more accurate position of up to 10 decimal places which is retained 'internally, with DMT or DMH used for display. This might produce very minor track kinking, but probably not so much that you'd notice.

Yours truly is guilty of writing programmes for the LNAV portion of some FMC units, and I have always opted for the exact position used for navigation, and the 'rounded off' position (DMT or DMH) used for display. (Do you REALLY want to know that your latitude is 61.3752398231 degrees North? I think not.)

Bear in mind also that FMC/FMS units have internal prioritisation. The highest of all priorities is the acquisition of the aircraft position. The LAST of the unit's priorities is map display, which, with several seconds between display refresh, may occasionally produce a slightly kinked track display, which LOOKS like map shift as screen refreshes occur, but is not.

Regards,

Old Smokey
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Old 22nd Oct 2009, 08:24
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Hey guys,

Thank you very much for all the interesting replies. Really learn a lot of things from you guys, hope to fly with you one day, Old Smokey :P.

I totally understand that there will be a change in the IMT for each great circle with a change of latitude.

So can I safely say, the FMC will fly a Great Circle track from point A to Abeam point B and another Great Circle Track from abeam point B to point C and the distance travelled is THE SAME as the Great Circle Track from point A directly to point C ?

Just to conclude the discussion. To make sure I got the concept right.

Thanks guys !!

Cheers.
lion-g
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Old 22nd Oct 2009, 10:33
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Yup! You have it right! Just to remove one slightly lingering doubt from one of your comments, I quote your comment with the worrying word emboldened, and my suggestion for revision with my suggestion also emboldened -

"FMC will fly a Great Circle track from point A to Abeam point B and another Great Circle Track from abeam point B to point C and the distance travelled is THE SAME as the Great Circle Track from point A directly to point C ?"

Changed to -

"FMC will fly a Great Circle track from point A to Abeam point B and CONTINUE ON THE SAME Great Circle Track from abeam point B to point C and the distance travelled is THE SAME as the Great Circle Track from point A directly to point C ?"

Just being pedantic to be absolutely sure!

Whilst still in pedantic mode, you mentioned Latitude change as the primary cause of GC continuous Track change. Actually it is a combination of Latitude AND Longitude if you revert back to the basic Spherical Trigonometry computations.

BTW, The map projection displayed is Antipodeal Stereographic, which can make some straight lines look 'bent' at times. Gnomonic would be better in this respect, but presents unacceptable scale expansion at greater distances.

Regards,

Old Smokey

Last edited by Old Smokey; 22nd Oct 2009 at 10:43.
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Old 22nd Oct 2009, 11:56
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So can I safely say, the FMC will fly a Great Circle track from point A to Abeam point B and another Great Circle Track from abeam point B to point C and the distance travelled is THE SAME as the Great Circle Track from point A directly to point C ?
Not only will the distance be the same, the track will be the same.

You are essentially asking (albeit in a spherical geometry): if I walk in a straight line from A to C, stopping at a point B along the way, will the distance travelled be different from walking directly from A to C without stopping?

In other words - whether or not you program it in to the FMC, you will ALWAYS pass the point B that is both abeam to the fix and on the Great Circle Track from A to C, and you will always travel the same distance regardless. And unless there is a bug, the FMC will know this.

In fact, you always pass over infinitely many other points along your track. Each of them are, in turn, abeam infinitely many other points lying on a line through that point and perpendicular to the Great Circle Track.

Last edited by dtaylor1984; 22nd Oct 2009 at 12:09. Reason: try to remove some of the confusion, and probably fail
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Old 22nd Oct 2009, 16:27
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" The displacement of track from the N60 Latitude would be proportional to the Sine of the Change of Longditude. The Sine of 60 degrees is 0.5; therefore, the Convergency is 30 degrees. The Initial True Track is half the convergency = 285 degrees. "

In case it wasn't clear, that's an approximation. (Also a typo.) Initial track for a change in longitude of 60 degrees will actually be 296.6 deg; for a change of 40 deg in longitude it will be 287.5 deg.
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Old 23rd Oct 2009, 06:05
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Thanks a lot guys for sharing your wonderful knowledge with me.

What an eye opener !!

Thank you
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Old 23rd Oct 2009, 09:01
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My question is, if we go direct from point A to point C, bypassing point B, as compared to point A to Point C with abeam point of point B inserted into the FMC, will the distance travelled by the a/c be different?

My logic being the FMC will calculate the great circle track from point A to ABEAM point B then another great circle track from ABEAM point B to point C. Will the recalculation differs from a direct great circle track from point A to C ?
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 or we follow radio signals which of course always follow the great circle track.

Based on the question posed it is clear that A, B and C are not on a great circle track - in simple terms, they would not be on the centerline of an airway defined by a VOR for example. If they were then changing the route from A-B-C to A-C would still cause the aircraft to overfly B.

Since there is a change in routing overhead B then the following applies-

The FMS will calculate a great circle route between A and C - just like if you tracked outbound from a VOR at A on a radial towards C. At some place on this great circle the aircraft will be abeam B and this point on the great circle between A and C is marked at the point abeam B.

If this was not the case, how would the FMS decide where to locate the "abeam B" waypoint?

One will find that the abeam point is a point on the surface of the earth where a great circle passing through A and C intersects a great circle passing through B at an angle of 90 degrees.
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Old 23rd Oct 2009, 15:10
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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
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Old 23rd Oct 2009, 17:46
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spherical and speheroidal

Hey Smokey!
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Old 23rd Oct 2009, 20:34
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Old Smokey,

You misunderstand the point I was trying to make.

As I said the abeam point will be the point on the surface of the earth where the great circle from A to C intersects a great circle passing through B at an angle of 90 degrees. There are only two possible places on the earth where that happens and generally one will be a lot further from A than the other!!.

I was responding in simple terms to the point that there could be a GC from A to the abeam point and then a second different great circle from the abeam point to C. If this were the case then the abeam point could be in a number of places at various distances from A, B and C but each of the unlimited possible points satisfies the "Abeam B" requirement. Therefore that is not a good idea.

Simple non-trig tables answer.

PS You are correct with the chart however if you want to be pedantic, man started navigating by great circle from the time that his fellow man was able to point the direction out i.e Over there = great circle. It was only the invention of the compass and lat/long that complicated matters.
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Old 23rd Oct 2009, 21:50
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Sorry if my post seemed critical DFC, you have a history of fine and respectable posts, my pedantic self zeroed in on the embedded question in your post, "how would the FMS decide?" My sincere apologies for apparent criticism of what was a very sound and constructive post.

enicalyth, where've you been? Your erudite words on these pages have been sorely missed, don't stay away too long again, at least without getting a note first! I must admit that as I scrolled down the page and saw your name in the header, I thought "Oh Sh!t, enicalyth is going to blow me out of the water with a WGS84 response and I'll have to scuttle off and hide somewhere!!! J_T's holding the fort well with a good OEI during SID thread, I might have to scuttle off and make mischeif there, particularly as Mutt hasn't emerged from the desert dust to intervene!

Best Regards to two fine posters,

Old Smokey
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Old 24th Oct 2009, 21:09
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Kindly expand.

Could you kindly expand given the following:
Flying directly from N60.00.00 W010.00.00 to N70.00.00 W050.00.00, in other words where the latitude changes.
Thanks,
Alex
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Old 24th Oct 2009, 21:17
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In the example given, the Latitude changes continuously. The only Great Circle Track where the Latitude does not change is flying the Equator.

Regards,

Old Smokey
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Old 26th Oct 2009, 21:38
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"Could you kindly expand given the following:
Flying directly from N60.00.00 W010.00.00 to N70.00.00 W050.00.00, in other words where the latitude changes."

We can expand to your heart's content. What's the question?
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