Flying a rhumb line track

[StudentATP]

Flying a rhumb line track (amended)

Good day,

Could anyone help me with this question.

an aircraft is tracking a great circle track, there is a large bearing change between departure and arrival points. If the aircraft were to fly the rhumb line heading of the track at the departure point, would it end up arriving overhead the destination airport?

Thank you all in advance.

Sorry about the confusion.

what I'd like to ask is:-

if an aircraft is tracking a great circle track, neither the equator nor the meridians, it will need to change heading constantly as far as I know.

halfway of this great circle track, we find a rhumb line heading, if I were to use this heading I found halfway of the great circle, and take off and maintain such heading at the departure airport.

would I end up arriving at the destination airport?

[AndiKunzi]

The rhumb line will also bring you there, but you'd fly a continuous, shallow turn to maintain the (constant) true track (loxodrome = const. TT) (provided there's no wind). Following the orthodrome (great circle): no bank

[India Four Two]

Andi,

I think you misread the question. StudentATP was asking what happens if you fly the initial track heading and don't change it.

Unless you are flying a track along the equator, you would eventually end up at the North or South Pole, if you had enough fuel. This is easy to see if you plot a rhumb line on a Mercator map of the world, where a rhumb line is of course a straight line. You will never get to your destination.

[DFC]

Exccept for either

two places that are on the equator (both great circle and rhumb line),

two places that are on the same meridian or one meridian and it's antemeridian (eg 10W and 170E) (both great circle and rhumbline),

two places that are at diametrically opposite points (eg N pole and S pole) (can be joined by infinite number of great circles which are also rhumb lines)

then the direction of a rhumb line joining those points and the direction of a great circle joining those points will be different except at the point half way between them.

At the departure point or destination the direction of the great circle will always be closer to the nearest pole than the direction of the rhumb line.

So since they are in different directions, starting in the direction of the great circle and flying a rhumb line will cause the aircraft to not only fail to arrive at the intended destination but to also not follow the great circle.

[rudderrudderrat]

the direction of a great circle joining those points will be different except at the point half way between them.

I think that was what StudentATP was asking. If I steered the equivalent direction of the average great circle track between two points, (from the starting point) then I would make my destination by a slightly longer route as Andi pointed out.
e.g. steering due West (true) across the Atlantic from London, would get me to Newfoundland and I'd never get to the N. pole. (i.e. along a line of Latitude)

[Tim Zukas]

For any two points, there is at least one constant-direction track leading from one to the other. (Constant direction relative to true north, anyway, and maybe constant magnetic as well.) Given the lat-lons of the two points, it will take you a couple minutes to calculate the direction of that line-- is that what you mean by "the rhumb line heading of the track at the departure point"?

In other words, which "track" are you asking about?

(You know about straight lines on Mercator charts being rhumb lines?)

[unusualAtitude]

Rhumb Line

Hi,

my initial thoughts to your Q is yes. The benefit of the Rhumb line is to be
able to navigate from A to B using the one heading(still wind).

A rhumb line crosses all meridians at the same angle.
A great circle crosses all meridians at different angles.

The part about the large change in heading(great circle) is just to throw you.

Ua..

[DFC]

Perhaps the easiest way to asnwer this is to think of radio waves which of course always follow a great circle path between points.

So ignoring range limitations and variation/deviation and with no wind effect think of the following.

VOR B is on the 300 radial at 3000nm when measured at VOR A

So the initial great circle track at VOR A is 300.

If the pilot follows the radial continuously, the no wind true heading (rhumb line track) will change becuse we are following a great circle.

If however, we maintain a constant heading (rhumbline track) of 300, one can easily picture how we will diverge from the VOR radial (depending on hemisphere).

[bfisk]

an aircraft is tracking a great circle track, there is a large bearing change between departure and arrival points. If the aircraft were to fly the rhumb line heading of the track at the departure point, would it end up arriving overhead the destination airport?

The answer is it depends on wheter the equator is crossed, and how

The reason for this is as follows:

an aircraft is tracking a great circle track, there is a large bearing change between departure and arrival points.

This indicates we are dealing with a great circle which is not along the equator, nor along a meridian. (This can be proven mathematically; since the difference in a great circle track and a rhumb line track in either end is +- 0,5*dlong*sin mlat, either dlong or sin mlat have to be zero for the function to be zero (ie for there to be no "bearing change"); dlong=0 occurs on a track exactly north/south (along a meridian), and sin mlat=0 occurs where the mean latitude is zero. This is along the equator (on a east-west track along the equator), but also at any other track crossing the equator at the semispan)

If the aircraft were to fly the rhumb line heading of the track at the departure point, would it end up arriving overhead the destination airport?

I assume this to mean "if the aircraft were to continously fly the GCTTinitial track, to make a rhumb line".

Initally, the answer would then be no: because we know there's a difference between the rhumb line track and the great circle track, then logically if one leads to the destination then the other will not, if they are diverging tracks (they start out on the same heading, one goes straight and the other one curves away, they are diverging.)

However, there is again one possibility; a track that has an east-west componet, and a north-south component, where the arrival and departure are in different hemispheres: is it possible to come up with two places that have both the same true rhumb line track and great circle true track initial?

Let's how about 4500S 00000E to 5800N 13554E -- the great circle true track starts at 48,07 degrees and ends at 96,29 degrees -- however the rhumb line track between them is also 48,07 degrees! We see they are not on the equator, they are not equidistant from the equator, and they are not diametrically opposite, and they are not on the same meridians, or antimeridians.

Two other points however, may or may not have the same features. However, you will not find such a track where both arrival and departure points are in the same hemisphere.

Edit: the precise tracks and thus the precice points I've chosen vary slightly from calculator to calculator, so you may get slightly different results when trying it out, but move the destination around a little bit and you'll see the principle still remains.

[BOAC]

This indicates we are dealing with a great circle which is not along the equator, nor along a meridian

- actually, bfisk first part right, second wrong. All that part of the question is telling you is that the route is NOT along the equator in which case rhumb and g circle are identical and you WOULD arrive at destination.

For Student ATP - the clue is there from 'unusualAtitude' - find yourself a globe and trace an imaginary line anywhere other than around the equator. Try maintaining the same angle of crossing of the meridians from, say London to N York. There's your answer.

[Pugilistic Animus]

Oh C'mon just follow the magenta line; it knows everything

[bfisk]

This indicates we are dealing with a great circle which is not along the equator, nor along a meridian

- actually, bfisk first part right, second wrong. All that part of the question is telling you is that the route is NOT along the equator in which case rhumb and g circle are identical and you WOULD arrive at destination.

an aircraft is tracking a great circle track, there is a large bearing change between departure and arrival points

If there's a "large bearding change between departure and destination", then how can said great circle be along a meridian? Surely a track along a meridian has no bearing change (it can only be 180/360?)?

Perhaps we are speaking past each other...

[BOAC]

Fly north along a meridian (Greenwich perhaps?), and tell me what heading you have when you depart the north pole on the same track? Is that a "large bearing change"?

[bfisk]

I'd beg to differ; the meridian ends at the north pole. By defintion you would be continuing on the antimeridian.

Given the following example however: a flight from 60N 000E to 60N 180E would have a GCTTin of 360T, however the rhumb line track would be either 090T or 270T, not 360, because it would then take you to the north pole, but not further, so it does not fulfill the requirements of the original question.

I do appreciate your points however and I am willing to admit I'm splitting semantic hairs right now. But that's kinda what General Nav is these days, rather than finding your way, isn't it...

[Old Smokey]

Nothing wrong with Rhumb line Tracks, they worked well for centuries

To address the original question, ABSOLUTELY NO, maintaining the initial Heading will get you nowhere (or at least to the wrong place). Magnetic Variation constantly changes, and even in a nil wind no drift situation, Heading will have to be constantly changed.

Of Course, if you have the facility to fly TRUE Tracks and/or Headings, it would be a piece of cake!

Regards,

Old Smokey

[BOAC]

Of Course, if you have the facility to fly TRUE Tracks and/or Headings, it would be a piece of cake!

- but you would still 'miss' unless you live on a flat earth

Student - now you have altered your question, I think the answer is now yes but I cannot recall the proof.

[Dan Winterland]

I once flew a rhumb line from Gatwick to Barbados.






Or was it a rum line?

[kijangnim]

Greetings
You will follow the rhumb line spiraling around the earth and reach your destination, the trip distance will be far greater of course.
Magenta line is Great circle (so is the Green)

[bookworm]

halfway of this great circle track, we find a rhumb line heading, if I were to use this heading I found halfway of the great circle, and take off and maintain such heading at the departure airport.

The rhumb line track that takes you to the destination airport will be somewhere between the start track for the great circle and the end track for the great circle. It's only the track at the midpoint in special cases (e.g. dep and dest on the same parallel).

[bfisk]

I still maintain that if you cross the equator, in certain conditions, the true rhumb line track will be the same as the initial true great circle track.

We know that a great circle is a straight line on the globe, and that a rhumb line spirals towards the poles (ie spirals away from the equator). However, as the rhumb line crosses the equator, it will start to spiral towards the opposite pole; giving the rhumb line an S shape on the globe.

Try it for yourselves; I've given sample coordinates already (4500S 00000E to approximately 5800N 13554E - move it slightly around to get it accurate depending on which calculator you use.)