>Though he has his magnetic courses and headings a little mixed up, Robert
>Allardyce has the right idea. If you extend a VOR radial, say 400 miles,
>you will find that, due to magnetic variation, a varying mag heading would
>be required to maintain a centered CDI if it were possible to fly a single
>VOR radial for 400 miles. This radial would describe a straight line over
>the suface of the earth, but it would not be a great circle course.

Magnetic variation has *nothing* to do with whether or not a course is a
great circle.  Nor can you assume that all great circles cause the true or
magnetic track to change over a given distance.  For example, any flight
along a single meridian will have a constant TRUE track, as will any
flight following the equator.  Any flight that is proceeding directly
toward or away from the magnetic pole will have a constant magnetic (north
or south) heading even though the variation and true course are both
changing.  All of these examples are great-circle routes, if we ignore the
effects of local geology on the magnetic flux lines.

The *only* thing that matters when describing a great circle route is
whether or not the track follows the surface of the earth along a single
plane through the center of the earth.  Radio waves, by their nature, go
in straight lines.  If you draw a straight line on the surface of a globe
between any two locations, that line will be a great circle.  If you were
to put VORs at those two locations, the radial from one VOR that passes
over the other would plot a great-circle track, regardless of the distance
between them.

There appears to be something about VORs that you missed– the VOR radials
are NOT magnetic headings or bearings, but *radials*.  The 360-radial is
simply the radial on which the omni-directional signal and the directional
signal are in phase.  The "90 degree" radial refers *not* to East, but to
the azimuth at which the two radio signals are out of phase with each
other by 90 degrees.  That "in-phase" radial could be pointed in any
direction, but the VORs are normally aligned with the local magnetic north
as an aid to pilots so that they can use their compass to help visualize
the aircraft’s position relative to the VOR and assist in finding/tracking
the desired radial.  In places where the compass isn’t any good for this
purpose, the VOR is aligned in an arbitrary direction (usually either true
north or grid north).

>Now, for example, if I depart IND to fly to DEN using the GPS in the
>"direct" mode with that mode engaged upon departing IND, and the
>autopilot is immediately coupled and left to fly the aircraft
>uninterrupted to DEN based upon headings calculate by the GPS, a
>great circle route will result. This is due to the fact that we’re not
>stuck with a straight line drawn from IND to DEN as if we had followed
>a VOR radial the whole way.

If the VOR at DEN was strong enough to be heard at IND, you’d find that
your GPS direct route (changing headings and all) would track inbound to
DEN on one radial, and that radial wouldn’t change for the entire trip.

> Also, the GPS uses great circle
>routes as they are the most direct, therefore the most efficient. Upon
>observation of the aircraft heading as it proceeds along this route, it
>is readily apparent that the aircraft is flying what is a long
>curve (to our flat map oriented minds, anyway). If you were to plot the actual ground track, it would, in fact, be
>a curved line accross the map.
>Remember this: A great circle route drawn on a chart will look to
>you like a curve and a straight line drawn on a (sectional or WAC) chart
>is not really the most direct route from "A" to "B".

That depends on what map you are using.  No map (other than a globe) will
perfectly plot all great circles as a straight line.  Some are better than
others, however, and most aviation maps (including sectionals, WACs, VNCs,
etc.) use the Lambert Conformal projection because it has a low percentage
of error at a reasonable scale.  It *approximates* great circles as a
straight line, whether the great circle represented is a VFR plot or a VOR
radial airway.  A straight line drawn on an aviation chart will plot a
great circle route far more acurately than most pilots are capable of
flying, so the small percentage of error isn’t really a factor.

—
Chris Rasley   <http://www.mi.net/dialin/cpr&gt;
Moncton, New Brunswick, Canada.  High-Enroute ATC (CZQM), PP-ASEL

///
.  Any flight that is proceeding directly

>toward or away from the magnetic pole will have a constant magnetic (north
>or south) heading even though the variation and true course are both
>changing.  All of these examples are great-circle routes, if we ignore the
>effects of local geology on the magnetic flux lines.
///
>Chris Rasley   <http://www.mi.net/dialin/cpr&gt;
>Moncton, New Brunswick, Canada.  High-Enroute ATC (CZQM), PP-ASEL

No shiny cyber-nickel for you I’m afraid.
Look over the section I quoted.

brian

In any case, discussions of this nature are one of the reasons I so
greatly enjoy the newsgroups. I often find food for thought or some
comment that makes me go way back and think about something basic that
I have not thought about for a long time. Back to my roots, as it were.

Don

- Hide quoted text — Show quoted text -

>e…@alaska.net (R Wood) wrote:

>>d…@cp.duluth.mn.us (don) wrote:

>>(CLIP)
>>>This radial would describe a straight line over
>>>the suface of the earth, but it would not be a great circle course.
>>(CLIP)
>>>Don

>>Wouldn’t ANY straight line over the surface of the earth be a great
>>circle?
>OOPS…..
>I meant PART of a geat circle.

I agree. I think some are confusing constantly changing heading to mean a
constant lateral curve over the earth. Its not. A straight line is still
the shortest distance between two points. In order to navigate this
straight line one needs to constantly change heading as longitude
changes. If you were to have a gyro that didn’t precess and operated it
in free gyro mode, flying one heading on the gyro would be a great circle
although in realty your true and magnetic "tracks" are changing as
longitude changes. Grid nav is an example of this. Airlines today have
their auto pilot/flight director systems slaved to the INS or whatever
system and the proper headings are flown automatically. In other words
the airplane is kept straight as the compass heading changes.

OOPS…..
I meant PART of a geat circle.

Rhea Wood
N3489Y C-185
Alaska-Based Floatplane

>Though he has his magnetic courses and headings a little mixed up, Robert
>Allardyce has the right idea. If you extend a VOR radial, say 400 miles,
>you will find that, due to magnetic variation, a varying mag heading would
>be required to maintain a centered CDI if it were possible to fly a single
>VOR radial for 400 miles. This radial would describe a straight line over
>the suface of the earth, but it would not be a great circle course.

Yes and No.

You are correct in saying that a varying magnetic heading would be
required due to changes in magnetic variation between the VOR and the
aircraft. After all, the orientation of the VOR station is based on the
variation at the station.

But you are incorrect to say that that the straight line described by
the radial would not be a great circle. The other factor which need to
be considered is Earth Convergency, i.e. the fact that lines of
longitude are not parallel (except at the equator).

The basic problem which we have is that the convergency on the earth
will not equal the convergency on the chart at all places.

Before discussing this any further it’s probably worth saying that given
the inaccuracies of a VOR at long range, what I’m about to say is pretty
well academic rather than practical.

The best way to visualise Earth Convergency (EC) is to imagine that we
are going to fly very near to the north pole. The reason for this is
that EC is maximum at the poles. We’ll keep very close to the pole, so
we can view the earth as being flat, a bit like a record on a record
player.

Also I want to take magnetic variation out of the equation, by having
magnetic north and true north conincident. I accept that this is a bit
contrived, but it means we can concentrate on the EC since we already
agree that changes in variation between VOR and aircraft are relevant.

Right. Lets say we are going the fly from point ‘A’ to point ‘B’.

Point ‘A’ is on the Grenwich Meridian, 1 mile south of the pole.
Point ‘B’ is on the 180 meridian, also 1 mile south of the pole.

The Rhumb Line (RL) Track is 090 (or 270!), and is 1 x PI = 3.14 miles
long whereas the Great Circle (GC) Track is due north to the Pole, and
then due south to B. This route is only 2 miles long. So that
illustrates the benefit of flying GC rather than RL.

I expect you knew that already, the reason I’ve said it is to set the
scene for considering VOR navigation. Let us say that there is a VOR at
both ‘A’ and ‘B’.

To fly from A to B:

Using VOR ‘A’ we would fly along the 000 radial, i.e. OBS set to 000,
needle centred, ‘away’ flag showing. After 2 miles we would be at ‘B’.
The fact that we’ve flown over the pole half way and our true track
changed from 000 to 180 is irrelevant as far as the radio waves are
concerned.

Alternatively, using VOR ‘B’, we would also fly along the 000 radial,
but this time with OBS set to 180 and the ‘TO’ flag showing. Again,
after 2 miles we would be at ‘B’, with a track change half way along.

In fact, at either pole the EC between two points, i.e. the difference
in GC track orientations at each point, is equal to the change in
longitude between the points. In this example the change in longitude is
180 so the difference between the GC track measured at A and that
measured at B is also 180.

O.K. That’s enough freezing our backsides off. Lets go somewhere warm,
like the equator.

If we wanted to fly between 2 points on the equator which were 2 miles
apart, it clearly would not make sense to go via the north pole! In
fact, if we were precisely at the equator, RL track would be equal to GC
track, the equator being the only GC which is also a RL. So, there is no
Earth Convergency (EC) at the equator.

We have seen that:

        At the poles, EC = Change In Longitude (Ch Long)

        At the equator, EC = 0

At intermediate latitudes, you get progressively more EC the closer you
are to the poles, and progressively less the closer to the equator.
Around now you should start to feel a trig function coming on, in fact
it is SINE, giving zero EC when latitude is zero, and full EC when
latitude is 90 (i.e. the poles).

When the two points are at different latitudes, a working approximation
can be made by taking the mean average of the latitudes. So, bringing
that all together:

        EC between two points = Ch Long x Sine mean lat

- Hide quoted text — Show quoted text -

>Now, for example, if I depart IND to fly to DEN using the GPS in the
>"direct" mode with that mode engaged upon departing IND, and the
>autopilot is immediately coupled and left to fly the aircraft
>uninterrupted to DEN based upon headings calculate by the GPS, a
>great circle route will result. This is due to the fact that we’re not
>stuck with a straight line drawn from IND to DEN as if we had followed
>a VOR radial the whole way. Also, the GPS uses great circle
>routes as they are the most direct, therefore the most efficient. Upon
>observation of the aircraft heading as it proceeds along this route, it
>is readily apparent that the aircraft is flying what is a long
>curve (to our flat map oriented minds, anyway). If you were to plot the actual
>ground track, it would, in fact, be
>a curved line accross the map.

>Remember this: A great circle route drawn on a chart will look to
>you like a curve and a straight line drawn on a (sectional or WAC) chart
>is not really the most direct route from "A" to "B".

Whether RLs or GCs (or neither!) are shown as straight lines is
dependant on chart type. But at the end of the day the type of chart you
are holding does not influence the real world Navigation problem.

If I assume that on your chart a RL approximates a straight line, then
you could measure the rhumb line track off the map. I don’t know where
IND and DEN are, so for the sake of arguement I’ll assume that DEN is
true west of IND, and that they are both at 42 North, 3 degrees of
longitude separated.

(I’ve chosen those numbers because it give a nice round figure of 2
degrees for the EC between them)

So, the True RL track from IND to DEN is 270 (which ever end you measure
it). To fly the GC track, you would have to take account of the 2
degrees EC. This GC track would be a curve, lying to the north of the
RL. Half way along the GC, you would have a true track of 270. So, if
there is going to be a 2 degree change in measured GC track between IND
and DEN, then it must be 1 degree to the right at IND and 1 degree to
the left at DEN.

I.E. your initial true track would be 271, measured at IND,
progressively changing to 269 true, measured at DEN.

To do this with VORs, fly the 271 True ‘FROM’ IND and / or the 269 True
‘TO’ DEN (whatever that equates to in magnetic terms bearing in mind
local variation).

The fundamental problem is that although on the chart DEN will appear to
be under the 270 radial from IND, remember as you pointed out, Great
Circles look like curves on the chart. So the propagation path followed
by the VOR 270 radial, if marked on the chart, would curve to *miss*
DEN, whereas the 271 radial would curve to pass overhead.

In summary then, VOR radials *are* Great Circle, but you must make sure
you follow the right one!

As I mentioned at the begining, this is largely academic unless you are
at extreme latitudes, bearing in mind that VOR accuracy is as bad as +/-
5 degrees 95% of the time and can be as poor as +/- 7.5 degrees.

Thank Goodness for GPS! Now all we need is for the CAA to accept it!  

Bill Chivers
‘my other signature file has something funny at the bottom of it’

(CLIP)

>This radial would describe a straight line over
>the suface of the earth, but it would not be a great circle course.
(CLIP)
>Don

Wouldn’t ANY straight line over the surface of the earth be a great
circle?

Rhea Wood
N3489Y C-185
Alaska-Based Floatplane