Magnetic Compass Errors
Tom Hubin
I need to build some corrections, such as variation and deviation, into
software for magnetic compass data. Earth location and time variation
corrections are available from the US Gov't but deviation is unique to
each platform.
I would like to find some discussion or math for determining and
correcting magnetic compass errors. The compass types would be typical
for a 30 foot sailboat or larger. Usually magnetic needle damped in a
fluid. But Hall effect and other electromagnetic compasses might be
used.
In particular I would like to know how to compensate for fixed metal
objects and constant magnetic fields located on the boat.
Is there a newsgroup that includes magnetic field discussion or
navigation technical info?
Tom Hubin
thu...@clark.net
Sam Wormley
Try: http://www.cnde.iastate.edu/staff/swormley/maps.html#declination
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Paul Lutus
news:3A249A...@clark.net...
> Hello,
>
> I need to build some corrections, such as variation and deviation, into
> software for magnetic compass data. Earth location and time variation
> corrections are available from the US Gov't but deviation is unique to
> each platform.
What? Each platform? Don't you mean each location?
> I would like to find some discussion or math for determining and
> correcting magnetic compass errors. The compass types would be typical
> for a 30 foot sailboat or larger. Usually magnetic needle damped in a
> fluid. But Hall effect and other electromagnetic compasses might be
> used.
>
> In particular I would like to know how to compensate for fixed metal
> objects and constant magnetic fields located on the boat.
This is very difficult. You haven't started addressing the primary issue,
which is deviation WRT geographic location.
What is your goal? The navigator is still going to be looking at a compass
that has variations caused by local disturbances.
[ long pause ... ]
Unless you're discussing compasses that only talk to a computer. That's a
different story.
Do you know what "swinging a compass" means? It is a procedure for obtaining
the variations in magnetic bearing caused by local disturbances. The boat's
pilot turns a slow, steady circle in calm waters while the computer gathers
data from a compass that is able to talk to the computer. The difference
between the boat's presumably steady rotation and the compass bearings
should reveal any local errors.
This procedure is a common preliminary step for electronic tillers used on
sailboats. And it is not at all exotic -- the program assumes the boat is
being swung at a constant angular rate, and gathers a full circle of data.
The final step is for the pilot to tell the computer which of its relative
bearings corresponds to an actual compass bearing. Once this final step has
been completed, the computer has a good idea of the pattern of errors caused
by local stray fields and distortions.
--
Paul Lutus
www.arachnoid.com
Ron Hardin
iteratively take half the error out over a number of headings,
and the result is a compass correction card. Nobody uses the
compass as a precision instrument though, since
winds make you determine the right heading experimentally anyway.
--
Ron Hardin
rhha...@mindspring.com
On the internet, nobody knows you're a jerk.
Richard Herring
> "Tom Hubin" <thu...@clark.net> wrote in message
> news:3A249A...@clark.net...
> >
> > I need to build some corrections, such as variation and deviation, into
> > software for magnetic compass data. Earth location and time variation
> > corrections are available from the US Gov't but deviation is unique to
> > each platform.
> What? Each platform? Don't you mean each location?
Pay no attention to Lutus. Evidently the self-proclaimed lone
yachtsman doesn't know the difference between variation and deviation.
[Variation is the difference between true and magnetic north as a
function of position and time. Deviation is the difference between
magnetic and compass north, which depends on the compass heading and
nearby (on-board) magnetic materials]
> > I would like to find some discussion or math for determining and
> > correcting magnetic compass errors. The compass types would be typical
> > for a 30 foot sailboat or larger. Usually magnetic needle damped in a
> > fluid. But Hall effect and other electromagnetic compasses might be
> > used.
> >
> > In particular I would like to know how to compensate for fixed metal
> > objects and constant magnetic fields located on the boat.
The traditional method is simply to create a deviation card.
Align the vessel on a series of known true bearings
(radar is useful for this, or you could measure horizontal
sextant angles) and measure the indicated magnetic heading. The
difference is the deviation. Of course, this assumes you know
the local variation accurately. If not, you'll have to average over
several locations and assume the mean charted variation is accurate.
Generally the deviation card shows a roughly sinusoidal variation
of deviation with heading; if there's a constant offset it might
well represent an error in the predicted variation.
--
Richard Herring | <richard...@baesystems.com>
failure to respond may imply broken news server, not agreement :-(
Richard Herring
> With airplanes we used to park over the airport compass rose and
> iteratively take half the error out over a number of headings,
> and the result is a compass correction card. Nobody uses the
> compass as a precision instrument though, since
> winds make you determine the right heading experimentally anyway.
That's where sailors sometimes have the advantage: tides are
relatively predictable. Before now I've set a compass course
and 12 hours later seen the Barfleur lighthouse appear over the
horizon right on the nose. In that 60-mile passage the tide
carried us sideways 20 miles one way and back 10 miles the other.
Yaakov Eisenberg
> [Variation is the difference between true and magnetic north as a
> function of position and time. Deviation is the difference between
> magnetic and compass north, which depends on the compass heading and
> nearby (on-board) magnetic materials]
Do you mean that the difference is
some function of heading, and which
function it is depends on nearby
magnetic materials?
At first I read your statement as
treating "compass heading" and
"nearby magnetic materials" equivalently,
so that even with no magnetic materials
nearby, there might still be a difference
that depends on heading. But I couldn't
figure out how that could be true.
Richard Herring
> Richard Herring <r...@gmrc.gecm.com> wrote in message news:3a24e0b2$1...@pull.gecm.com...
> > [Variation is the difference between true and magnetic north as a
> > function of position and time. Deviation is the difference between
> > magnetic and compass north, which depends on the compass heading and
> > nearby (on-board) magnetic materials]
> Do you mean that the difference is
> some function of heading, and which
> function it is depends on nearby
> magnetic materials?
Yes. The nearby materials may have a permanent field, or their
higher permeability may produce an additional magnetisation field
which will not generally be parallel to the inducing field.
> At first I read your statement as
> treating "compass heading" and
> "nearby magnetic materials" equivalently,
No; I meant by "and" that both factors must be combined, not
that either will cause the effect on its own.
> so that even with no magnetic materials
> nearby, there might still be a difference
> that depends on heading. But I couldn't
> figure out how that could be true.
It's not too hard to invent possibilities, but (apart from the
effects of electric currents) none I can think of that would
occur in practice. Compass needle suspended on a torsion
wire?
Bob Moore
>Richard Herring <r...@gmrc.gecm.com> wrote in message
> news:3a24e0b2$1...@pull.gecm.com...
>> [Variation is the difference between true and magnetic north as a
>> function of position and time. Deviation is the difference between
>> magnetic and compass north, which depends on the compass heading and
>> nearby (on-board) magnetic materials]
>
>Do you mean that the difference is
>some function of heading, and which
>function it is depends on nearby
>magnetic materials?
Just the opposite....
Deviation is some function of nearby ferrous/magnetic materials and
that function depends on heading.
Aircraft compass deviation cards are entered with the heading as the
"known" element.
Bob Moore
Paul Lutus
(nos...@nosite.com) wrote:
> > "Tom Hubin" <thu...@clark.net> wrote in message
> > news:3A249A...@clark.net...
> > >
> > > I need to build some corrections, such as variation and deviation,
into
> > > software for magnetic compass data. Earth location and time variation
> > > corrections are available from the US Gov't but deviation is unique to
> > > each platform.
>
>
> Pay no attention to Lutus.
Don't let your personal feelings get in the way of reality. In my post I
provided a detailed, up-to-date approach to solving the poster's problem.
But, because his terminology was unclear, I addressed that first.
And deviation is in fact unique to each location. A simple fact check would
have prevented your gaffe.
--
Paul Lutus
www.arachnoid.com
James Hunter
Paul Lutus wrote:
> "Richard Herring" <r...@gmrc.gecm.com> wrote in message
> news:3a24e0b2$1...@pull.gecm.com...
> > In article <3J3V5.20528$IP1.6...@news1.giganews.com>, Paul Lutus
> (nos...@nosite.com) wrote:
> > > "Tom Hubin" <thu...@clark.net> wrote in message
> > > news:3A249A...@clark.net...
> > > > Hello,
> > > >
> > > > I need to build some corrections, such as variation and deviation,
> into
> > > > software for magnetic compass data. Earth location and time variation
> > > > corrections are available from the US Gov't but deviation is unique to
> > > > each platform.
> >
> > > What? Each platform? Don't you mean each location?
> >
> > Pay no attention to Lutus.
>
> Don't let your personal feelings get in the way of reality. In my post I
> provided a detailed, up-to-date approach to solving the poster's problem.
> But, because his terminology was unclear, I addressed that first.
>
> And deviation is in fact unique to each location. A simple fact check would
> have prevented your gaffe.
If you know anything about compasses, platform means PLATFORM
not some sort of effeminate "science" dork "location".
Bob Moore
>And deviation is in fact unique to each location. A simple fact check would
>have prevented your gaffe.
I trust that you mean "mounting location" on/in a vehicle and not location
on the earth's surface. My aircraft compass deviation is a factor of the
ferrous metals near its mounting location, variable with the actual heading
and does not change as the aircraft flies from one earth location to another.
Most GPS'ers take "location" to mean location on the earth's surface.
Bob Moore
Joe Fischer
: > nearby (on-board) magnetic materials]
:
: Do you mean that the difference is
: some function of heading, and which
: function it is depends on nearby
: magnetic materials?
The "on-board" is more important than the "nearby",
if the craft doesn't have modern electronics, computers,
gyro or radio compass, then the magnetic compass needs
to be subjected to a process called "swinging the compass".
A trained technician should do the actual placement
of trim metals to compensate for proximity of installed
metal components or devices on the dashboard, or even in
the engine compartment.
Airports have a compass painted on a paved area,
and the aircraft is swung a few degrees, and the compass
reading checked, and metal trim placed to compensate for
the error, and this is done at least for 720 degrees, to
get the best reading at all points on the compass.
It is a kludge fix, but better than nothing.
Joe Fischer
--
3
Dave Martindale
>Do you mean that the difference is
>some function of heading, and which
>function it is depends on nearby
>magnetic materials?
The problem is that large boats are made of steel, and they tend to
be long and narrow not disc-shaped. This causes local distortions in
the earth's magnetic field so a magnetic compass mounted on the ship
does not point in quite the same direction as a compass floating on
the water in the same spot with no metal nearby.
If you ever see an old-fashioned marine compass mounting, you'll
see that it has several adjustable "arms" that carry chunks of
iron mounted near the compass itself. The process of aligning the
compass on that ship involves moving these masses of iron (which are
just iron or steel, not magnets) in position to compensate for some
of the ship's distortions.
In addition, the boat itself may be somewhat magnetized, and have
current-carrying wires that have magnetic fields (this is true even
of fibreglass boats). These further distort the field.
Dave
Tom Hubin
>
> Tom Hubin wrote:
> >
> > Hello,
> >
> > I need to build some corrections, such as variation and deviation, into
> > software for magnetic compass data. Earth location and time variation
> > corrections are available from the US Gov't but deviation is unique to
> > each platform.
> >
> > I would like to find some discussion or math for determining and
> > correcting magnetic compass errors. The compass types would be typical
> > for a 30 foot sailboat or larger. Usually magnetic needle damped in a
> > fluid. But Hall effect and other electromagnetic compasses might be
> > used.
> >
> > In particular I would like to know how to compensate for fixed metal
> > objects and constant magnetic fields located on the boat.
> >
> > Is there a newsgroup that includes magnetic field discussion or
> > navigation technical info?
> >
> > Tom Hubin
> > thu...@clark.net
>
> Try: http://www.cnde.iastate.edu/staff/swormley/maps.html#declination
Thanx for the lead but the URL does not work. Maybe a typo?
Tom Hubin
thu...@clark.net
Tom Hubin
I did not mean to start a flaming war. I just wanted some theoretical
info. So let me clarify.
I know how to swing a compass and how it is done by sailors.
I want to create a software method for doing something equivalent with
electronic data from a compass. To do this I need to end up fitting some
equations. Most writings point toward a sinusoidal deviation around the
compass. So fitting a sinusoid should work fairly well.
But I would also like to have a better handle on the theory.
The earth has a magnetic field which is constant enough for a sailing
season in my neighborhood. The variation over the planet is predictable
for a few years and that data is readily available from the US Gov't.
Earth's magnetic anomalies are not a concern to me for sailing near
Annapolis MD.
Items on a fiberglass boat, such as the engine in the stern and the
magnets on speakers, contribute to errors on the compass. I realize that
the compass must be located as far as practical from metals and
especially from magnets. But these items are on the boat and, like it or
not, usually within a few feet of any compass.
I am looking for theory that would give values and vectors that would
tell me which way a compass needle would point in the presence of the
earth magnetic field, metal objects on the boat, and fixed magnetic
fields on the boat. Fixed magnetic fields would include DC wiring in use
at all times and speaker magnets. Intermittent magnetic fields do not
concern me nor do AC magnetic fields.
I can probably combine fixed magnetic fields on the boat with the
earth's magnetic field by using vectors to get a resultant magnetic
field at the compass. But I am not sure how to combine this with nearby
ferrous metals. Ultimately it must all be converted to forces on the
needle where it is at equilibrium.
I expect that compass needles will behave slightly differently than
electromagnetic sensors. A metal object close to a needle could cause a
bistable condition. That is, whichever end of the needle happens to be
closer to the metal object could be held closer to the object. So the
deviation could be a function of direction of rotation of the boat.
Petty details to be sure, but I am curious.
None of this needs to be practical at the moment. Just educational.
Tom Hubin
thu...@clark.net
Paul Lutus
news:V0bV5.77839$vc3.16...@typhoon.tampabay.rr.com...
> "Paul Lutus" <nos...@nosite.com> wrote:
>
> >And deviation is in fact unique to each location. A simple fact check
would
> >have prevented your gaffe.
>
> I trust that you mean "mounting location" on/in a vehicle and not location
> on the earth's surface.
Both. Nautical charts contain warnings about local magnetic deviations, some
affecting quite small areas. These would be bad choices for a compass swing
meant to reveal vessel deviations.
For obvious reasons this is less a factor for aircraft than for land and
marine applications.
--
Paul Lutus
www.arachnoid.com
Paul Lutus
> I am looking for theory that would give values and vectors that would
> tell me which way a compass needle would point in the presence of the
> earth magnetic field, metal objects on the boat, and fixed magnetic
> fields on the boat.
You can wade into this, but the deeper you go, the more you will realize the
math is extremely complex and has very little to do with the original
problem.
The software need only record the compass readings WRT time, and produce a
correction function (or a table of data) that relates the expected readings
with the observations. The only problem is the boat's constant turning rate
is assumed, not measured. This is why sailors are advised to find a calm
body of water, undisturbed by currents, and perform the swing at a windless
time of day.
> I can probably combine fixed magnetic fields on the boat with the
> earth's magnetic field by using vectors to get a resultant magnetic
> field at the compass. But I am not sure how to combine this with nearby
> ferrous metals. Ultimately it must all be converted to forces on the
> needle where it is at equilibrium.
Good luck in this project.
--
Paul Lutus
www.arachnoid.com
Yaakov Eisenberg
> Sam Wormley wrote:
> > Try: http://www.cnde.iastate.edu/staff/swormley/maps.html#declination
>
> Thanx for the lead but the URL does not work. Maybe a typo?
http://www.cnde.iastate.edu/staff/swormley/maps/maps.html#declination
Sam Wormley
Dave Martindale
>I can probably combine fixed magnetic fields on the boat with the
>earth's magnetic field by using vectors to get a resultant magnetic
>field at the compass. But I am not sure how to combine this with nearby
>ferrous metals. Ultimately it must all be converted to forces on the
>needle where it is at equilibrium.
The problem is that each piece of magnetic metal (which may or may not
be magnetized itself) "captures" some of the magnetic field from the
other sources of magnetism, including the earth, and tends to align
that magnetic field with the object. Thus, the net magnetic field
at the compass depends on all of the sources of magnetism *and their
interaction with nearby magnetic material*.
To the extent that all of the objects on the boat remain in the same
orientation to each other, you can probably reduce this to a single
vector that biases the field at the compass. But the angle between
all these pieces of magnetic material and the earth's field varies
as the boat heading changes. You probably can't model all of the
interactions reasonably. The best you can do is likely to calculate
a table of corrections.
Dave
Paul Rubin
> I would like to find some discussion or math for determining and
> correcting magnetic compass errors. The compass types would be typical
> for a 30 foot sailboat or larger. Usually magnetic needle damped in a
> fluid. But Hall effect and other electromagnetic compasses might be
> used.
>
> In particular I would like to know how to compensate for fixed metal
> objects and constant magnetic fields located on the boat.
>
> Is there a newsgroup that includes magnetic field discussion or
> navigation technical info?
There are two kinds of magnetic distortion caused by metal stuff in
the boat, called hard and soft iron distortion. Hard-iron distortion
can be modelled by imagining a bunch of magnets placed around the
boat, adding a constant offset to the B field that the compass sees,
so simple digital compasses can compensate for it. Soft-iron
distortion is more complicated and only the fancier digital compasses
compensate for it.
See www.precisionnav.com for some compasses of both types.
Ajw29585
"sectional charts," used by pilots mostly flying visual flight rules, show
magnetic deviation on the chart. If you can't find a theory from which to
derive deviation, you may just what to use a curve fitting method to match the
imperical data. So, the routine would be to swing the compass so you know what
its errors are with respect to magnetic north, then using that corrected
reading determine magnetic north, and from a look -up table or the like, deduce
true north.
Richard Herring
> "Richard Herring" <r...@gmrc.gecm.com> wrote in message
> news:3a24e0b2$1...@pull.gecm.com...
> > In article <3J3V5.20528$IP1.6...@news1.giganews.com>, Paul Lutus
> (nos...@nosite.com) wrote:
> > > > Hello,
> > > >
> > > > I need to build some corrections, such as variation and deviation,
> into
> > > > software for magnetic compass data. Earth location and time variation
> > > > corrections are available from the US Gov't but deviation is unique to
> > > > each platform.
> >
> >
> > Pay no attention to Lutus.
> Don't let your personal feelings get in the way of reality. In my post I
> provided a detailed, up-to-date approach to solving the poster's problem.
> But, because his terminology was unclear, I addressed that first.
His terminology was perfectly clear. "Variation" and "deviation" are
well defined quantities. I provided a reasonable paraphrase.
> And deviation is in fact unique to each location.
In principle, maybe. In so far as deviation is due to induced
magnetic fields rather than permanent ones, there may be a small
correction due to the strength and declination of the geomagnetic
field, which do of course vary with location. In practice it's
generally ignored. A ship's deviation card gives the correction
as a function of heading, not of any other variables.
To first order, deviation is a property of the platform,
not of the location.
> A simple fact check would have prevented your gaffe.
Likewise to you.
Richard Herring
> "Bob Moore" <rmoo...@tampabay.rr.com> wrote in message
> news:V0bV5.77839$vc3.16...@typhoon.tampabay.rr.com...
> > "Paul Lutus" <nos...@nosite.com> wrote:
> >
> would
> > >have prevented your gaffe.
> >
> > on the earth's surface.
> Both. Nautical charts contain warnings about local magnetic deviations, some
> affecting quite small areas.
I think you'll find they avoid the word "deviation", preferring "anomaly".
Richard Herring
> I suspect you already know this (or it may have already been posted) but
> "sectional charts," used by pilots mostly flying visual flight rules, show
> magnetic deviation on the chart.
That's *variation*, not deviation.
Rich Johnson
the compass. Also afaik it is still a reqwuirement, under many countries
shipping regs, to swing the compass every couple of years, regardless of
the number of "modern" nav aids.
Regards Rich
Paul Lutus
> > Both. Nautical charts contain warnings about local magnetic deviations,
some
> > affecting quite small areas.
>
> I think you'll find they avoid the word "deviation", preferring "anomaly".
This varies from place to place, life the magnetic field itself. :)
--
Paul Lutus
www.arachnoid.com
Paul Lutus
> > A simple fact check would have prevented your gaffe.
>
> Likewise to you.
Had you performed the fact check, you would have discovered the term
"deviation" is used for both cases under discussion. Instead you called me
to task for misusing the term.
--
Paul Lutus
www.arachnoid.com
Paul Lutus
> > "sectional charts," used by pilots mostly flying visual flight rules,
show
> > magnetic deviation on the chart.
>
Richard, come on. The terms are used interchangeably. Will you now argue
that flammable and inflammable should rightly refer to different things? I
happen to agree, but a word's definition is provided by how people use it,
not by an arbitrary standard.
To say it a more concise way, dictionaries describe, they don't prescribe.
This is why the words "literally" and "figuratively" are listed as synonyms,
at least on this side of the Atlantic.
--
Paul Lutus
www.arachnoid.com
Hamish Reid
Richard Herring <richard...@baesystems.com> wrote:
>In article <15dV5.64050$DG3.1...@news2.giganews.com>, Paul Lutus (nos...@nosite.com) wrote:
>> Both. Nautical charts contain warnings about local magnetic deviations, some
>> affecting quite small areas.
>I think you'll find they avoid the word "deviation", preferring "anomaly".
Who's "they"? My much-used collection of nautical and aeronautical
charts for two continents has both phrases....
Hamish
Richard Herring
> "Richard Herring" <r...@gmrc.gecm.com> wrote in message
> news:3a2640b3$1...@pull.gecm.com...
> > In article <20001129221601...@ng-cr1.aol.com>, Ajw29585
> (ajw2...@aol.com) wrote:
> > > "sectional charts," used by pilots mostly flying visual flight rules,
> show
> > > magnetic deviation on the chart.
> >
> Richard, come on. The terms are used interchangeably.
I'll willingly grant that many people use them so. Having been
brought up on mnemonics like "true virgins make dull company",
it's not a mistake that I make.
In any case, it was quite clear that the original poster understood
the distinction and was asking about the platform-dependent effect,
not the geographic one, so your initial put-down response was
inappropriate.
...
> a word's definition is provided by how people use it,
> not by an arbitrary standard.
If we're talking about everyday uage, then I entirely agree.
But if it's a specific technical term, then it should be used
only with its standard meaning.
> To say it a more concise way, dictionaries describe, they don't prescribe.
And you've never posted a dictionary "definition" in order to
make a point?
Paul Lutus
> > > That's *variation*, not deviation.
>
> > Richard, come on. The terms are used interchangeably.
>
> I'll willingly grant that many people use them so. Having been
> brought up on mnemonics like "true virgins make dull company",
> it's not a mistake that I make.
>
> In any case, it was quite clear that the original poster understood
> the distinction and was asking about the platform-dependent effect,
> not the geographic one, so your initial put-down response was
> inappropriate.
It wasn't a put-down. I asked "What? Each platform? Don't you mean each
location?" I asked this because trying to analyze a vessel's magnetic
anomalies is, in a practical sense, unheard of. One simply swings the
compass and writes down the differences between the compass readings and
"reality." Or one lets a computer do it.
My post went on to describe just that procedure. It was clear that the
poster hoped to be able to analyze a vessel's influence on the compass a
priori, rather than simply measuring the resulting effects.
No put-down was intended, and none was posted. My question was just what it
seemed -- a question.
> > a word's definition is provided by how people use it,
> > not by an arbitrary standard.
>
> If we're talking about everyday uage, then I entirely agree.
> But if it's a specific technical term, then it should be used
> only with its standard meaning.
True, I agree completely, but that standard doesn't apply in this case.
There are a lot of words than can be used to describe the difference between
(1) where geographical north is, and (2) where a compass points. I am sure I
have heard a good percentage of them, and most have been used
interchangeably.
It is why I asked my question -- in fairness to the original correspondent,
I wanted to get him to tell me what he actually meant.
On that topic, if you haven't sailed below the equator (I don't actually
know), you haven't seen your northern-hemisphere compass tilt at a wild
angle, making it difficult to use. This comes about because of yet another
magnetic-field anomaly called "dip." Compasses that are to be sold and used
un the northern hemisphere are weighted (or their sensing magnets are
tilted) to produce a level display. But the magnetic field itself isn't
level (except near the equator). So those who sail across the equator begin
to see their compasses assume a rakish angle to the horizon.
> > To say it a more concise way, dictionaries describe, they don't
prescribe.
>
> And you've never posted a dictionary "definition" in order to
> make a point?
Very regularly, in discussions of how people use words, information a
dictionary can reliably provide.
--
Paul Lutus
www.arachnoid.com
Karl Andrews
>
> I did not mean to start a flaming war. I just wanted some theoretical
> info. So let me clarify.
>
>
Chapter 6 of Bowditch (American Practical Navigator) has a pretty
comprehensive discussion of the theory. You can find it online at
http://www.irbs.com/bowditch.
- Karl (Ancient Mariner)
--
Linux: because rebooting is for adding new hardware.
Marshall Price
use them for celestial navigation?
"The sea is sissy and crass." ==>
c = c c + c s s ==>
cos a = cos b cos c + cos A sin b sin c
Karl Andrews wrote:
: Chapter 6 of Bowditch (American Practical Navigator) has a pretty
: comprehensive discussion of the theory. You can find it online at
: http://www.irbs.com/bowditch.
--
Marshall Price of Miami, Florida
d021...@dc.seflin.org : "Oh, to unfree one's heaven!"