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Bill wrote, about extracting further information than azimuth from the
rectangluar to polar conversion I had quoted

 >I feared I was too hasty thinking about it in
> transit.  Returning home, I found all but the below was pure gibberish as
> I
> managed to get caught in my own circular argument.
>
> While there is a high correlation between the first value returned from
> R-to-P and cosine of Hc, they are not nearly close enough for any
> meaningful
> use in navigation.
>
> Back to the drawing board.

==============================

Alas, he is not the only one to be writing nonsense. Nearly all of my own
last posting on this topicwas complete hokum. Please tear it up and throw it
in the bin.

It's true that when you compute (R,A), in polar coordinates, using the
function

ATAN2  or POL ( tan dec cos lat - cos LHA sin lat  , - sin LHA)

the resulting angle A gives the true azimuth. This is simple, useful, and
direct, providing azimuth in the range -180, through zero, to +180. If you
prefer, you can add 360 to a negative azimuth to put it into the range 0 to
360.

=========================

In addition to an azimuth A, that function also returns another result, R.
What Bill asked was whether the resulting R value provided any useful
information., or just had to be discarded. A sensible question.

I suggested (rather tentatively, because I couldn't see how to prove it)
that the R value appeared to correspond to Tan (90 - altitude), just from a
few numerical test-results.

I've now checked more thoroughly, and can now state that that conclusion
applied only in certain special circumstances, and is NOT generally valid.
So now, that suggestion must be withdrawn. It just ain't so. Sorry about
that.

So my answer to Bill's question is that as far as I can see it, the R-value
that emerges from use of that rectangular to polar conversion function is of
no use at all, and has to be discarded.

=========================

To compute the altitude of the body, Meeus' formula 13.6 should be used,
which is-

alt = arcsin (sin lat sin dec + cos lat cos dec cos LHA)

===========================.

The equivalent formulae, for getting great-circle course and distance, from
position 1 (lat1, long1) to position 2 (lat2 long2), involves working out
dlong = long2 - long1 (and getting the sign right; see below)

Then apply the function

atan2 or POL ( tan lat2 cos lat1 - cos dlong sin lat1, - sin dlong)

which gives two values, R and A. Discard R, and A is the required course. If
it's negative, add 360.

And for distance, in miles, use

dist = 60* (arc cos( sin lat1 sin lat2 + cos lat1 cos lat2 cos dlong))

this is adapted from Meeus 13.6.

In these expressions, the signs have to be taken seriously, North positive,
South negative; longitudes and hour angles are positive Westwards. Then
azimuth and course are positive clockwise from North.

George.

contact George Huxtable at george@huxtable.u-net.com
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
.

   

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