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Subject: [Nml] Re: Mercator Projection
From: Mike Wescott (mike.wescott@XXX.XXX)
Date: Thu Mar 11 1999 - 12:51:33 EST
H. T. Feuerhelm wrote:
> - Sailing on the Chart, we use Mercator Projection, right? Now I found
> out that for some countries, they use different formulae for the
> "Mercator Bit":
> - US uses an ellipse (one or another "Correction" to the main part of
> the formula) as model of the earth.
> - In Germany, at least the navigation "hobbyists" use the formulae
> which correspond to a sphere (no ... additions to the "main" formula).
> - I dont know much about other countries, but at least there seem to be
> more and different "Mercator Projections" floating around, for example
> for geodetic (??) purposes like actual chart making.
> Now comes the problem:
>
> - In Silicon Sea i learned a few years ago, that the american use of an
> ellipse-model would be more precise (as you can also find out when you
> look at a GPS display, where "Chart Datum" is very improtant), which
> caused me some headaches to squeeze both versions into a spreadsheet
> for solving silicon sea problems.
>
> BTW, I have found that working german navigation exam problems with
> those "ellipse type formulae" would produce errors which could result
> to a FAIL in an actual test, therefore, those differences are at least
> somewhere important...
>
> - However, in astronavigation, we all use a spherical model !
>
> 1) Does that make sense instead of being more simple for calculation ?
> 2) Has anybody ever tried to evaluate the deviations / errors produced
> when using those different formulae (I would suspect those
> differences to be negligible, but one never knows....)
> 3) Are there any programs out there to make printouts for plotting
> sheets in which you could adjust for the different formulae, and
> would that make sense at all ?
> 4) Would we not, at least in principle, have to correct positions
> obtained by astronavigation to the correct chart datum to the chart
> which is used on the ship ?
>
> 1) Does that [sphereical model] make sense instead of being more
> simple for calculation ?
The general formula for a Mercator projection is:
M(lat) = (360*60)/(2*PI) *
[ ln tan (45 + lat/2) +
e/2 * ln ((1-e*sin lat)/(1+e*sin lat))]
where e is the eccentricity of the ellipsoid
ln means natural log
lat is latitude
PI = 3.14159...
Now in general e is a small number (WGS ellipsoid has e = 0.0818188).
Moreover e only has an effect in the latter part of the equation and
that effect is small when lat is small. And if the spherical model is
used, e = 0 and the exact formula is much reduced.
> 2) Has anybody ever tried to evaluate the deviations / errors produced
> when using those different formulae (I would suspect those
> differences to be negligible, but one never knows....)
A quickly written perl program gives us:
M(lat) M(lat)
Lat e=0818118 e=0 diff diff/M
==== ========= ======= ====== =========
0.0 0.00 0.00 0.00 0.000000
1.0 59.60 60.00 0.40 0.006739
10.0 599.07 603.07 4.00 0.006671
11.0 659.70 664.09 4.39 0.006657
20.0 1217.27 1225.14 7.87 0.006468
21.0 1280.95 1289.20 8.25 0.006440
30.0 1876.86 1888.38 11.51 0.006134
31.0 1946.15 1958.01 11.86 0.006094
40.0 2607.88 2622.69 14.81 0.005678
41.0 2686.49 2701.60 15.11 0.005625
50.0 3456.82 3474.47 17.65 0.005107
51.0 3550.90 3568.81 17.91 0.005043
60.0 4507.40 4527.37 19.96 0.004429
61.0 4629.06 4649.23 20.16 0.004356
70.0 5944.25 5965.92 21.67 0.003645
71.0 6123.90 6145.70 21.80 0.003560
80.0 8352.48 8375.20 22.71 0.002719
81.0 8716.28 8739.06 22.78 0.002613
89.0 16276.49 16299.56 23.06 0.001417
89.9 24192.28 24215.35 23.06 0.000953
The differences aren't large. Always less than 1%. Moreover, in navigation
we almost always use the difference in values of M(lat). And for short
distances the difference in the two methods is even smaller.
So the short answer is yes. The differences are for all practical purposes
negligible.
> 3) Are there any programs out there to make printouts for plotting
> sheets in which you could adjust for the different formulae, and
> would that make sense at all ?
If my analysis above is correct, I don't think it makes sense.
> 4) Would we not, at least in principle, have to correct positions
> obtained by astronavigation to the correct chart datum to the chart
> which is used on the ship ?
By the time you get close enough for this to matter, you should be piloting. One should be aware that charts are not
always accurate and that there can be
noticeable differences between chart, celestial nav results, and GPS for
whatever datum.
-- -Mike Wescott mike.wescott@XXX.XXX=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=--=-= =-= TO UNSUBSCRIBE, send this message to majordomo@XXX.XXX: =-= =-= unsubscribe navigation =-= =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=--=-=
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