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    Re: Mercator Projection
    From: Mike Wescott
    Date: 1999 Mar 11, 12:51 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.
    <PRE>
    --
    	-Mike Wescott
    	 mike.wescott{at}XXX.XXX
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