# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=--=-= =-= TO UNSUBSCRIBE, send this message to majordomo{at}XXX.XXX: =-= =-= unsubscribe navigation =-= =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=--=-= </PRE>