Mercator Sailing: Course and Distance

08282018, 08:53 AM
(This post was last modified: 08282018 05:15 PM by Dieter.)
Post: #9




RE: Mercator Sailing: Course and Distance
(08282018 12:46 AM)Gene222 Wrote: M = a ln(10) log tan (45 + L/2)  a (e^2 sin L + e^4 / 3 sin^3 L + e^6/5 sin^5 L ... Thank you very much for this formula. (08282018 12:46 AM)Gene222 Wrote: a is the equitorial radius of the earth expressed in minutes of arc of the equator = 3437.74677078. This is the factor 60*180/pi in my formula. Circumference at equator in arc minutes = 60 · 360° = 21600' Radius at equator = circumference / 2\(\pi\) = 60 · 180 / \(\pi\) = 3437,7467707849392526... Now let's compare the Bowditch formula with the one I posted, assuming e=0,081819191 for WGS84. The Bowditch formula was used with exact (12digit) coefficients and one more term. L=5° Bowditch: 298,375698376 Artanh: 298,375698374 L=10° Bowditch: 599,073043655 Artanh: 599,073043671 L=20° Bowditch: 1217,26588915 Artanh: 1217,26588917 L=30° Bowditch: 1876,86220651 Artanh: 1876,86220652 L=40° Bowditch: 2607,88368513 Artanh: 2607,88368514 L=50° Bowditch: 3456,82030073 Artanh: 3456,82030073 L=60° Bowditch: 4507,40395347 Artanh: 4507,40395348 L=70° Bowditch: 5944,24941337 Artanh: 5944,24941337 L=80° Bowditch: 8352,48380806 Artanh: 8352,48380805 L=85° Bowditch: 10741,6440576 Artanh: 10741,6440579 L=87° Bowditch: 12499,0738936 Artanh: 12499,0738943 L=89° Bowditch: 16276,4947743 Artanh: 16276,4947699 Convinced ?) I think we can safely assume that the slight differences beyond the 10th digit are due to roundoff errors, especially as L approaches 90°. That's why with 12digit precision the 89° artanh result is slightly off. The 34s here returns 16276,49477437... ;) Edit: I have now compared both formulas in Excel (15 digits). The more terms you add to the Bowditch formula the closer it matches the result of the artanh formula. With terms up to sin(L)^11 and exact coefficients the results agree within a few ULP. So the Bowditch formula seems to be a series expansion of the artanh formula. The latter essentially has two advantages: you don't need half a dozen of numeric constants but just one, the eccentricity e, and the whole thing is much shorter. Dieter 

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