Mercator Sailing: Course and Distance

[Dieter]

(08-28-2018 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.

(08-28-2018 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 (12-digit) 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 12-digit 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