Mercator Sailing: Course and Distance

[Dieter]

(08-28-2018 08:53 AM)Dieter Wrote:  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.

Indeed, this is actually is the case here:

e² sin(L) + e⁴ sin³(L)/3 + e⁶ sin⁵(L)/5 + e⁸ sin⁷(L)/7 + ...  =  e · artanh(e · sin(L))

As already mentioned, also

7915,7044678978...· lg(tan(45°+L/2))  =  60·180/\(\pi\) · ln(tan(45°+L/2))  =  60·180/\(\pi\) · artanh(sin(L))  =  60·180/\(\pi\) · arsinh(tan(L))

Note: the 12-digit value of the first constant is 7915,70446790 and not ...87

In other words: the Bowditch formula and the artanh version I posted are mathematically equivalent. They give the same results. The artanh formula just is much more compact and requires merely one single constant, the eccentricity e. I like it. ;-)

BTW, close to 90° (where Mercator's projection is problematic anyway) the original formula loses some accuracy. But this can be improved by modifying the formula slightly:

M  =  60·180/\(\pi\) · [arsinh(tan(lat)) – e·artanh(e·sin(lat))]

Dieter