[Proj] Bessel and the geodesic problem
Karney, Charles
ckarney at Sarnoff.com
Fri Apr 24 15:00:54 EST 2009
> The peculiar thing is that this technique seems to have been promptly
> forgotten. Certainly most of the 20th century literature (Rainsford,
> Vicenty, Bowring, Rapp) doesn't use it.
No, I'm wrong about Vincenty...
Vincenty's 1975 paper [Survey Review 23(176), 88-93 (Apr. 1975)] does
indeed ignore Bessel's method for halving the number of terms in the
series
However, in a subsequent letter to the journal [Survey Review 23(180),
294 (Apr. 1976)], he gives shorter expressions for his constants A and
B, eqs (3) and (4), using Bessel's variable (citing Helmert's book).
His new formulas for A and B are:
eps = (sqrt(u^2 + 1) - 1) / (sqrt(u^2 + 1) + 1)
A = (1 + eps^2/4) / (1 - eps)
B = eps * (1 - 3*eps^2/8)
except that he uses Helmert's notation eps = k_1. I would also
improve the expression for eps with
eps = u^2/(2 + 2*sqrt(u^2+1) + u^2)
As far as I can tell, implementors of Vincenty's method have overlooked
this optimization. (In fact, it will improve the accuracy of the method
as well.)
--
Charles Karney <ckarney at sarnoff.com>
Sarnoff Corporation, Princeton, NJ 08543-5300
Tel: +1 609 734 2312
Fax: +1 609 734 2662
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