[Proj] Bessel and the geodesic problem

[Karney, Charles]

> 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