[Proj] Troubles with Newton-Raphson inverse projections

[Gerald I. Evenden]

On Saturday 18 October 2008 2:10:18 pm OvV_HN wrote:
> Some time ago I programmed the inverse of the Winkel-Tripel according to
> the Turkish paper.
> I found no serious difficulties apart from the following.
> The lat_0 must be smaller than for instance 80d absolute and larger than
> say 1e-6 deg.
> In these circumstances the maximum number of iterations in the inverse
> should be smaller than 150 or so.
> If a larger value of lat_0 must be used: in one test I came up with 579
> iterations at a
> lat_0 of 89.9d.

LOL, wow!!  I cutoff at 10 iterations and running at a moderately loose 
tolerance of 10^-8 radians.

> For even larger values, the correct value of the longitude could not be
> obtained from the x, y.

With limited adjustment of the initial estimates I can go out to 180 lon.
But as lat increases so do the problems.

> I tested some values with lat(itude) running from -89.99 to +89.99 d.
> Note that the Winkel-Tripel works with a fixed value of lat_0 = 50d28m, so
> difficulties with lat_0 at extreme latitudes could be ignored.
>
> It is possible that Aitoff and Hammer will pose more difficulties, I dunno,
> merely used the algorithm for inverse Winkel-Tripel.

Just visual examination suggests that Winkel Tripel should have fewer 
problems.  But I'm going to bash the tough ones first.

> Oscar van Vlijmen
>
	...
-- 
The whole religious complexion of the modern world is due
to the absence from Jerusalem of a lunatic asylum.
-- Havelock Ellis (1859-1939) British psychologist