[Proj] Troubles with Newton-Raphson inverse projections

[OvV_HN]

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

Oscar van Vlijmen




----- Original Message ----- 
From: "Gerald I. Evenden" <geraldi.evenden at gmail.com>
To: "PROJ.4 and general Projections Discussions" <proj at lists.maptools.org>
Sent: Saturday, October 18, 2008 5:48 PM
Subject: [Proj] Troubles with Newton-Raphson inverse projections


>I finished the basics of the Newton-Raphson general inverse projection
>method
> (as described in the Turkish paper) about a week ago.
>
> One additional problem with the Turkish paper is that they understated the
> problem of making an appropriate initial estimation of the lon/lat
> solution.
> If the estimate is sufficiently poor, the looping process can follow the
> wrong path to a solution or simply fail to converge.  Also, the poorer the
> estimation, more loops are required to converge to a solution.  At the
> moment
> I am pursuing developing a low degree polynomial estimation function which
> will hopefully improve the initial estimate process.
>
> Secondly, finding the root may also be difficult when the precision of the
> derivatives start to fail.  This is the case near the poles of both the
> Hammer and Aitoff projections.  At a latitude greater than 89° the method
> fails for both these projections.  One also gets suspicious that this will
> occur when the nature of the curve flattens out like the top of a sine
> curve.
> I have not tried the flat pole Winkel Triplel yet.
>
> This is just an alert to the readership that the Turkish method *is not* a
> cureall for determining the inverse projection.

>