[Proj] Troubles with Newton-Raphson inverse projections

[OvV_HN]

One shouldn't laugh too soon.
With Winkel-Tripel on a sphere with a radius, equal to the major axis of the 
WGS84 ellipsoid, I got the following results.

Starting with: lat = 0.1 deg;  lon = 14 deg (east); lat_0 = 89.9 deg;
Full round trip forward - inverse needed 286 iterations to get lat/lon back 
within an error of 1e-8 in the iteration loop. The loop needed 579 
iterations for an error of 1e-14.
So it really needed that lot of iterations.

Two things are rather stupid though.
1) How silly is this type of projection with a lat_0 near the pole, being 
interested in locations near the equator? Locations near the equator are not 
extremely silly, because the Winkel-Tripel will be probably used for 
thematic maps of the whole world. But a lat_0 near the pole is a bit 
strange.

2) The Newton-Raphson method fails miserably in the known mathematical 
areas. We've already seen this happening in the Transverse Mercator 
projection, performed with complex elliptic functions, and near the 
mathematical boundaries.
One should use another method in the difficult areas or restrict or warn the 
user for these areas.

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 9:17 PM
Subject: Re: [Proj] Troubles with Newton-Raphson inverse projections


> 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.
....