[Proj] Troubles with Newton-Raphson inverse projections
ovv at hetnet.nl
Sat Oct 18 14:10:18 EDT 2008
Some time ago I programmed the inverse of the Winkel-Tripel according to the
I found no serious difficulties apart from the following.
The lat_0 must be smaller than for instance 80d absolute and larger than say
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
> (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
> 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
> 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
> 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.
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