As far as the Dutch projection is concerned (EPSG:28992), Peter's
summary is correct: it first projects the ellipsoid to the sphere and
then projects the sphere stereographically to the plane. I don't know if
that is the way PROJ4 computes the transformation, but I can garantee
that using "sterea" as parameter gives correct results here.
Jan
On 2011-02-07 12:32, Peter J Halls wrote:
I've kept out of this debate so far - partly waiting until I had my
copy of Snyder's 'Working Manual' (USGS Professional Paper 1395) in
front of me. I've also got ESRI's 'Understanding Map Projections' by
Kennedy and Kopp (my copy is dated 2000; ESRI Press) to hand. I had a
hazy memory that this was a complicated area ... as it indeed seems to
be.
According to Kennedy and Kopp, the ESRI Stereographic projection uses
the ellipsoid directly, whilst the Double Stereographic projection
takes the ellipsoid to a Gaussian sphere before calculating the planar
form.
According to Snyder (pp154-157) the basic Stereographic projection is
an azimuthal conformal perspective projection on the sphere but only
azimuthal conformal in the polar form on the ellipsoid, when it 'is
not perspective'. There are also modified forms (Snyder p 203) that
are not symmetrical and in which meridians and parallels are complex
curves. Snyder provides separate formulae for the spherical and
ellipsoidal forms, implying that the spherical form is not a simple
special case of the ellipsoidal.
In both Bugayevskiy & Snyder (1995, Map Projections: A Reference
Manual, London: Taylor & Francis) and Yang, Snyder and Tobler (2000,
Map Projection Transformation: Principles and Applications, London:
Taylor & Francis) a "double" projection is defined as one in which the
ellipsoidal form is first projected onto a sphere and then the sphere
to planar form. They also separate the spherical and ellipsoidal
forms for the stereographic formulae. This suggests that Kennedy &
Kopp do give the detail necessary when they mention projection first
onto the Gaussian sphere.
I have not examined the code of GDAL, nor of Proj4, so do not know
whether these have the facility to transform from ellipsoid to sphere
to planar to replicate the 'double' or whether the spherical or
ellipsoidal form is provided - let alone as to whether such a double
method is supported. However, I suspect that Even is correct to
surmise that the issue is not straightforward.
Best wishes,
Peter
Maciej Sieczka wrote:
W dniu 05.02.2011 14:47, Hermann Peifer pisze:
On 05/02/2011 13:12, Even Rouault wrote:
I've just had a look, but not being neither Dutch nor a specialist
of (stereographic) projections, I don't feel competent enough to do
any action on it. Those tickets plus the reading of
http://udig.refractions.net/files/docs/api-
geotools/org/geotools/referencing/operation/projection/Stereographic.html
make me deeply confused and wondering if r21627 was really
appropriate.
Hmm. This page confirms what I mentioned in ticket #2869:
"Double_Stereographic" is simply the ESRI alias of
"Oblique_Stereographic" (EPSG code 9809).
About r21627: I am afraid I do not understand enough about the inner
workings of GDAL/OGR to make a judgement about whether this change
was appropriate or not.
But by the end of the day and from what I remember when looking
somewhat deeper into the issue, about 2 years ago: there is one
stereographic projection (known as "the USGS formula" or "the
approach given by Snyder") which translates into +proj=stere, and
there is the other "Double_Stereographic", aka
"Oblique_Stereographic" which is expected to end up as +proj=sterea.
I'm not an expert too, but Hermann findings are in accordance with my
findings for Polish stereographic CRS (from over 5 years ago):
http://lists.maptools.org/pipermail/proj/2005-March/001524.html
The Polish CRSs which use +proj=sterea version of stereographic are
3120, 2172-2174, 3328.
See also Gerald Evendeen's notes:
http://lists.maptools.org/pipermail/proj/2005-March/001526.html
Maciek
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