[MetaCRS] Stereographic Projections: stere and sterea

[Mikael Rittri]

> The Proj.4 documenation doesn't mention the "sterea" projection,
> but pages found by Google suggest it is a recent alternative addition?
> This would explain its absence from the original Snyder documentation
> and addenda.
 
The sterea formulas are documented in the libproj4 documentation 
at http://dl.maptools.org/dl/libproj4 ; I guess it was ported 
from libproj4 to Proj.4. 
 
stere and sterea coincide in the polar aspect, so there is no trouble
if you never want an oblique or equatorial aspect. 
 
However, in the oblique Netherlands grid, for example, they differ 
by up to 8 meters.  

Both are valid projections, in the sense that both are exactly
conformal mappings from ellipsoid to plane.  But only sterea has
the nice property that the local scale factor is minimal at the
projection center.  In oblique stere, the point of minimal scale
factor is displaced towards the equator by up to 0.4 degrees. 

sterea has less distortion (near the center), but more complicated
formulas.  On the other hand, stere is continuous globally, while
sterea is discontiuous along the antipode of the central meridian,
where the map overlaps itself along a strip that can be up to 1.2°
wide, so sterea should not be used more than 90° of longitude from
its central meridian.  

stere is never used for data storage, as far as I know.  
In fact, I doubt that it is used at all (except in its
polar aspect, where it coincides with sterea anyway).
It is a bit of a mystery why Snyder described stere but
not sterea.  

sterea is used for data storage in Canada, Romania,
and the Netherlands. (Confirmed by Oscar van Vlijmen, 
http://lists.maptools.org/pipermail/proj/2009-April/004603.html )

Poland uses the Roussilhe version of stereographic projection,
+proj=rouss, which is an approximation of sterea intended to
be easier to calculate (I don't think it is exactly conformal).
When I checked the difference between rouss and sterea, I found
that the difference was at most 5 centimeters within 350 km 
from the projection center, at least in my test case.
(I am not sure whether +proj=rouss uses exactly the same 
Roussilhe formulas as they do in Poland.) 

Best regards,

--
Mikael Rittri
Carmenta AB
SWEDEN
www.carmenta.com 

________________________________

From: metacrs-bounces at lists.osgeo.org [mailto:metacrs-bounces at lists.osgeo.org] On Behalf Of Richard Marsden
Sent: den 6 november 2009 00:50
To: metacrs
Subject: [MetaCRS] Stereographic Projections: stere and sterea


Following on from equal-area-maps.com <http://equal-area-maps.com/> , I am putting together some samples for polar projections. Eventually I'll publish a couple of articles at GeoWebGuru.com.  I'm still at the survey stage, but it looks like there are two stereographic projections: "stere" and "sterea" and it isn't clear what the difference is.

The Proj.4 documenation doesn't mention the "sterea" projection, but pages found by Google suggest it is a recent alternative addition?
This would explain its absence from the original Snyder documentation and addenda.

The best description I have found is on the GeoTIFF site which says that they are two different but valid approaches, with "stere" being the original (Snyder) approach. However it notes that the two approaches produce different results away from the centre! To me this suggests that it is unlikely that are both are equally valid.

Considering both were ported to Proj4JS, I assume this issue has already come up? 
What does "sterea" offer over "stere", and which is usually recommended?

Best regards,

Richard Marsden
Winwaed Software Technology LLC
http://www.winwaed.com <http://www.winwaed.com/> 
http://www.mapping-tools.com <http://www.mapping-tools.com/> 
http://www.geowebguru.com <http://www.geowebguru.com/>