[Proj] "Double ellipsoid" case?

Noel Zinn ndzinn at comcast.net
Mon Dec 1 07:02:53 EST 2008



Thanks for the useful links, which I’ll follow later in the day (off to work


The only difference between the spherical and ellipsoidal Mercator is that
the eccentricity squared is zero in the former case.  I agree that both are
valid.  Intuition tells me that the spherical case is conformal, too, though
I haven’t checked.  Of course, the LL<=>NE mapping is different, largely
because the meridional arc is so different. My example shows that, so I used
them relatively.  


My objection (well, one of my objections) is the implicit expectation that
one can do relative computations (whatever one does on a map) on a spherical
Mercator and “the resulting lat/long coordinates are intended to be treated
as WGS84 after that” (in Frank’s paraphrasing of what the Google Maps model
implies).  That’s not true.  


Datums CANNOT be switched under the projection, the whole issue of this
“double ellipsoid” thread.  The Google Sphere is NOT WGS84.  Yes, it can be
made to look similar at a point, but wander away from that point and it’s
very different.  My intention was merely to quantify the error with a
numerical example.  


Regarding the EPSG, I respect their integrity in resisting bogus geodesy.
Misrepresentations aside, the Google Sphere is not WGS84.






From: proj-bounces at lists.maptools.org
[mailto:proj-bounces at lists.maptools.org] On Behalf Of Mikael Rittri
Sent: Monday, December 01, 2008 3:44 AM
To: PROJ.4 and general Projections Discussions
Subject: RE: [Proj] "Double ellipsoid" case?



I haven't checked your results, but there is no reason to expect that the
two results would be the same.  


As I see it, Ellipsoid Mercator is one projection, and Sphere Mercator is a
different one. 

Each one is internally consistent, but the two are not the same.  That is,
since the formulas 

are different, the (Easting,Northing) values are not the same for the same
(Lon,Lat).  But 

for each projection, you can use its inverse to get exactly back to the
original (Lon,Lat). 


The Ellipsoid Mercator is exactly conformal while Sphere Mercator is only
approximately conformal

(if you regard the ellipsoid as the true shape of the Earth).  I think the
maximal angle distortion 

for Sphere Mercator is 0.2 degrees, so for visual display it is hard to


Since the projections are distinct, we should not expect that analoguous
calculations done 

in their projected planes should give identical results.  


So, the situation is similar to the French truncated Lambert Conformal
Conic, which is 

not exactly conformal, and is a different projection than the true Lambert
Conformal Conic. 


It is tricky to create WKT to represent the Sphere Mercator projection
together with an ellipsoid datum. 

But that is a failure of WKT, not of the Sphere Mercator.  One way to
express it in WKT has 

been suggested by Martin Desruisseaux of GeoTools, another way by Melita
Kennedy of ESRI. 

See the August 2008 archives for the MetaCRS mailing list, 

http://lists.osgeo.org/pipermail/metacrs/2008-August/thread.html . 


And it is not only Google that uses spherical projections.  So does the
National Snow and Ice Data Center, USA. 



I think it is unfortunate that EPSG insists that a spherical projection can
only be used together with a spherical datum, because 

a) it is not true,

b) spherical datums do not exist for the Earth


Best regards,


Mikael Rittri
Carmenta AB
Box 11354
SE-404 28 Göteborg
Visitors: Sankt Eriksgatan 5
Tel: +46-31-775 57 37
Mob: +46-703-60 34 07
mikael.rittri at carmenta.com




From: proj-bounces at lists.maptools.org
[mailto:proj-bounces at lists.maptools.org] On Behalf Of Noel Zinn
Sent: den 30 november 2008 06:05
To: 'PROJ.4 and general Projections Discussions'
Subject: RE: [Proj] "Double ellipsoid" case?

Not that Frank is responsible for the geodesy and cartography in Google Maps
(or their abuse therein), but the phrase “the resulting lat/long coordinates
are intended to be treated as WGS84 after that” so troubles me that I am
sympathetic to Cliff’s sentiments.  So, let’s quantify the offense with an
experiment that anyone can duplicate, perhaps in Proj4 (I work in Matlab).


Consider the following Mercator grid parameters defined on both WGS84 and
the Google Sphere (whose radius equals the semi-major axis of WGS84): CM =
95W, FN=FE=0.  That’s all we need.


Now consider at point at 30N / 95W.  That’s somewhere near Houston, Texas,
where I live.  Convert to Mercator in both systems.  Then traverse to the NE
about 141km in both systems by adding 100km to both the Northing and Easting
(in Mercator).  Convert the resulting Mercator coordinates back to
geographicals (lat / lon).


Here’s what I get for WGS84:  30N / 95W is N3,503,549.84350437m / E0m with a
point scale (Mercator is conformal) of 1.15470053837925 and at the end of
the traverse the geographicals are 30-46-29.63568N /


For the Google Sphere the results are:  30N / 95W is N3,482,189.08540862m /
E0m with a point scale of 1.15373388324025 and at the end of the traverse
the geographicals are 30-46-43.56897N /  94-06-06.06498W.


Pardon the excessive precision; it’s just what I get.  And I hope that I
haven’t blundered in my haste to respond.  Perhaps someone can confirm.


Interestingly, the longitudes are the same, but the latitudes are very
different (more than might be accounted for by the scale differences, a
cartographic subtlety likely beyond the ken of most Google Maps users
anyway).  I don’t believe the resulting lat/long coordinates can be treated
as WGS84 at all.  


Noel Zinn



From: proj-bounces at lists.maptools.org
[mailto:proj-bounces at lists.maptools.org] On Behalf Of Clifford J Mugnier
Sent: Saturday, November 29, 2008 8:15 AM
To: PROJ.4 and general Projections Discussions
Subject: RE: [Proj] "Double ellipsoid" case?




Ah, there's a BIG difference between the true coordinate system relations of
geodesy used by national governments and one cooked up by an ignoramus at
Google Maps that did not know what they were doing ... I guess there's a lot
of that going around, too.


I suppose even twits help contribute to keep knowledgeable consultants in


Cliff Mugnier




From: proj-bounces at lists.maptools.org on behalf of Frank Warmerdam
Sent: Fri 28-Nov-08 10:52
To: PROJ.4 and general Projections Discussions
Subject: Re: [Proj] "Double ellipsoid" case?

support.mn at elisanet.fi wrote:
> Hello,
> I just started to think about a situation where there might be a double
> ellipsoid case.
> 1) projection uses ellipsoid A independently
> 2) datum shift uses ellipsoid B
> Is this possible to be handled with proj.4? Since there is only one
> ellipsoid definition available


PROJ.4 does not currently handle this situation conveniently.  One special
case is where the projection uses a particular ellipsoid, but datum shifts
should treat the corresponding lat/long values as being WGS84. In that
special case you can use +nadgrids=@null to effectively say the datum shift
to WGS84 is a no-op.

I have at times contemplated having a way of having a datum/ellipsoid
definition used for datum shifting purposes that is independent of the
normal ellipsoid used by the projection functions but I have not pursued it.

Clifford J Mugnier wrote:
 > I've seen a few projections and datums in my day, and I've never come
 > accross that.  For there to be a different ellipsoid of reference for a
 > projection than there is for a datum is a contradiction in terms.


The very common case we see a lot these days is the google maps mercator
projection.  The mercator calculations are done based on a particular
spherical earth model, but the resulting lat/long coordinates are intended
to be treated as WGS84 after that.

Best regards,
I set the clouds in motion - turn up   | Frank Warmerdam,
warmerdam at pobox.com
light and sound - activate the windows | http://pobox.com/~warmerdam
and watch the world go round - Rush    | Geospatial Programmer for Rent

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