[Proj] Re: "Double ellipsoid" case?

[strebe]

Duncan:

Thank you for the notes, with which I largely agree.

>Finally, the Mercator on the sphere is exactly conformal, as is the 
>Mercator on an ellipsoid, as defined mathematically. This result,
>coming from differential geometry, is independent of the eccentricity.

Yes, and I should have been more clear when I described the Google Maps projection as "not conformal". If you ignore the geodetic origins of the data, then the projection is conformal. If you wish to preserve conformal relationships between the data as they exist on the ellipsoid that they were originally surveyed on, you would be obliged to apply a conformal sphere mapping of the geodetic coordinates before projecting the results to the spherical Mercator. Or, use the ellipsoidal Mercator development formulæ. Or, perform a datum shift from WGS84 to "Google Sphere", before projecting via the spherical Mercator (though it appears Clifford wants to dispute that the "Google Sphere" could be a datum). The first two results would be identical to each other to round-off precision. The last result would be close to the first two, with the closeness dependent upon the datum shift mechanism.

Regards,
— daan Strebe


On Dec 2, 2008, at 9:38:47 AM, "Duncan Agnew" <dagnew at ucsd.edu> wrote:
From:   "Duncan Agnew" <dagnew at ucsd.edu>
Subject:    Re: [Proj] Re: "Double ellipsoid" case?
Date:   December 2, 2008 9:38:47 AM PST
To: "PROJ.4 and general Projections Discussions" <proj at lists.maptools.org>
I've been following the debate about Google Earth (GE) and Google Maps, 
and
wanted to contribute that I've seen the same kind of misfit between
geodetic-grade GPS coordinates and those from GE (for points I've 
visited and
so can locate on the photos): a few meters, up to about 10 m--and higher
elevations do tend to have larger errors. I'd agree that for a small 
area most
of this error can be removed by a local x-y shift.

But I am baffled by the discussion about what kind of projection GE 
uses,
and especially the aspersions cast by several participants. There seem 
to be
three different issues:

1. What *coordinates* does GE produce? The answer would seem to be, 
geographic
coordinates in a global datum, presumably something close to WGS-84 (in 
its
various incarnations) or ITRF (in its). That is, what you get from GE 
is the
same as what you will get from a GPS receiver.  Viewing GE as a device 
for
producing pairs of numbers from where you put the cursor, the ellipsoid 
used in
irrelevant: there is just some kind of algorithm from pixels to 
coordinates,
which is probably done by local interpolation.  Registration of the 
imagery is
much more important than anything else. Whether or not the WGS84 
ellipsoid
"fits" the projection used to make a rendering is beside the point when 
it
comes to lat and long, since there is no projection. Whether that 
ellipsoid
"fits" the Earth (or a part thereof) is even more beside the point, for 
this
application.

2. What shape is used for making the oblique views? Clearly a sphere,
and orthographic, is fine for GE's purely visual presentation.

3. What algorithm is used to determine distance when using the ruler 
tool?
That is, how is distance found from two pairs of coordinates? I don't 
know, but
again would say that using a sphere would be fine for the accuracy that 
anyone
should expect from a viewing tool: for casual applications, an error
of 1 in 300 isn't going to be an issue. Note that the projection used 
to render
the map on the screen could be Mercator on the sphere, while the tools 
could
use an ellipsoidal algorithm. Does anyone know what is used?

I think the same ideas apply to OSM; any inadequacies of their rendering
engine (which, for a Web application, seem minor) need not affect the
quality of the underlying data.

Finally, the Mercator on the sphere is exactly conformal, as is the 
Mercator
on an ellipsoid, as defined mathematically. This result, coming from
differential geometry, is independent of the eccentricity.

Duncan Agnew

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