[Proj] Re: "Double ellipsoid" case?

[Mikael Rittri]

Richard Greenwood wrote:

> I for one, am finding this an interesting, and pertinent thread. 
> Let's not quash it, and let's keep it civil.

Thanks, Richard. 

I thought daan explained the matter clearly, and I have not much to add.   

daan Strebe wrote:
> The practical consequence is that they do not have a conformal projection; 
> hence local distances measured from a point are not quite uniform in all 
> directions, and neither are azimuths quite equally spaced radially from 
> that point. So?

I would add that these deviations from conformality of course can be 
quantified and bounded.  For example, I wrote that "I think the maximal 
angle distortion for Sphere Mercator is 0.2 degrees".  This was wrong; 
I was thinking of the maximal azimuth error, and the maximal angle 
distortion is twice that, so make it 0.4 degrees.  

It is not surgical precision, and it is 16th century technology, 
but I wouldn't say that a 0.4 degree error is "absurd".
(Well, unless my application required exact conformality for some reason.) 
The benefit is simpler formulas and faster execution times.  

Noel Zinn wrote:
> If I could just switch the WGS84 ellipsoid in the WGS84 datum with 
> the Google Sphere (as you suggest), why couldn't I switch 
> the International Ellipsoid in ED50 with Clarke 1866?  Or any other 
> switch for that matter?  

Let's see if I understand you correctly. You are asking: if I have data 
in the datum ED50, why can't I define a projection which, by definition, 
treats (Lon,Lat) from ED50 as if it were (Lon,Lat) on the Clarke 1866 
ellipsoid?

Well, I am forced to say that this is quite possible. The mathematics works,
and it is not illegal.  Again, the result would be a map that is not exactly
conformal, relative to the shape of ED50.  Because this naive mapping 
from (Lon,Lat) on the International Ellipsoid to (Lon,Lat) on the Clarke 1866 
ellipsoid - without changing the numerical values - is not conformal. 
At least, I don't think so. 

But I am just saying that it is possible. I think it would be silly 
and meaningless, because I would give up the exact conformality 
(relative to the shape of ED50) in return for nothing.  I would not 
get the benefit of simpler formulas or faster execution times, 
since I would have to use ellipsoid formulas to go from Clarke 1866 
to the plane.  

Noel: 
> In addition to worse fits mathematically 
> (since the adjustment was done on the defining ellipsoid), 

Yes, spherical formulas give worse maps for large scale maps. 
But I would say, only slightly worse. Not absurdly worse. 

> we'd open the door to uncertainty and crisis. 
> You are proposing (and Google has introduced) the geodetic 
> equivalent of sub-prime mortgages in the financial market.  
> Don't do it!

Well, we are talking about a maximal 0.4 degree error for angles, 
and a variation of the local scale in different direction 
through a point, which would be at most 0.5 percent, I think.  

If economists could do predictions with 0.5 percent accuracy, 
the world economy would be in a better state! 

--
Mikael Rittri
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