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<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
11.0pt;font-family:Arial;color:navy'>Mikael,<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
11.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
11.0pt;font-family:Arial;color:navy'>Tonight rereading my post with yours
concatenated below I see that you anticipated my point. You wrote:<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
11.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=black face=Arial><span style='font-size:
11.0pt;font-family:Arial;color:black'>“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.
“<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=black face=Arial><span style='font-size:
11.0pt;font-family:Arial;color:black'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=black face=Arial><span style='font-size:
11.0pt;font-family:Arial;color:black'>My apologies for being obtuse. My 1.0067
scale and your 0.5 percent are about the same (and it’s a function of
latitude), but how differently we react to knowledge of this fact. The angular
distortion and the scale change are related, of course. 100 meters times the tangent
of 0.4 degrees is about 70 centimeters. That’s 7m per km and 70m per
10km.<br>
<br>
</span></font><font size=2 color=navy face=Arial><span style='font-size:11.0pt;
font-family:Arial;color:navy'><o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=black face=Arial><span style='font-size:
11.0pt;font-family:Arial;color:black'>For me (meters on the ground) it’s
a big deal, while you just shrug it off. I find it hopeful that you’ve actually
quantified these distortions, but I wonder how many developers (to say nothing
of users) are that savvy, that able to compensate for them, almost 20 times
worse in scale than the worst in a UTM zone. Too few, I think. <o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=black face=Arial><span style='font-size:
11.0pt;font-family:Arial;color:black'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=black face=Arial><span style='font-size:
11.0pt;font-family:Arial;color:black'>Regards,<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=black face=Arial><span style='font-size:
11.0pt;font-family:Arial;color:black'>Noel</span></font><font size=2
color=navy face=Arial><span style='font-size:10.0pt;font-family:Arial;
color:navy'><o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
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<p class=MsoNormal><b><font size=2 face=Tahoma><span style='font-size:10.0pt;
font-family:Tahoma;font-weight:bold'>From:</span></font></b><font size=2
face=Tahoma><span style='font-size:10.0pt;font-family:Tahoma'>
proj-bounces@lists.maptools.org [mailto:proj-bounces@lists.maptools.org] <b><span
style='font-weight:bold'>On Behalf Of </span></b>ndzinn@comcast.net<br>
<b><span style='font-weight:bold'>Sent:</span></b> Wednesday, December 03, 2008
3:00 PM<br>
<b><span style='font-weight:bold'>To:</span></b> <st1:PersonName w:st="on">PROJ.4
and general Projections Discussions</st1:PersonName><br>
<b><span style='font-weight:bold'>Subject:</span></b> Re: [Proj] Re:
"Double ellipsoid" case?</span></font><o:p></o:p></p>
</div>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'><o:p> </o:p></span></font></p>
<div>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'>Mikael,<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'>Thanks for your revised assessment of
maximum angular distortion in the Google Maps Projection (GMP). <o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'>Are you aware that the scale distortion in
GMP is very large and unstated? In the vicinity of the Equator and it's about
1.0067 (that's 67cm per 100m). You can't compensate for it with the point
scale factor equations of the Mercator projection in the second stage of the
GMP (your alternative A) because it's embedded in the first stage of the GMP,
the non-conformal mapping from WGS84 to the Google Sphere (GS). Here's how you
can demonstrate this to yourself. <o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'>Choose any meridian. We're going to
traverse north up that meridian (a great circle geodesic) from the Equator to 1
degree North on the Google Sphere (GS) and on the GMP. Consequently, the
Eastings are irrelevant; they're constant. Here's the raw data.<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'>The inverse geodesic distance along the
meridian on the GS from the Equator to 1 degree North is 111319.4908m. The GMP
Northing on the Equator is 0m and the point scale is 1 (unity). The GMP
Northing at 30 minutes North is 55,660.4519m and the point scale factor is
1.00003808. The GMP Northing at 1 degree North is 111,325.1429m and the point
scale factor is 1.00015233. <o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'>To convert from grid distances
(111,325.1429m in this case) to geodesic distances surveyors use Simpson's
Approximation to compute the line scale factor. That's the sum of the
end-point point scale factors plus 4 times the mid-point point scale factor
divided by 6, or 1.000050775 in this case. Divide the grid distance by line
scale factor and you'll see that we have a sub-millimeter tie with the geodesic
distance. <o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'>If there were a physical monument on the
Equator and one at 1 degree North (same meridian), what would be the distance
between them? Well, that would be the geodesic distance on the WGS84
ellipsoid. This is the WGS84 datum, isn't it? Well, that distance is
110,574.3886m. The ratio of these distances is the hidden (and, therefore, insidious)
scale distortion of the GMP in this vicinity of the world. <o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'>An underlying and yet unmentioned
distinction in this thread is that geodesists think in terms of meters on the
ground and GIS cartographers think in terms of pixels on a monitor. It's a different
frame of reference, a different tolerance for error. If I surveyed football
pitch (about 100m), a 67cm error would concern me. My question for you is: How
long a distance would it take for the hidden GMP scale distortion to move your
image to the wrong pixel on your monitor? <o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'>Regards,<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 color=black face="Times New Roman"><span
style='font-size:12.0pt;color:black'>Noel Zinn<o:p></o:p></span></font></p>
<p><font size=3 color=black face=Arial><span style='font-size:12.0pt;
font-family:Arial;color:black'><br>
----- Original Message -----<br>
From: "Mikael Rittri" <Mikael.Rittri@carmenta.com><br>
To: "<st1:PersonName w:st="on">PROJ.4 and general Projections Discussions</st1:PersonName>"
<proj@lists.maptools.org><br>
Sent: Tuesday, December 2, 2008 10:13:22 AM GMT -06:00 <st1:country-region
w:st="on">US</st1:country-region>/<st1:country-region w:st="on"><st1:place
w:st="on">Canada</st1:place></st1:country-region> Central<br>
Subject: RE: [Proj] Re: "Double ellipsoid" case?<br>
<br>
Richard Greenwood wrote:<br>
<br>
> I for one, am finding this an interesting, and pertinent thread. <br>
> Let's not quash it, and let's keep it civil.<br>
<br>
Thanks, Richard. <br>
<br>
I thought daan explained the matter clearly, and I have not much to add.
<br>
<br>
daan Strebe wrote:<br>
> The practical consequence is that they do not have a conformal projection;
<br>
> hence local distances measured from a point are not quite uniform in all <br>
> directions, and neither are azimuths quite equally spaced radially from <br>
> that point. So?<br>
<br>
I would add that these deviations from conformality of course can be <br>
quantified and bounded. For example, I wrote that "I think the
maximal <br>
angle distortion for Sphere Mercator is 0.2 degrees". This was
wrong; <br>
I was thinking of the maximal azimuth error, and the maximal angle <br>
distortion is twice that, so make it 0.4 degrees. <br>
<br>
It is not surgical precision, and it is 16th century technology, <br>
but I wouldn't say that a 0.4 degree error is "absurd".<br>
(Well, unless my application required exact conformality for some reason.) <br>
The benefit is simpler formulas and faster execution times. <br>
<br>
Noel Zinn wrote:<br>
> If I could just switch the WGS84 ellipsoid in the WGS84 datum with <br>
> the Google Sphere (as you suggest), why couldn't I switch <br>
> the International Ellipsoid in ED50 with Clarke 1866? Or any other <br>
> switch for that matter? <br>
<br>
Let's see if I understand you correctly. You are asking: if I have data <br>
in the datum ED50, why can't I define a projection which, by definition, <br>
treats (Lon,Lat) from ED50 as if it were (Lon,Lat) on the Clarke 1866 <br>
ellipsoid?<br>
<br>
Well, I am forced to say that this is quite possible. The mathematics works,<br>
and it is not illegal. Again, the result would be a map that is not
exactly<br>
conformal, relative to the shape of ED50. Because this naive mapping <br>
from (Lon,Lat) on the International Ellipsoid to (Lon,Lat) on the Clarke 1866 <br>
ellipsoid - without changing the numerical values - is not conformal. <br>
At least, I don't think so. <br>
<br>
But I am just saying that it is possible. I think it would be silly <br>
and meaningless, because I would give up the exact conformality <br>
(relative to the shape of ED50) in return for nothing. I would not <br>
get the benefit of simpler formulas or faster execution times, <br>
since I would have to use ellipsoid formulas to go from Clarke 1866 <br>
to the plane. <br>
<br>
Noel: <br>
> In addition to worse fits mathematically <br>
> (since the adjustment was done on the defining ellipsoid), <br>
<br>
Yes, spherical formulas give worse maps for large scale maps. <br>
But I would say, only slightly worse. Not absurdly worse. <br>
<br>
> we'd open the door to uncertainty and crisis. <br>
> You are proposing (and Google has introduced) the geodetic <br>
> equivalent of sub-prime mortgages in the financial market. <br>
> Don't do it!<br>
<br>
Well, we are talking about a maximal 0.4 degree error for angles, <br>
and a variation of the local scale in different direction <br>
through a point, which would be at most 0.5 percent, I think. <br>
<br>
If economists could do predictions with 0.5 percent accuracy, <br>
the world economy would be in a better state! <br>
<br>
--<br>
Mikael Rittri<br>
<st1:place w:st="on"><st1:City w:st="on">Carmenta</st1:City> <st1:State w:st="on">AB</st1:State></st1:place><br>
<st1:address w:st="on"><st1:Street w:st="on">Box</st1:Street> 11354</st1:address><br>
SE-404 28 Göteborg<br>
Visitors: Sankt Eriksgatan 5<br>
<st1:country-region w:st="on"><st1:place w:st="on">SWEDEN</st1:place></st1:country-region><br>
Tel: +46-31-775 57 37<br>
Mob: +46-703-60 34 07 <br>
mikael.rittri@carmenta.com<br>
www.carmenta.com<br>
<br>
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