<html><head><style type='text/css'>p { margin: 0; }</style></head><body><div style='font-family: Arial; font-size: 12pt; color: #000000'><P class=MsoNormal style="MARGIN: 0in 0in 0pt"><FONT face="Times New Roman">Mikael,</FONT></P>
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<P class=MsoNormal style="MARGIN: 0in 0in 0pt"><FONT face="Times New Roman">Thanks for your revised assessment of maximum angular distortion in the Google Maps Projection (GMP).<SPAN style="mso-spacerun: yes"> </SPAN></FONT></P>
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<P class=MsoNormal style="MARGIN: 0in 0in 0pt"><FONT face="Times New Roman">Are you aware that the scale distortion in GMP is very large and unstated?<SPAN style="mso-spacerun: yes"> </SPAN>In the vicinity of the Equator and it's about 1.0067 (that's 67cm per 100m).<SPAN style="mso-spacerun: yes"> </SPAN>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).<SPAN style="mso-spacerun: yes"> </SPAN>Here's how you can demonstrate this to yourself.<SPAN style="mso-spacerun: yes"> </SPAN></FONT></P>
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<P class=MsoNormal style="MARGIN: 0in 0in 0pt"><FONT face="Times New Roman">Choose any meridian.<SPAN style="mso-spacerun: yes"> </SPAN>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.<SPAN style="mso-spacerun: yes"> </SPAN>Consequently, the Eastings are irrelevant; they're constant.<SPAN style="mso-spacerun: yes"> </SPAN>Here's the raw data.</FONT></P>
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<P class=MsoNormal style="MARGIN: 0in 0in 0pt"><FONT face="Times New Roman">The inverse geodesic distance along the meridian on the GS from the Equator to 1 degree North is 111319.4908m.<SPAN style="mso-spacerun: yes"> </SPAN>The GMP Northing on the Equator is 0m and the point scale is 1 (unity). <SPAN style="mso-spacerun: yes"> </SPAN>The GMP Northing at 30 minutes North is 55,660.4519m and the point scale factor is 1.00003808.<SPAN style="mso-spacerun: yes"> </SPAN>The GMP Northing at 1 degree North is 111,325.1429m and the point scale factor is 1.00015233.<SPAN style="mso-spacerun: yes"> </SPAN></FONT></P>
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<P class=MsoNormal style="MARGIN: 0in 0in 0pt"><FONT face="Times New Roman">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.<SPAN style="mso-spacerun: yes"> </SPAN>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.<SPAN style="mso-spacerun: yes"> </SPAN>Divide the grid distance by line scale factor and you'll see that we have a sub-millimeter tie with the geodesic distance.<SPAN style="mso-spacerun: yes"> </SPAN></FONT></P>
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<P class=MsoNormal style="MARGIN: 0in 0in 0pt"><FONT face="Times New Roman">If there were a physical monument on the Equator and one at 1 degree North (same meridian), what would be the distance between them?<SPAN style="mso-spacerun: yes"> </SPAN>Well, that would be the geodesic distance on the WGS84 ellipsoid.<SPAN style="mso-spacerun: yes"> </SPAN>This is the WGS84 datum, isn't it?<SPAN style="mso-spacerun: yes"> </SPAN>Well, that distance is 110,574.3886m.<SPAN style="mso-spacerun: yes"> </SPAN>The ratio of these distances is the hidden (and, therefore, insidious) scale distortion of the GMP in this vicinity of the world.<SPAN style="mso-spacerun: yes"> </SPAN></FONT></P>
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<P class=MsoNormal style="MARGIN: 0in 0in 0pt"><FONT face="Times New Roman">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.<SPAN style="mso-spacerun: yes"> </SPAN>It's a different frame of reference, a different tolerance for error.<SPAN style="mso-spacerun: yes"> </SPAN>If I surveyed football pitch (about 100m), a 67cm error would concern me.<SPAN style="mso-spacerun: yes"> </SPAN>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?<SPAN style="mso-spacerun: yes"> </SPAN></FONT></P>
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<P class=MsoNormal style="MARGIN: 0in 0in 0pt"><FONT face="Times New Roman">Regards,</FONT></P>
<P class=MsoNormal style="MARGIN: 0in 0in 0pt"><FONT face="Times New Roman">Noel Zinn</FONT></P>
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<P><BR>----- Original Message -----<BR>From: "Mikael Rittri" <Mikael.Rittri@carmenta.com><BR>To: "PROJ.4 and general Projections Discussions" <proj@lists.maptools.org><BR>Sent: Tuesday, December 2, 2008 10:13:22 AM GMT -06:00 US/Canada 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>Carmenta AB<BR>Box 11354<BR>SE-404 28 Göteborg<BR>Visitors: Sankt Eriksgatan 5<BR>SWEDEN<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>_______________________________________________<BR>Proj mailing list<BR>Proj@lists.maptools.org<BR>http://lists.maptools.org/mailman/listinfo/proj<BR></P></P></div></body></html>