So Monster meets Turnip - who wins?<div><br clear="all"><span style="font-size:large"><i><font face="garamond, serif"><b><font color="#FF6600">Hilmy</font></b></font></i></span><br>
<br><br><div class="gmail_quote">On Thu, Apr 28, 2011 at 10:02 PM, Clifford J Mugnier <span dir="ltr"><<a href="mailto:cjmce@lsu.edu">cjmce@lsu.edu</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
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<div dir="ltr"><font face="Times New Roman" color="#000000" size="2">I've read Hotine's series in <em>Empire Survey Review</em>. I tell my students that his Aposphere is shaped like a turnip.</font></div>
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<div class="MsoNormal" style="margin:0in 0in 0pt"><span lang="DE" style="font-size:10pt">Clifford J. Mugnier, C.P., C.M.S.</span></div>
<div class="MsoNormal" style="margin:0in 0in 0pt"><span style="font-size:10pt">Chief of Geodesy,</span></div>
<div class="MsoNormal" style="margin:0in 0in 0pt"><b><span style="font-size:10pt;font-variant:small-caps">Center for GeoInformatics</span></b></div>
<div class="MsoNormal" style="margin:0in 0in 0pt"><span style="font-size:10pt">Department of Civil Engineering </span></div>
<div class="MsoNormal" style="margin:0in 0in 0pt"><span style="font-size:10pt">Patrick F. Taylor Hall 3223A</span></div>
<div class="MsoNormal" style="margin:0in 0in 0pt"><b><span style="font-size:10pt">LOUISIANA STATE UNIVERSITY </span></b></div>
<div class="MsoNormal" style="margin:0in 0in 0pt"><span style="font-size:10pt">Baton Rouge, LA<span> </span>70803</span></div>
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<font face="Tahoma" size="2"><b>From:</b> <a href="mailto:proj-bounces@lists.maptools.org" target="_blank">proj-bounces@lists.maptools.org</a> on behalf of Charles Karney<br><b>Sent:</b> Thu 28-Apr-11 06:35<br><b>To:</b> PROJ.4 and general Projections Discussions<br>
<b>Cc:</b> Hilmy Hashim<br><b>Subject:</b> Re: [Proj] Meaning of aposphere<br></font><br></div>
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<p><font size="2"></font></p><div><div><font size="2"></font></div><div class="h5"><font size="2">Well, I puzzled by how an aposphere could be different from a sphere.<br>However there are surfaces with constant curvature which are not<br>
spheres. I think if you impose additional conditions, e.g., that the<br>surface is closed and nowhere singular you end up with a sphere. A<br>simple example of a non-spherical surface is what you get if you<br>partially folded up a swimming cap. A tractrix rotated about its<br>
aymptote gives you a surface of constant negative curvature. I'm<br>uncertain whether any of these are really needed to develop map<br>projections.<br><br>For pictures see Eisenhardt (1909), Chap 8, Figs. 26-30:<br>
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<a href="http://books.google.com/books?id=hkENAAAAYAAJ&pg=PA270" target="_blank">http://books.google.com/books?id=hkENAAAAYAAJ&pg=PA270</a><br><br>On 04/28/11 07:20, Mikael Rittri wrote:<br>> Yes, I understand that the aposphere is some kind<br>
> of intermediate surface.<br>><br>> But it's the phrase "sphere of constant total curvature"<br>> that bothers me. Most people who describes the Hotine cites<br>> this phrase; I think it's from Snyder. I tried to look up<br>
> "total curvature", and if I remember rightly, it has at least<br>> two meanings:<br>> In one meaning, every surface that is topologically equivalent<br>> to a sphere has the same total curvature (4*pi or something like<br>
> that). That's probably not what Snyder meant...<br>> In another meaning, total curvature refers to Gaussian curvature<br>> at a point of a surface. But in this meaning, every sphere has<br>> constant total curvature, so the aposphere seems to be a sphere,<br>
> no more and no less. So, how does it differ from the Gaussian<br>> sphere that is used in some other projections, like Swiss Oblique<br>> Mercator, Krovak, and Oblique Sterographic?<br>> Or are there other surfaces than spheres that can have a constant<br>
> Gaussian curvature at every point? (I think there is some trumpet-shaped<br>> surface that has constant Gaussian curvature, but curvature like a saddle;<br>> is that positive or negative curvature? But apart from that.)<br>
><br>> Well, I shouldn't complain but try to read Hotine's original paper,<br>> but rumors say it's very dense and difficult to follow. I suspect<br>> I wouldn't understand it.<br>><br>> But if someone knows a snappy explanation of the aposphere...?<br>
> Just curious,<br>><br>> Mikael Rittri<br><br>--<br>Charles Karney <<a href="mailto:charles.karney@sri.com" target="_blank">charles.karney@sri.com</a>><br>SRI International, Princeton, NJ 08543-5300<br>Tel: <a href="tel:%2B1%20609%20734%202312" value="+16097342312" target="_blank">+1 609 734 2312</a><br>
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