<HTML dir=ltr><HEAD><TITLE>Re: [Proj] Meaning of aposphere</TITLE>
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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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<FONT face=Tahoma size=2><B>From:</B> proj-bounces@lists.maptools.org 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>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><BR> <A href="http://books.google.com/books?id=hkENAAAAYAAJ&pg=PA270">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 <charles.karney@sri.com><BR>SRI International, Princeton, NJ 08543-5300<BR>Tel: +1 609 734 2312<BR>_______________________________________________<BR>Proj mailing list<BR>Proj@lists.maptools.org<BR><A href="http://lists.maptools.org/mailman/listinfo/proj">http://lists.maptools.org/mailman/listinfo/proj</A><BR></FONT></P></DIV></BODY></HTML>