[Proj] Meaning of aposphere

[Hilmy Hashim]

So Monster meets Turnip - who wins?

*Hilmy*


On Thu, Apr 28, 2011 at 10:02 PM, Clifford J Mugnier <cjmce at lsu.edu> wrote:

>  I've read Hotine's series in *Empire Survey Review*.  I tell my students
> that his Aposphere is shaped like a turnip.
>
>  Clifford J. Mugnier, C.P., C.M.S.
> Chief of Geodesy,
> *Center for GeoInformatics*
> Department of Civil Engineering
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>
> ------------------------------
> *From:* proj-bounces at lists.maptools.org on behalf of Charles Karney
> *Sent:* Thu 28-Apr-11 06:35
> *To:* PROJ.4 and general Projections Discussions
> *Cc:* Hilmy Hashim
> *Subject:* Re: [Proj] Meaning of aposphere
>
>  Well, I puzzled by how an aposphere could be different from a sphere.
> However there are surfaces with constant curvature which are not
> spheres.  I think if you impose additional conditions, e.g., that the
> surface is closed and nowhere singular you end up with a sphere.  A
> simple example of a non-spherical surface is what you get if you
> partially folded up a swimming cap.  A tractrix rotated about its
> aymptote gives you a surface of constant negative curvature.  I'm
> uncertain whether any of these are really needed to develop map
> projections.
>
> For pictures see Eisenhardt (1909), Chap 8, Figs. 26-30:
>
>    http://books.google.com/books?id=hkENAAAAYAAJ&pg=PA270
>
> On 04/28/11 07:20, Mikael Rittri wrote:
> > Yes, I understand that the aposphere is some kind
> > of intermediate surface.
> >
> > But it's the phrase "sphere of constant total curvature"
> > that bothers me.  Most people who describes the Hotine cites
> > this phrase; I think it's from Snyder.  I tried to look up
> > "total curvature", and if I remember rightly, it has at least
> > two meanings:
> >    In one meaning, every surface that is topologically equivalent
> > to a sphere has the same total curvature (4*pi or something like
> > that). That's probably not what Snyder meant...
> >    In another meaning, total curvature refers to Gaussian curvature
> > at a point of a surface.  But in this meaning, every sphere has
> > constant total curvature, so the aposphere seems to be a sphere,
> > no more and no less.  So, how does it differ from the Gaussian
> > sphere that is used in some other projections, like Swiss Oblique
> > Mercator, Krovak, and Oblique Sterographic?
> >    Or are there other surfaces than spheres that can have a constant
> > Gaussian curvature at every point? (I think there is some trumpet-shaped
> > surface that has constant Gaussian curvature, but curvature like a
> saddle;
> > is that positive or negative curvature? But apart from that.)
> >
> > Well, I shouldn't complain but try to read Hotine's original paper,
> > but rumors say it's very dense and difficult to follow. I suspect
> > I wouldn't understand it.
> >
> > But if someone knows a snappy explanation of the aposphere...?
> > Just curious,
> >
> > Mikael Rittri
>
> --
> Charles Karney <charles.karney at sri.com>
> SRI International, Princeton, NJ 08543-5300
> Tel: +1 609 734 2312
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