[Proj] Meaning of aposphere (was: RE: RSO, +gamma and Hotine Oblique Mercator Variant A)

[Mikael Rittri]

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
Carmenta
Sweden
http://www.carmenta.com 

________________________________

From: Hilmy Hashim [mailto:hilmyh at gmail.com] 
Sent: den 28 april 2011 11:48
To: Mikael Rittri
Cc: PROJ.4 and general Projections Discussions
Subject: Re: [Proj] RSO, +gamma and Hotine Oblique Mercator Variant A


Mikael,


In http://trac.osgeo.org/proj/ticket/104, you said

A +no_uoff ("no u-offset") means that the origin is at the so-called natural origin, on the central oblique line of the projection, and near the ordinary equator (on the "aposphere equator", but I've never understood what an aposphere is).

>From my simplistic understanding of Hotine's method, during the first part of the forward projection, the coordinates from the ellipsoid are projected onto a sphere of "constant total curvature" or an aposphere. I suppose this makes the maths simpler? So the equator on this aposphere is slightly offset from the equator on the ellipsoid.

Regards

Hilmy