[Proj] Meaning of aposphere

[Noel Zinn (cc)]

For what it's worth, the Geodetic Glossary published by the (US) National
Geodetic Survey in September 1986 has a definition of aposphere:

"A surface of rotation whose meridional section is defined by the equation:
r = a sech [b (tau + c)], where a, b and c are constants, tau is the
isometric latitude, and r is the perpendicular distance from the axis of
rotation to the surface."

Also, "The constants are chosen so that the aposphere touches the ellipsoid
with which it has a common axis of rotation along some parallel that passes
through the center of the area for which the transformation is required"

Noel Zinn, Principal, Hydrometronics LLC
+1-832-539-1472 (office), +1-281-221-0051 (cell)
noel.zinn at hydrometronics.com (email)
http://www.hydrometronics.com (website)

-----Original Message----- 
From: Charles Karney
Sent: Thursday, April 28, 2011 6:35 AM
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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