[Proj] Meaning of aposphere

Mikael Rittri Mikael.Rittri at carmenta.com
Fri Apr 29 07:22:01 EST 2011

> his Aposphere is shaped like a turnip.

Like a swede, you mean? I'm flattered. 
(If the pun doesn't work in America, never mind.)

So, rotational symmetry, nearly like a sphere, but
perhaps somewhat pointy poles?  I suppose that makes
sense.  Thanks, all who answered.

Mikael Rittri

From: proj-bounces at lists.maptools.org [mailto:proj-bounces at lists.maptools.org] On Behalf Of Clifford J Mugnier
Sent: den 28 april 2011 16:03
To: PROJ.4 and general Projections Discussions; PROJ.4 and general Projections Discussions
Cc: Hilmy Hashim
Subject: Re: [Proj] Meaning of aposphere

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 
Patrick F. Taylor Hall 3223A
Baton Rouge, LA  70803
Voice and Facsimile:  (225) 578-8536 [Academic] 
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Member of the Americas Petroleum Survey Group


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

For pictures see Eisenhardt (1909), Chap 8, Figs. 26-30:


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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