[OSRS-PROJ] Concerning calumny and the ellipsoidal transverse Mercator

[Gerald I. Evenden]

On Fri, 2003-08-22 at 23:59, Strebe at aol.com wrote:
> In the "Orthographic Projections and MapServer" thread, Gerald I.
> Evenden <gerald.evenden at verizon.net> writes:
> 
> > What nonsense.  If there was a closed form of the transverse
> > Mercator
> > for the elliptical case, it would extend to infinity also.  The
> > basis
> > of the math requires it.
> 
> No, Mr. Evenden; I'm afraid it is your objection which is nonsense,
> though I certainly understand the skepticism: exceedingly few people
> know of the finite nature of the ellipsoidal transverse Mercator, and
> it certainly defies intuition. It was first brought to my attention by
> a Dr. David E. Wallis of Glendale, California, who is expert in
> elliptical integrals and harmonic functions. I gather he discovered
> that fact independently, though it hardly seems likely that Krüger or
> the great Gauss was unaware of it.
>
> The lower the eccentricity, the greater the extent of the map on the
> plane, until at zero eccentricity the map becomes infinite. For
> earthlike spheroids, however, the map is indeed finite and a very
> pleasing and unexpected shape, and its areal inflation is vastly less
> than any other conformal map of the entire sphere (or spheroid) in my
> acquaintance. I would be happy to introduce you to the mathematics. My
> own very modest contribution was to construct an iterative solution to
> the basic equations, one which can be computed to any desired
> accuracy. I'll publish that, as well as a large number of other papers
> on the topic of small scale map projections, after I finish rewriting
> Geocart.
> 
> Because the finite nature of the ellipsoidal transverse Mercator is so
> rarely known, and because no numeric solutions have ever been
> published, and because there is no geodetic need for such a
> projection, it is entirely likely that I have produced the only images
> of the projection ever made. I would be happy to make some available
> if anyone is interested.

It is difficult for me to refrain from becoming a bit sarcastic here
as your story sounds like a tabloid exposure.  I will remain a
sceptic until I see peer review publication.

> > Extending more than 4° to 5° is pointless anyway.  Distortion starts
> > to take over and the map becomes useless as a cadastral tool.
> 
> Naturally; that is why I clearly stated that my comments apply to
> small-scale projections.
> 
> > For world scale mapping Mercator in any form is terrible.
> 
> While it is true that the Mercator has been misused commonly for
> centuries, it seems hyperbole to dismiss it entirely. Firstly,
> straight rhumbs are interesting and useful even on world maps, as long
> as that reason for choosing Mercator is clear to the observer;
> secondly, if it is reasonable to produce an equal-area cylindrical
> world map (Peters) then it is also reasonable to produce a conformal
> cylindrical world map; and thirdly, as I mention above, the
> ellipsoidal transverse Mercator makes an unexpectedly good conformal
> world map.

Use of Mercator in a cartographic educational sense is OK, certainly
in demonstration of its distortion and rhumb lines.  At the moment I
can't think of any practical application of rhumb lines except in
large-scale manual navigation.

As for conformity, its application is limited to large-scale
cadastral mapping.  Conformity is a concept which only
applies to the infinitesimal region about a point.  At a distance
distortion is quickly apparent and eventually becomes extreme.
Conformal projection for global presentations is most useful to
demonstrate the limitation of conformal projections.

As for the Peter's projection (a perverse use of the well known
cylindrical equal area) it has been severely criticized by well
know cartographers.

There are reasons for picking certain projections but in general,
my basic recommendation is one of the equal area systems which preserve
the concept of region size.  There are also some good small scale
projections which minimize overall distortion but are neither
conformal nor equal area.

> > Many projections do not have mathematical limits other than the
> > polar limit and one can extent the longitude indefinitely.  It
> > is not the projections duty to supply "boundary" information other
> > than indicate when the limits of the projection are exceeded because
> > they have no meaning in non-graphical usage.
> 
> The fact that many projections have no mathematical bounds does not
> excuse the projection engine from supplying bounds for those many
> projections that do have mathematical bounds. It is the projection,
> and the projection only, that knows what those bounds are. It makes no
> sense to supply a projection to a graphics engine if the graphics
> engine cannot properly draw it due to incomplete information. I have
> no quarrel with PROJ's philosophy if its intent is purely cadastral.
> However, the topic arose in the context of the orthographic
> projection, which has no cadastral use. It is clear that graphics
> programs *are* having difficulty drawing small-scale maps, and they
> will continue having problems drawing small-scale maps if they are not
> supplied complete information. If you do not view that responsibility
> to be PROJ's, yet you do view the orthographic projection
> transformation to be the responsibility of PROJ's, then I'm completely
> confused about your philosophy, and I doubt from any failing in my
> intellect.

>From my own experience plotting global maps, the problem is not where
the boundary is but what do you do with it.  My mapping as been purely
vector so the effect of the edges more easily handled than the fill
problem.

The proj library does indicate when points are out of bounds.  In some
cases it would be very difficult to provide an analytic function
defining the projection boundary and the nature of this function
will mathematically.  Even if the function is known, the determination
of the intersect with the limiting function and the vector is just
about as complicated as determining the intersection by using the
projection itself.

Lastly, like the cadastral people supply their addenda to meet their
need, the datum conversion group add their addenda, then why shouldn't
the graphic types do the same.

	<deleted material>
> 
> Regards,
> 
> daan Strebe
> Geocart author
http://www.mapthematics.com/
-- 
Gerald I. Evenden <gerald.evenden at verizon.net>

----------------------------------------
PROJ.4 Discussion List
See http://www.remotesensing.org/proj for subscription, unsubscription
and other information.