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

[Strebe at aol.com]

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.

> 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.

> 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.

> In general, these comments are getting off track as I feel the basis
> of the thread was to solve the problems of "mapserve" and not advertise
> a graphics package.  A package, I might add, that package seems to be
> only available for Apple systems.
> 
I am disappointed that you have chosen to interpret my intent so cynically. 
The fact that Geocart is available only on Macintosh is irrelevant, since I am 
not advertising Geocart on this list. My comments, I believe, bore direct 
relevance to solving, or at least clarifying, mapserve's problem. Because I wrote 
Geocart and because Geocart has solved all these problems this thread has been 
discussing, it hardly seems inappropriate for me to invoke it. I've been a 
member of this list for several years now. If my intent were to spam you with 
advertising, it would have happened long ago.

> BTW, is the math for your Strebe-X projections published?
> 
It is not, and I don't consider most of them worth publishing. However, the 
one titled simply "Strebe" is a particular case of a completely general and 
heretofore unobserved method of generating useful equal-area projections. I will 
publish that one. The theory is simple enough: If Projection A is equal-area, 
and Projection B is equal-area, and Projection C is equal-area, then you may 
produce an equal-area projection D by applying the transformation

A->B

to C, insofar as you have first scaled C such that its range lies entirely 
within A's range. The same strategem can be employed in conformal maps.

Regards,

daan Strebe
Geocart author
http://www.mapthematics.com/
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