[Proj] Re: Comments of tmerc, etmerc and ftmerc errors

[Gerald I. Evenden]

On Friday 13 June 2008 12:42 am, strebe wrote:
> On Jun 12, 2008, at 6:47:53 PM, "Gerald I. Evenden"
> <geraldi.evenden at gmail.com> wrote: Are you saying scale and its associated
> error is not related to error of area?
>
> Snyder (1395, p. 24) says "s=hk" where s is areal scale factor and h and k
> are  the meridian and parallel scale factors (in the orthogonal case).
>
> I must be misinterpreting your words.(?)
> You are interpreting my words somewhat correctly, but seemingly
> misinterpreting their significance. "Somewhat" because you've selected a
> quotation from Snyder that deals only with projections whose graticules (in
> some magical aspect) are orthogonal (meaning, meridian and parallel meeting
> at right angles). That hardly seems "fair" or representative. Nonetheless,
> even restricting the discussion to orthogonal graticules, it still follows
> that any given scale factor at a point says nothing about the point's areal
> inflation.

Lets see.  I think I was originally only discussing conformal projections so 
the last I checked the graticule lines are orthogonal at any point and thus 
the statement I made about s=hk applied.

I fail to see what is "fair" or unfair about the statement.  Is it correct or 
incorrect.

> Firstly, the scale factor along the meridian or parallel is no indication
> of the scale factor in other directions unless the map is conformal.

I never said it was.

> It is 
> possible to create a map projection with unitary scale factor along all
> meridians and parallels across the entire map — yet the map still suffers
> diverse scale factors in other directions and even infinite and
> infinitesimal scale factors in some directions in some regions of the map.

I fail to understand how that applies to anything I said unless you are 
challenging Snyder's statement.  For non-orthogonal parallel and meridian the 
scale factor is more complex and I did not bother to quote it because I was 
only concerned with the conformal projection.

> Hence the scale factor along meridian and parallel are meaningless in
> characterizing what's going on at a point in the absence of other
> information.

I fail to see your point.

> Secondly, back to the orthogonal case: "s = hk". If h = 1000.0 and k =
> 0.001 then s = 1. If h = 1 and k = 1, then s = 1. Therefore, clearly the
> magnitude of any particular scale factor is irrelevant to area.

I fail to see what is irrelevant other than the map with such conditions would 
be *very* distorted.  The scale factor is still relevant and if the scale 
factor remained equal to 1 over the entire map then the map is an "equal 
area" map---distortion and all.  Equal area maps have  distortion problems 
like conformal maps.

My argument for the equal area map over the conformal map in small scale and 
global presentations is that I feel that the equal area gives a more 
realistic mental image of the world than the conformal image.  For example, a 
Mercator map distorts the apparent "size" of Greenland whereas an equal area 
map does not.  In thematic maps of, say, air masses, arctic air masses would 
seem to become much larger than equitorial masses and perhap give a mistaken 
bias or impression of their relative importance over other regions.

> Your original statement was: 
>
> "Extended geographic range usage of any conformal projection is a
> contentious issue as any resultant grid system has sufficiently large scale
> errors as to make the Cartesian usage of the grid very questionable.".
>
> My point is, every (reasonably) continuous projection will have enormous
> scale factors across large swaths of the map. There is no research to
> suggest conformal maps are any worse than (for example) equal-area maps in
> that regard. The reason conformal maps are frowned upon for global mapping
> is not for any problem with their scale factors, but for their areal
> inflation. In some applications the areal inflation might be preferable to
> the shearing inevitable in an equal-area map or even in some compromise
> projection.
>
> Regards,
> -- daan Strebe

-- 
The whole religious complexion of the modern world is due
to the absence from Jerusalem of a lunatic asylum.
-- Havelock Ellis (1859-1939)  British psychologist