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

[strebe]

Egads. 

>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.

>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.

Your thesis was not solely about conformal projections, so the math wouldn't be solely about them, either. Your thesis was that conformal projections are ill-suited to small-scale application, and that other projections (particularly equal-area) are better. How can you make a statement about the suitability of non-conformal projections, and how can I challenge that statement, by resorting only to the mathematics of conformal projections?

>>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.

Then why would you trot out mathematics that deals only with the scale factor in two specific (and arbitrary) directions?

>>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.

Indeed. Well. See above.

>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.

All right, we might be getting somewhere. There is some confusion in terminology here, and it seems to reflect confusion in the actual situations as well. Scale factor is not areal inflation or deflation; nor is it directly related to areal distortion. On a general map, scale factor requires both a point and a direction in order to mean anything. You cannot sensibly say a map has a "scale factor of 1 over the entire map". Unless the map is conformal, the scale factor at any point varies according to direction. And if the map IS conformal, the scale factor will vary continuously over the entire map anyway and therefore could not be equal to 1 over the entire map.

Hence an equal projection map does not have a scale factor of 1 over the entire map. It is distortion of area that it lacks, not distortion of "scale factor". Meanwhile, at every point on the map, there will be scale factors that range between the semimajor and semiminor axis of the Tissot ellipse, depending upon direction. You cannot sensibly say THE scale factor at a given point.

>Equal area maps have distortion problems like conformal maps.

Why yes, they do. As do all maps. So... why is it, again, that 

>Extended geographic range usage of any conformal projection is a
>contentious issue asany resultant grid system has sufficiently large
>scale errors as to make the Cartesian usage of the grid very
>questionable [from your original post]. 

applies only to conformal projections and not all projections? I think you try to answer that here:

>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.

Yes; it is generally agreed that thematic maps need to be equal-area except in certain highly specialized scenarios. But thematic maps are only one kind of small-scale map. It takes a lot more than just feelings to credibly dismiss conformal projections for small-scale use. In any case, the normal Mercator is only one of an infinitude of conformal projections. Its abuse is not a case for denigrating the entire category.

Most relevant to the discussion at hand, I do not find even faintly persuasive your argument that the ellipsoidal transverse Mercator should not be used for extended ranges in the absence of other considerations. All of your arguments point to ANY projection being bad for such use. And, well, it's true. All projections are bad for extended ranges. Just not the ellipsoidal transverse Mercator specifically or particularly or to a worse degree. While it's hard to make a case for a conformal global map (unless you interrupt it along two complete meridians at least), a global map is a far cry from the kinds of ranges that have been discussed here for the Gauß-Krüger.

Regards,
-- daan Strebe


On Jun 13, 2008, at 7:58:37 AM, "Gerald I. Evenden" <geraldi.evenden at gmail.com> wrote:
From:   "Gerald I. Evenden" <geraldi.evenden at gmail.com>
Subject:    Re: [Proj] Re: Comments of tmerc, etmerc and ftmerc errors
Date:   June 13, 2008 7:58:37 AM PDT
To: "PROJ.4 and general Projections Discussions" <proj at lists.maptools.org>
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

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