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

[strebe]

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.

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

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.

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

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