[Proj] Scale factor for Transverse Mercator

Gerald I. Evenden geraldi.evenden at gmail.com
Tue Sep 9 18:01:32 EDT 2008

On Tuesday 09 September 2008 9:49:13 am Charles Karney wrote:
> What you mean by "scale error" is, I believe, the amount by which the
> scale deviates from unity.  Whether this is an error or a problem
> depends on the application.

I agree that I may be a little free in my usage and perhaps "scale factor" is 
more appropriate.  But in the bottom line, it is what "screws up" any 
measurement process using the Cartesian grid unless the one making the 
measurements account for this factor.  I think that for the causal user or 
one not fully familiar with the factor the use of the term scale-error is 
preferred because it red-flags the usage of the coordinates without 
considering this factor.

> Sure, in the standard UTM zones, the scale is close to unity and this
> allows reasonably accurate distances to be computed directly.  In other
> applications it may only be important that the scale does not vary much
> over an area of interest (even though it's far from the central
> meridian).  And the regular Mercator projection is regularly used over
> large areas even though it then has a large scale variation (you
> remember the variable scale on the side of the map?).  Wikipedia claims
> that Google maps uses Mercator (presumably for its joint properties of
> conformality and zero meridian convergence).

Let me pose the question: why are all of the state plane zones in the US 
established to have very small scale errors (factors) over their extent?  
California divides the state into 6 zones to obviously ensure small scale 
error anywhere in the state.  Why did DOD create the 6 degree zone if it were 
not to minimize the scale error.  Of the non-US grid systems I have seen, the 
same philosophy is apparent: limit the maximum scale error of the grid 

> So I think it's incorrect to say that the TM is only "useful" +/- 3deg
> from the central meridian.  You have TM algorithms which have <1um error
> out to +/- 65 deg. from the central meridian.  I recommend that libproj
> provide access to these and leave it to the user to decide on the
> usefulness.

There is little point in mixing the computational error---what we have been 
thrashing over for the last eternity---with the scale error or factor.  They 
are wo different problems.  We have *no* control over the scale error because 
that is an inherent property of the projection (other than fudging it with 
k_0).  We can only control how accurately we can compute the forward/inverse 

The use of TM as a grid system is very much limited to the 6 degree zone (at 
the equator) and its use as a grid system is the only situation where I find 
any justification in concern over the computational accuracy of the 
projection process.  Computational accuracy to 1x10-4 meter should be 
sufficient and I think we have demonstated that is sufficient for the Kruger 
(Taylor series) method with the current length of the polynomials.  I believe 
we can certify the Kruger as the Gold standard in terms of speed and accuracy 
for grid class usage.

> It may well be that you need to stop before 90 deg. because an algorithm
> with reasonable accuracy is not available.  But here again I would hold
> off on statements like "so distorted as to be unrecognizable"; locally
> there is NO distortion --- the projection is conformal after all.

Oh indeed, quite true.  But I find it a bit difficult to shrink myself to an 
infinitesimal size in order to appreciate the conformality factor.  
Conformality is an empty claim where the rate of change of the scale factor 
is so large that a feature of any extent is distorted beyond recognition.  
Please remember, conformality is a mathematical term and does not mean there 
will not be distortion over a finite extent with varying scale factor that 
we, as mere mortals, will find objectionable.

Once one leaves the requirements of small scale-error grid systems the choice 
of projection is wide open and I find it hard to find any particular feature 
of TM that makes it a better choice over many other projection.  To use a 
projection outside its range of grid system usage as a means of measurement 
is very questionable.  And why make that a criteria?  The solution to 
determining distance and azimuth has been and always will be usage of a 
geodesic algorithm using the geographic coordinates of the two points of 
interest.  The main function of a projection in any graphic system is to get 
the geographic information to the Cartesian display and similarly convert 
display selected points back to geographic coordinates for a purpose such as 
distance determination.

In computer display systems there is rare excuse to even need an elliptical 

> (By 
> the way, it's not the case that the TM projection on an ellipsoid
> "stops" at 90deg from the central meridian; it can be extended
> continuously past this point.)

I think you need to try to explain to school children what is going on with 
this bizarre projection.  It is bad enough with the ordinary Mercator.

Pursuing the super-accurate TM is purely an academic problem and does not at 
the current time show any practical application.  But one needs a little fun.

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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