[Proj] Transverse Mercator algorithm
Charles Karney
ckarney at sarnoff.com
Sun Sep 7 14:53:23 EDT 2008
Gerald I. Evenden wrote:
> I appreciate your approach to error diagnosis in the comparison of the
> ?tmercs however I must take a more brute force approach and merely
> look at primitive results. This is due to the fact that I am *not* a
> theoretician but merely one looking for the bottom line.
It's probably good to have an idea of what the error might look like as
a function of position. The error will approximately be the first
dropped term in the series expansions. In the case of the 4-term JHS
154 expansion this is
~ e^10 sin(10 xi) cosh(10 eta)
~ e^10 sin(10 phi) cosh(10 x/a)
or some oscillating function of y multiplying an exponential function of
x. We are, of course, primarily interested in the potentially large
variation with x.
> I will get a 3D plot made sometime today.
There's a good chance that this will look messy because of the
oscillatory behavior of the error with latitude and the fact that
round-off errors dominate for small x.
My methodology is:
Generate a set of random latitudes (lat0), longitudes (lon0), exact
eastings (x0), and exact northings (y0). I posted a set of such data
(1/4 million points) at
http://charles.karney.info/geographic/TMcoords.dat.gz
For each geographic point (lat0,lon0), compute
(x,y) = F(lat0,lon0)
(lat,lon) = F^-1(x,y)
where F is the numerical transverse Mercator projection being
evaluated. Define
err = max( hypot(x-x0, y-y0),
1e7/90 * hypot(lat-lat0, cos(lat0)*(lon-lon0)) )
mu = asin( sin(lon0) * cos(lat0) )
The second term in err approximately converts the lat/lon discrepancy
into a distance. mu is the angular distance away from the central
meridian and I use this as an ersatz x of the purpose of error analysis.
You can get a feel for the behavior of the error by plotting err against
mu. E.g., in Matlab notation:
plot(mu, log10(err), 'x');
The tabulated error data I sent out yesterday was computed with, e.g.,
max(err(x0<5e5 & y0<96e5))
max(err(mu<25))
and so on. The second statement takes the maximum of the error for all
mu < 25 and this is useful because (from the plot) err is strongly
increasing with mu.
> Of course, the above has one critical assumption we must not forget:
> the maxima procedure is correct and god-like.
Quite so. The good news is that when the procedure fails, it fails
badly.
--
Charles Karney <ckarney at sarnoff.com>
Sarnoff Corporation, Princeton, NJ 08543-5300
URL: http://charles.karney.info
Tel: +1 609 734 2312
Fax: +1 609 734 2662
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