[Proj] Transverse Mercator algorithm

Charles Karney ckarney at sarnoff.com
Wed Sep 3 23:47:06 EDT 2008

 >> (As an example of why this is useful, you can compute accurate
 >> transformations from one UTM zone to the neighboring one which rely
 >> on the fact that the transformation is conformal.  A 4-point fit
 >> gives 2um accuracy over a 10km x 10km area.)

I checked my notes on this the UTM<->UTM error is 0.5um -- see below

 > I am not exactly sure that I follow what you are doing above.  In one
 > mental image I have of your description I feel you would run afoul of
 > scale error due to the non-equivalent distance that the particular
 > area is away from the respect CM of the two zones.
 > I really need a step-by-step description of your operation.

Here goes...  Let the coordinates of a point be

    x,y in zone 38


    x',y' in zone 39

The mapping x,y -> x',y' is conformal; thus the mapping can be written

    z' = f(z)

where z = x + iy, z' = x' + iy', and f is an analytic function.  We
approximate f(z) by a polynomial in z, g(z), and we pick the polynomial
to be of (n-1)th order such that

    g(z) = f(z)

at n points z = z1, z2, z3 ... zn.  The resulting polynomial is

    g(z) = sum_i f(zi) * product_j!=n (z-zj)/(zi-zj)

(This probably isn't a convenient form of the polynomial for evaluation
-- but never mind that.)

If four points z1, z2, z3, z4 are taken at the corners of a square of
side 100km then the difference |g(z) - f(z)| ~ 5mm within the square.
(I did this test with a square whose SW corner is at 36.12N 47.33E.)
For a 10km square the error is 0.5um.

The same technique can be used to convert from UTM to geographic
coordinates by approximating the conformal mapping from UTM to Mercator
followed by the conversion from isometric to geodetic latitude.  In this
case the error for a 100km square is 2cm.

Incidentally with just a two-point fit, the approximate transformation
(which is just a linear polynomial) accounts for the changes in scale
and meridian convergence.

This procedure depends on the fact that f is a conformal mapping.  For
this reason, it's important that numerical schemes preserve
conformalality to machine precision.

 > What really held me hostage on Lee's paper were the two equations 10
 > and 11 and I was not sure what to do at that point so I decided not to
 > waste any more aspirin.

Which paper of Lee's is this?

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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