[Proj] Re: Mercator Problem

Ron Russell ron at russfam.freeserve.co.uk
Wed Oct 19 14:10:25 EDT 2005

Daan, Clifford,

Of course, deep down, I knew that the scale varies along the central line of
an oblique Mercator projection and I suppose my suggestion was not
theoretically sound. I should also say that I get as much satisfaction as
anybody when I get the chance to use Vincenty!

Even Vincenty does not deal with antipodal points, however!

In my defence I would say that we were talking about points scaled off a
paper map!

In my days as a surveyor in the early 70s - do you remember Peter’s Tables
and Facit mechanical calculators? - we used to use the mean of the scale
factors at the end points of lines to reduce measured distances to the
projection, and for the very longest lines we sometimes used Simpson’s rule
(but I never remember it making a sensible difference). That was on at the
edge of UTM zones. In those days it was really important to be able compute
on the projection. I once computed a traverse using the Mid Latitude
formulae - it took me two days, and I got the wrong answer!

I bet the scale change on the central line of a Hotine Oblique Mercator
projection, even across California, could be treated the same way! Do you
have any figures?


Ron Russell
Tel : 01823 270308 email : ron at russfam.freeserve.co.uk


From: proj-bounces at lists.maptools.org
[mailto:proj-bounces at lists.maptools.org] On Behalf Of Strebe at aol.com
Sent: 17 October 2005 22:56
To: proj at lists.maptools.org
Subject: [Proj] Re: Mercator Problem

Ron's proposal is perfectly sound on a spherical earth. Since the oblique
Mercator central line runs through the two points of interest, and since the
scale is constant and correct along that great circle, there's no need for
integration or scale factor corrections. Obviously it's more problematic on
the ellipsoid, given that none of the usual oblique formulations carry
constant scale along the transformed equator (and if I recall correctly,
it's not possible whilst retaining conformality), but even so, Hotine is
close enough to constant scale that it might suffice, depending on the
accuracy needs and assuming the two points are reasonably short of

If other calculations that Ron mentions aren't needed then clearly Cliff's
suggestion is the way to go. Why go out on a... erm... tangent?

daan Strebe

In a message dated 10/17/2005 2:42:58 PM 太平洋夏時間, cjmce at lsu.edu writes:


You're proposing a piecewise-integration along a grid line.  With that,
you'd also have to include the correction for scale factor at a point, as
integrated by so many points of choice along the projected line.  Scale
factor along the Central Line of an Oblique Mercator is not going to
obviate that need.

It's far simpler to just call the subroutine in PROJ4 for an ellipsoidal
geodesic between the two end points.  (Once called the "Principal Problem
of Geodesy" in the 19th Century).

Assuming you know how to program the subroutine calls, it's easier done
than said.

Clifford J. Mugnier
Chief of Geodesy and
Associate Director,
Department of Civil Engineering
Baton Rouge, LA 70803
Voice and Facsimile: (225) 578-8536 [Academic]
Voice and Facsimile: (225) 578-4474 [Research]

What about using the Lat and Long of the endpoints to define the central
line of an Oblique Mercator, with a scale factor of 1.0 on the central
Then the distance can be calculated by Pythagoras, and other useful
operations can easily be calculated - mid point, distance of a third point
from the line and even the calculation of a buffer zone, all of which seem
horrendous when working on the ellipsoid. (OK, the mid point is not too

Ron Russell
Tel : 01823 270308 email : ron at russfam.freeserve.co.uk
-----Original Message-----
From: proj-bounces at lists.maptools.org
[mailto:proj-bounces at lists.maptools.org] On Behalf Of Clifford J Mugnier
Sent: 17 October 2005 20:52
To: PROJ.4 and general Projections Discussions
Subject: Re: [Proj] Mercator Problem

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