[Proj] Projecting a rectangle to another coord. system?

Clifford J Mugnier cjmce at lsu.edu
Fri Sep 10 11:17:46 EDT 2004


The problem is not as simple to solve at it appears because a rectangle is
comprised of straight lines.  A straight line on one projection is NOT a
straight line on another projection.

In the real world of geodetic transformations on the ellipsoid (projections
and datums), the problem is often encountered with offshore geophysical
exploration.  National governments lease offshore concessions defined by
their local datum and grid system on their respective continental shelf.
That same area of the offshore continental shelf is travelled by
international commercial shipping.  A country then guarantees that certain
"highways" in the ocean near their shores will be free of obstructions to
navigation - (e.g. no oil drilling platforms) - even if an exploration and
production concession has been awarded that covers that area!

These "highways" are called Traffic Separation Schemes or Safety Fairways.
They are ellipsoidal loxodromes (rhumb lines) that are straight only on a
normal Mercator projection - used for ship navigation.  The concessions are
defined on Lambert Conformal Conic, Transverse Mercator, Oblique Mercator,
Cassini-Soldner, Polyhedric, etc. projection grids that comprise the
national Grid for that nation.  Loxodromes are curved lines on those other

Finding the intersection of a loxodrome and a straight grid line is an
iterative solution first developed jointly by the late John P. Snyder and I
back in the early 1980s.  It can be done in a general case if
object-oriented programming is used to define the algorithms for X and for
Y for both projections as calling arguments to a generalized subroutine.
John and I did it in Fortran 77 back then.

Oil companies that have ignored this problem in the past have had expensive
lessons learned - to the tune of tens of millions of dollars per mistake!
This commonly occurs in the North Sea, the Gulf of Mexico, South China Sea,

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

I am new to this list.

My problem is:
- I have a rectangle given in coordinate system A
(defined by its southwest and northeast corners).
- In coordinate system B, I need to find one, or possibly
two, rectangles that cover all of the original rectangle.
The new rectangle(s) don't have to be as small as possible,
although this would be nice.  (They shouldn't be much too
large, though.)
(By a "coordinate system", I mean a geodetic datum plus a projection.)

Can proj4 (or some other tool) solve this problem in the general case?

I would also be interested in the algoritms used. Do you know any
paper that has been published?

(The motivation is, of course, that a GIS system displays data in
coordinate system A, but the data is stored in coordinate system B,
so the bounding box in A (defined by the window limits) must be
translated to one or two query rectangles in B, for the geo-database.)

Best regards,
Mikael Rittri

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