[Proj] Converting EPSG (Hotine) Oblique Mercator definitions to PROJ lines

[Mikael Rittri]

Hello,
I have thought about the relation between the Hotine Oblique Mercators
of EPSG and PROJ.4.  
I'll explain my current understanding in the form of two design
questions.
(This concerns only Hotine Oblique Mercator == Rectified Skew
Orthomorphic; not 
 Swiss/Hungarian Oblique Mercator or Laborde Oblique Mercator.)

1) Where is the point located, whose projected coordinates are x_0, y_0
(false easting, false northing)?  
   This question has two possible answers.
a) At the projection center (which has the central longitude and the
central latitude, and is
   on the central line).  
   EPSG calls this choice "Oblique Mercator", with coordinate operation
method code 9815,
   and the x_0 and y_0 are called EC and NC (Easting at projection
center, Northing at projection center).
   In PROJ.4, this corresponds to +proj=omerc, without using +no_uoff.

b) At the so-called natural origin, which is the nearest intersection of
the central line and the
   aposphere equator.  (It is enough to know that this will be pretty
near the ordinary equator of the earth.)
   This answer was apparently preferred by Hotine, as it simplifies the
formulas a little, and the countries
   he mapped were near the equator anyway.  
   EPSG calls this choice "Hotine Oblique Mercator", with coordinate
operation method code 9812, 
   and the x_0 and y_0 are called FE and FN (False Easting and False
Northing at natural origin).
   In PROJ.4, this corresponds to +proj=omerc, with +no_uoff.  

2) What shall the grid azimuth be for the central line?  That is to say,
what is the angle between
   a line that goes upwards on the map, and the projected image of the
central line (which is an 
   (approximate?) great circle on earth).  
   This has an infinite number of answers, but they can be classified
like this: 
   a) zero.  In EPSG, this corresponds to a Rectified-To-Skew-Grid angle
of zero.  
      In PROJ.4, this corresponds to +proj=omerc, with +no_rot.
   b) alpha_c, which is the geographic azimuth of the central line at
the projection center.  
      This choice means that grid north equals true north an the
projection center.  
      In EPSG, this corresponds to a Rectified-To-Skew-Grid angle that
equals alpha_c.  
      In PROJ.4, this corresponds to +proj=omerc, without +no_rot, and
without +rot_conv. 
   c) gamma_0, which is the geographic azimuth of the central line at
the so-called natural 
      origin (see 1 b above).  This choice means that grid north equals
true north at 
      the natural origin.  In EPSG, this corresponds to a
Rectified-To-Skew-Grid angle 
      that has been carefully chosen to be equal to gamma_0.  
      In PROJ.4, this corresponds to +proj=omerc, without +no_rot, but
with +rot_conv.  
   d) An arbitrary number gamma.  This choice, if ever used by EPSG,
would be hard to 
      distinguish from gamma_0 in an automatic script, since gamma_0 is
non-trivial 
      to compute from alpha_c etc.  I don't think this choice is
supported by PROJ.4.
      However, I believe it is possible in libproj4, by setting the
gamma parameter.  

I don't know whether choice d) is actually used in any official map
projection.  
But I suppose so, since Gerald Evenden made the effort to implement the
general gamma 
parameter in libproj4.

Best regards,
   Mikael Rittri, Carmenta, Sweden (www.carmenta.se) 
(These thoughts are provided "as is", without warranty of any kind,
expressed or implied...) 


-----Original Message-----
From: proj-bounces at lists.maptools.org
[mailto:proj-bounces at lists.maptools.org] On Behalf Of Melita Kennedy
Sent: den 30 januari 2007 22:24
To: proj at lists.maptools.org
Subject: [Proj] Re: URGENT Non-existent Projection Definition
(ObliqueMercator)

EPSG already has codes for 2 versions of Michigan GeoRef, although they
call them "Michigan Oblique Mercator". 

3078 is based upon NAD83 and uses meters. 
3079 is based upon NAD83(HARN) and also uses meters. 

The method listed is "Hotine Oblique Mercator" (9812) with these
parameters (note the azimuth):

latitude of projection center: 45 18 33.0 N longitude of projection
center: 86 0 0.0 W Azimuth of initial line: 337 15 20.016 Angle of
rectified to skew grid: 337 15 20.016 scale factor on initial line:
0.9996 false easting: 2546731.496 false northing: -4354009.816

EPSG Note: If using Oblique Mercator (9815), Ec = 499840.252 m and Nc =
528600.303 m

Melita
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