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The Oblique Mercator projection uses a (U,V) coordinate system, with the U axis along the central line. During the forward projection, coordinates from the ellipsoid are projected conformally to a sphere of constant total curvature, called the "aposphere", before being projected onto the plane. The projection coordinates are further convented to a (X,Y) coordinate system by rotating the calculated (u,v) coordinates to give output (x,y) coordinates. The rotation value is usually the same as the projection azimuth (the angle, east of north, of the central line), but some cases allow a separate rotation parameter.

There are two forms of the oblique mercator, differing in the origin of their grid coordinates. The (EPSG code 9812) has grid coordinates start at the intersection of the central line and the equator of the aposphere. The (EPSG code 9815) is the same, except the grid coordinates begin at the central point (where the latitude of center and central line intersect). ESRI separates these two case by appending "Natural_Origin" (for the "Hotine_Oblique_Mercator" ) and "Center" (for the "Oblique_Mercator" ) to the projection names.

Two different methods are used to specify the central line for the oblique mercator: 1) a central point and an azimuth, east of north, describing the central line and 2) two points on the central line. The EPSG does not use the two point method, while ESRI separates the two cases by putting "Azimuth" and "Two_Point" in their projection names. Both cases use the point where the "latitude_of_center" parameter crosses the central line as the projection's central point. The is not a projection parameter, and is instead calculated as the intersection between the central line and the equator of the aposphere.

For the azimuth method, the central latitude cannot be ±90.0 degrees and the central line cannot be at a maximum or minimum latitude at the central point. In the two point method, the latitude of the first and second points cannot be equal. Also, the latitude of the first point and central point cannot be ±90.0 degrees. Furthermore, the latitude of the first point cannot be 0.0 and the latitude of the second point cannot be -90.0 degrees. A change of 10^-7 radians can allow calculation at these special cases. Snyder's restriction of the central latitude being 0.0 has been removed, since the equations appear to work correctly in this case.

Azimuth values of 0.0 and ±90.0 degrees are allowed (and used in Hungary and Switzerland), though these cases would usually use a Mercator or Transverse Mercator projection instead. Azimuth values > 90 degrees cause errors in the equations.

The oblique mercator is also called the "Rectified Skew Orthomorphic" (RSO). It appears is that the only difference from the oblique mercator is that the RSO allows the rotation from the (U,V) to (X,Y) coordinate system to be different from the azimuth. This separate parameter is called "rectified_grid_angle" (or "XY_Plane_Rotation" by ESRI) and is also included in the EPSG's parameters for the Oblique Mercator and Hotine Oblique Mercator. The rotation parameter is optional in all the non-two point projections and will be set to the azimuth if not specified.

Projection cases and aliases implemented by the ObliqueMercator are:

• Oblique_Mercator (EPSG code 9815) - grid coordinates begin at the central point, has "rectified_grid_angle" parameter.
• Hotine_Oblique_Mercator_Azimuth_Center (ESRI) - grid coordinates begin at the central point.
• Rectified_Skew_Orthomorphic_Center (ESRI) - grid coordinates begin at the central point, has "rectified_grid_angle" parameter.
• Hotine_Oblique_Mercator (EPSG code 9812) - grid coordinates begin at the interseciton of the central line and aposphere equator, has "rectified_grid_angle" parameter.
• Hotine_Oblique_Mercator_Azimuth_Natural_Origin (ESRI) - grid coordinates begin at the interseciton of the central line and aposphere equator.
• Rectified_Skew_Orthomorphic_Natural_Origin (ESRI) - grid coordinates begin at the interseciton of the central line and aposphere equator, has "rectified_grid_angle" parameter.
• Hotine_Oblique_Mercator_Two_Point_Center (ESRI) - grid coordinates begin at the central point.
• Hotine_Oblique_Mercator_Two_Point_Natural_Origin (ESRI) - grid coordinates begin at the interseciton of the central line and aposphere equator.

• libproj4 is available at libproj4 Miscellanea
  Relevent files are: PJ_omerc.c , pj_tsfn.c , pj_fwd.c , pj_inv.c and lib_proj.h
• John P. Snyder (Map Projections - A Working Manual, U.S. Geological Survey Professional Paper 1395, 1987)
• "Coordinate Conversions and Transformations including Formulas", EPSG Guidence Note Number 7 part 2, Version 24.
• Gerald Evenden, 2004, Documentation of revised Oblique Mercator

This parameter applies to the "azimuth" case only. It is set to Double.NaN NaN for the "two points" case.