Oblique Mercator Projection. A conformal, oblique, cylindrical projection with the cylinder
touching the ellipsoid (or sphere) along a great circle path (the central line). The
Mercator and
TransverseMercator projections can be thought of
as special cases of the oblique mercator, where the central line is along the equator or a
meridian, respectively. The Oblique Mercator projection has been used in Switzerland, Hungary,
Madagascar, Malaysia, Borneo and the panhandle of Alaska.
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
HotineObliqueMercator (EPSG code 9812) has
grid coordinates start at the intersection of the central line and the equator of the aposphere.
The
ObliqueMercator (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
#centralMeridian 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.
References:
-
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