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 Transverse Mercator 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 Hotine_Oblique_Mercator (EPSG code 9812) has grid
coordinates start at the intersection of the central line and the equator of the
aposphere. The Oblique_Mercator (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 Obique_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 central meridian 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 equaitons 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
IdrLabordeMadagascar 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