3.4.4.5 Oblique Mercator and Hotine Oblique Mercator

It has been noted that the Transverse Mercator projection is employed for the topographical mapping of longitudinal bands of territories, limiting the amount of scale distortion by limiting the extent of the projection either side of the central meridian. Sometimes the shape, general trend and extent of some countries makes it preferable to apply a single zone of the same kind of projection but with its central line aligned with the trend of the territory concerned rather than with a meridian. So, instead of a meridian forming this true scale central line for one of the various forms of Transverse Mercator, or the equator forming the line for the Mercator, a line with a particular azimuth traversing the territory is chosen and the same principles of construction are applied to derive what is now an Oblique Mercator. Such a single zone projection suits areas which have a large extent in one direction but limited extent in the perpendicular direction and whose trend is oblique to the bisecting meridian - such as East and West Malaysia, Madagascar, and the Alaskan panhandle. It was also originally applied to Hungary in the 1970’s and, at the beginning of the 20^th century, by Rosenmund to the mapping of Switzerland. The projection's initial line may be selected as a line with a particular azimuth through a single point, normally at the center of the mapped area, or as the geodesic line (the shortest line between two points on the ellipsoid) between two selected points. The latter approach is not currently followed by EPSG/POSC; it has been applied to mapping space imagery or, more frequently, for applying a geographical graticule to the imagery. However, the repeated path of the imaging satellite does not actually follow the centre lines of successive oblique cylindrical projections so a projection was derived whose centre line does follow the satellite path. This is known as the Space Oblique Mercator Projection and although it closely resembles an oblique cylindrical it is not quite conformal and has no application other than for space imagery.

Hotine projected the ellipsoid conformally onto a sphere of constant total curvature, called the ‘aposphere’, before projection onto the plane. This projection is sometimes referred to as the Rectified Skew Orthomorphic. Formulas, involving hyperbolic functions, were derived by Hotine. Snyder adapted these formulas to use exponential functions, thus avoiding use of Hotine's hyperbolic expressions. Alternative formulas derived by projecting the ellipsoid onto the ‘conformal’ sphere give identical results within the practical limits of the use of the formulas.

EPSG identifies two forms of the oblique Mercator projection, differentiated only by the point at which false grid coordinates are defined. If the false grid coordinates are defined at the intersection of the initial line and the aposphere the projection is known as the Hotine Oblique Mercator; if the false grid coordinates are defined at the projection centre the projection is known as the Oblique Mercator.

The co-ordinate system is defined by:

The initial line central to the map area of given azimuth a _c passes through a defined center of the projection (j _c, l _c ) . The point where the projection of this line cuts the equator on the aposphere is the origin of the (u , v) co-ordinate system The u axis is parallel to the center line and the v axis is perpendicular to (90° clockwise from) this line.

In applying the formulas for the (Hotine) Oblique Mercator the first set of co-ordinates computed are referred to the (u, v) co-ordinate axes defined with respect to the azimuth of the center line. These co-ordinates are then ‘rectified’ to the usual Easting and Northing by applying an orthogonal transformation. Hence the alternative name as the Rectified Skew Orthomorphic. In the special case of the projection covering the Alaskan panhandle the azimuth of the line at the natural origin is taken to be identical to the azimuth of the initial line at the projection centre. This results in grid and true north coinciding at the projection centre rather than at the natural origin as is more usual.

The formulas can be used for the following cases:

Alaska State Plane Zone 1

Hungary EOV

Madagascar Laborde Grid

East and West Malaysia Rectified Skew Orthormorphic grids

Swiss Cylindrical projection

The Swiss and Hungarian systems are a special case where the azimuth of the line through the projection center is 90 degrees. These therefore gives similar but not exactly the same results as a conventional transverse mercator projection.

Specific references for the formulas originally used in the individual cases of these projections are:

Switzerland:	"Die Änderung des Projektionssystems der schweizerischen Landesvermessung." M. Rosenmund 1903. Also "Die projecktionen der Schweizerischen Plan und Kartenwerke." J. Bollinger 1967.
Madagascar:	"La nouvelle projection du Service Geographique de Madagascar". J. Laborde 1928.
Malaysia:	Series of Articles in numbers 62-66 of the Empire Survey Review of 1946 and 1947 by M. Hotine.

The defining parameters for the oblique mercator projection are:

j_c = latitude of center of the projection.

l_c = longitude of center of the projection.

a_c = azimuth (true) of the center line passing through the center of the projection.

g_c = rectified bearing of the center line.

k_c = scale factor at the center of the projection.

and either,

for the Oblique Mercator

E_c = False Easting at the center of the projection

N_c = False Northing at the center of the projection

or for the Hotine Oblique Mercator

FE = False Easting at the natural origin.

FN = False Northing at the natural origin.

From these defining parameters the following constants for the projection may be calculated:

To avoid problems with computation of F, if D<1, make D² = 1.

Then compute the (u_c, v_c) co-ordinates for the center point (f_c, l_c).

In general

But note that for the special cases where a_c = 90 degrees (e.g. Hungary, Switzerland) then

Forward Case: To compute (E, N) from a given (j, l):

For the Hotine Oblique Mercator (where the FE and FN values have been specified with respect to the origin of the (u, v) axes):

For Oblique Mercator where the false easting and northing values (E_c, N_c) have been specified with respect to the center of the projection (f_c, l_c) then:

For the special cases where a_c = 90 degrees care must be taken with the sign of the u_c in this formula because the center of the projection is equidistant from the two points at which the center line cuts the equator on the aposphere.

The rectified skew co-ordinates are then derived from:

Reverse case: Compute (f, l) from a given (E, N):

For the Hotine Oblique Mercator:

For the Oblique Mercator:

Then the other parameters can be calculated.

Example: Timbalai 1948 / R.S.O. Borneo m. (Note: Later versions of the EPSG guide have modified the meanings and values of the input parameters, and have worked out a different example).

Latitude of Projection Center	jc	4°00'00"N	=	0.069813170 rad
Longitude of Projection Center	lc	115°00'00"E	=	2.007128640 rad
Azimuth of Central Line	ac	53°18'56.9537"N	=	0.930536611 rad
Rectified to skew grid	gc	53°07'48.3685"N	=	0.927295218 rad
Scale factor	kc	0.99984
False Easting	FE	0.00 m
False Northing	FN	0.00 m

Forward calculation for:

Latitude	j	4°39'20.783"N	=	0.081258569 rad
Longitude	l	114°28'10.539"E	=	1.997871312 rad 1

B =	1.003303209				F =	1.07212156
A =	6376278.686				H =	1.00000299
to =	0.932946976				go =	0.92729522
D =	1.002425787				lo =	1.91437347
D2 =	1.004857458
uc =	738096.09				vc =	0.00
t =	0.922369529				Q =	1.084456854
S =	0.081168129				T =	1.003288725
V =	0.0836757 1				U =	0.014680803
v =	-93307.40				u =	734236.558
					u - uc =	-3859.536

Then

Easting (E) = 531404.81 m

Northing (N) = 515187.85 m

Reverse calculations for the same easting and northing first gives:

v' =	-93307.40				u' =	734236.558
u' + uc =	1.472332.652				Q' =	1.014790165
S' =	0.014682385				T' =	1.000107780
V' =	0.115274794				U' =	0.080902065
t' =	0.922369529				c =	0.080721539

Then

Latitude (j) = 4°39'20.783"N

Longitude (l) = 114°28'10.539"E

¹ Two values were corrected from the original release. Corrections made in 2006-01

Last modified: 12 July 2000
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