Hotine Oblique Mercator

Name Hotine Oblique Mercator
EPSG Code 9812
GeoTIFF Code CT_ObliqueMercator (3)
CT_ObliqueMercator_Hotine (3)
OGC WKT Name hotine_oblique_mercator
Supported By EPSG, GeoTIFF, PROJ.4, OGC WKT

Projection Parameters

Name EPSG # GeoTIFF ID OGC WKT Units Notes
Latitude of projection center 1 CenterLat latitude_of_center Angular
Longitude of projection center 2 CenterLong longitude_of_center Angular
Azimuth of initial line 3 AzimuthAngle azimuth Angular
Angle from Rectified to Skew Grid 4 RectifiedGridAngle rectified_grid_angle Angular
Scale factor on initial line 5 ScaleAtCenter scale_factor Unitless
False Easting 6 FalseEasting false_easting Linear
False Northing 7 FalseNorthing false_northing Linear

Notes

The EPSG Projection Oblique Mercator with code 9815 indicates that it's formula is really just that of Hotine Oblique Mercator despite the slightly different naming of the false easting and northing. I am treating that projection as an alias for this one, but with a page of it's own since it has it's own codes, at least in EPSG.

Most packages do not support the angle from rectified to skewed grid, and GeoTIFF didn't used to have a tag to carry it (I have added it for libgeotiff 1.1.2). 90 degrees seems to be the default value (removing it's effect in the equations below).

PROJ.4 Organization

While PROJ.4 dosn't have a Angle from Rectified to Skew Grid I think the +no_rot flag is related to this parameter.

  +proj=omerc +lat_0=Latitude of projection center 
              +lonc=Longitude of projection center
              +alpha=Azimuth of initial line
              +k_0=Scale factor on initial line 
              +x_0=False Easting
              +y_0=False Northing
The PROJ.4 initialization function fails if Azimuth is zero (or 90 degrees). I am not clear if this is fundamental to the oblique mercator projection or a peculiarity of PROJ.4. It is inconvenient because some of the GeotIFF sample files have an azimuth of zero.

Clarification From the EPSG

Via email from Roger Lott of the EPSG working group:

EPSG considers the Oblique Mercator and Hotine Oblique Mercator projections as two very similar but separate methods because there is a subtle difference regarding the point at which the rectification from skew grid to map grid and where the false grid coordinates are applied. It is possible to interchange the two possible points, but to do so requires that the three parameters (skew angle, false grid coordinates) have different values. Conversely, it is important that a set of published parameter values are applied at the correct point. It is the method that determines this. As always, parameters and method must be consistent with each other. This is somewhat analogous to the Lambert Conic Conformal projection where it is possible to inter-relate parameters and appropriate parameter values between the 1- and 2-standard parallel cases. In both cases EPSG publishes a single formula with if statements to accommodate the different methods where necessary. An alternative approach would be to use two independent algorithms in which much code could or would be duplicated. (Note: some folks call the Lambert 1SP case 'Lambert Tangential' which is only true if the scale factor value is unity - which it generally isn't).

EPSG Notes

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 (Hotine Oblique Mercator), Madagascar (Laborde Oblique Mercator) and the Alaskan panhandle. It was also originally applied to Hungary in the 1970's and, at the beginning of the 20th century, by Rosenmund to the mapping of Switzerland. This projection is sometimes referred to as the Rectified Skew Orthomorphic projection or the Hotine Oblique Mercator. Hotine projected the ellipsoid conformally onto a sphere of constant total curvature, called the `aposhere', before projection onto the plane. Alternative formulae derived by projecting the ellipsoid onto the `conformal' sphere give identical results within the practical limits of the use of the formulae.

The co-ordinate system is defined by:

An initial line central to the map area of given azimuth ac passes through a defined centre of the projection (jc , lc ) . 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 centre line and the v axis is perpendicular to this line.

The projection's initial line may be selected as a line with a particular azimuth through a single point, - normally at the centre 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 in Epicentre. 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 other application than for space imagery.

In applying the formulae 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 centre 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.

To ensure that all co-ordinates in the map area have positive grid values, false co-ordinates are applied. These may be given values (Ec , Nc) if applied at the projection centre or be applied as false easting (FE) and false northing (FN) at the natural origin.

Formulas for the oblique Mercator, involving hyperbolic functions, were derived by Hotine. Snyder adapted these formula using exponential functions, thus avoiding use of Hotine's hyperbolic expressions. As in the case of the several varieties of Transverse Mercator, the choice of the co-ordinate transformation parameters distinguish the co-ordinate transformation within the Oblique Mercator Co-ordinate Transformation method. The formulae can be used for the following cases: 

	Alaska Zone 1
	Hungary EOV
	Madagascar
	East and West Malaysia
	Switzerland
The Swiss and Hungarian systems are a special case where the azimuth of the line through the projection centre is 90 degrees. This therefore gives similar but not exactly the same results as a conventional transverse mercator.

Epicentre supports the formulae to cover the co-ordinate transformation methods for the following versions of the Oblique Mercator:

  1. USGS (Snyder) formulae where the false origin is defined at the natural origin
  2. USGS (Snyder) formulae where the false origin is defined at the centre of the projection
  3. Swiss Oblique Cylindrical, using polynomial equations developed by Bolliger.
The two versions of the Snyder formulae could be unified by introducing an extra parameter to define which origin is used for the application of false co-ordinates. The Swiss Oblique Cylindrical is a special case, as the equations given by Bolliger may deviate from the rigorous formulae for points at some distance from the origin. Throughout Switzerland the differences are always less than one metre.

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

Switzerland:
	""Die Žnderung des Projektionssystems der schweizerischen Landesvermessung."" M. Rosenmund 1903.
	""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: 

fc 	= latitude of centre of the projection
lc 	= longitude of centre of the projection
ac 	= azimuth (true) of the centre line passing through the centre of the projection
gc 	= rectified bearing of the centre line
kc	= scale factor at the centre of the projection
and either
Ec	= False Easting at the centre of projection
Nc	= False Northing at the centre of projection
or
FE	= False Easting at the natural origin
FN	= False Northing at the natural origin
From these the following constants for the projection may be calculated : 

B	=	(1 + e2 cos4(fc) / (1 - e2 ))0.5
A	=	a B kc (1 - e2 )0.5 / ( 1 - e2 sin2 (fc))
t0	=	tan(p / 4 - fc / 2) / ((1 - e sin (fc)) / (1 + e sin (fc)))e/2
D	=	B (1 - e2 )0.5  / (cos(fc) ( 1 - e2 sin2 (fc))0.5)
Dsq	=	D2  = 1 if D*D < 1 to avoid problems with computation of F
F	=	D + (Dsq - 1)0.5 . SIGN(fc)
H	=	F t0B
G	=	(F - 1 / F) / 2
g0	=	asin(sin (ac) / D)
l0	=	lc - (asin(G tan(g0))) / B
Then compute the (uc , vc) co-ordinates for the centre point (fc , lc). In general


uc	=	(A / B) atan((Dsq - 1)0.5 / cos (ac) ). SIGN(fc)
vc	=	0
But note that for the special cases where ac = 90 degrees (e.g. Hungary, Switzerland) then 

uc	=	A (lc - l0 )  
vc	=	0
Forward case: To compute (E,N) from a given (f,l) : 

t	=	tan(p / 4 - f / 2) / ((1 - e sin (f)) / (1 + e sin (f)))e/2
Q	=	H / tB
S	=	(Q - 1 / Q) / 2
T	=	(Q + 1 / Q) / 2
V	=	sin(B (l - l0))
U	=	(- V cos(g0) + S sin(g0)) / T
v	=	A ln((1 - U) / (1 + U)) / 2 B
u	=	A atan((S cos(g0) + V sin(g0)) / cos(B (l - l0 ))) / B
The value of u from the above equation assumes that the FE and FN values have been specified with respect to the origin of the (u , v) axes - see diagram.

For projection parameters where the false easting and northing values (Ec , Nc) have been specified with respect to the centre of the projection (fc , lc) then :

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

The rectified skew co-ordinates are then derived from: 

E	= v cos(gc) + u sin(gc) + (FE or Ec)
N 	= u cos(gc) - v sin(gc) + (FN or Nc)
Reverse case: Compute (f,l) from a given (E,N) : 

v'	=	(E - (FE or Ec)) cos(gc) - (N - (FN or Nc)) sin(gc)
u'	=	(N - (FN or Nc)) cos(gc) + (E - (FE or Ec)) sin(gc)
For projection parameters where the false easting and northing values (Ec , Nc) have been specified with respect to the centre of the projection (fc , lc) then : 

u'	=	u' + uc

Q'	=	e- (B v `/ A)  where e is the base of natural logarithms.
S'	=	(Q' - 1 / Q') / 2
T'	=	(Q' + 1 / Q') / 2
V'	=	sin (B u' / A)
U'	=	(V' cos(gc) + S' sin(gc)) / T'
t'	=	(H / ((1 + U') / (1 - U'))0.5)1 / B

c	= p / 2 - 2 atan(t')

f	= c + sin(2c).( e2 / 2 + 5 e4 / 24 + e6 / 12 + 13 e8 / 360) + 
	   sin(4c).( 7 e4 /48 + 29 e6 / 240 + 811 e8 / 11520) +  sin(6c).( 7 e6 / 120 + 81 e8 / 1120) + 	   sin(8c).(4279 e8 / 161280)

l	=  l0  - atan ((S' cos(gc) - V' sin(gc)) / cos(B u' / A)) / B","For Projected Coordinate System  Timbalai 1948 / R.S.O. Borneo (m)

Parameters:
	Ellipsoid:  Everest 1830 (1967 Definition)
				a = 6377298.556 metres 	1/f = 300.8017
				then	e = 0.081472981	e2 = 0.006637847
					
	Latitude Projection Centre	fc  	4o00'00""N		=	0.069813170 rad
	Longitude Projection Centre	lc  	115o00'00""E	=	2.007128640 rad
	Azimuth of central line	ac	53o18'56.9537""	=	0.930536611 rad
	Rectified to skew	gc	53o07'48.3685""	=	0.927295218 rad
	Scale factor	ko	 0.99984
	False Eastings	FE	 0.00 m
	False Northings	FN	 0.00 m

Forward calculation for: 
	Latitude		f	4o39'20.783""N		=	0.081258569 rad
	Longitude		l	114o28'10.539""E	=	1.997871312 rad
		
B = 	1.003303209		F =	1.07212156
A =	6376278.686		H = 	1.00000299
		to = 	0.932946976		g0 = 	0.92729522
		D = 	1.002425787		l0 = 	1.91437347
		D2 =	1.004857458
	
		uc =	738096.09		vc =	0.00
		t =	0.922369529		Q =	1.084456854
		S =	0.081168129		T = 	1.003288725
		V =	0.83675700		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 same easting and northing first gives :
		v' =   	-93307.40		u' =	734236.558
		u'+uc = 1472332.652		Q' =	1.014790165
		S'  =	0.014682385		T' =	1.000107780	
		V' =	0.115274794		U' = 	0.080902065
		t' = 	0.922369529		c =	0.080721539		
				
Then	Latitude    	 f  =	4o39'20.783""N		
	Longitude	l   =	114o28'10.539""E"