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 |

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 | |

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=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

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).

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 SwitzerlandThe 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:

- USGS (Snyder) formulae where the false origin is defined at the natural origin
- USGS (Snyder) formulae where the false origin is defined at the centre of the projection
- Swiss Oblique Cylindrical, using polynomial equations developed by Bolliger.

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 originFrom 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))) / BThen 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 = 0But note that for the special cases where ac = 90 degrees (e.g. Hungary, Switzerland) then

uc = A (lc - l0 ) vc = 0Forward 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 ))) / BThe 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 - ucFor 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"