EPSG Dataset :: v12.003

Hotine Oblique Mercator (variant B)

Coordinate Operation Method Details [VALID]

Name:

Hotine Oblique Mercator (variant B)

Code:

9815

Operation is Reversible:

Yes

Formula:

Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

The following constants for the projection may be calculated :

B = {1 + [e^2 * cos^4(latc) / (1 - e^2 )]}^0.5
A = a * B * kc *(1 - e^2 )^0.5 / ( 1 - e^2 * sin^2(latc))
to = tan(pi/4 - latc/2) / ((1 - e*sin(latc)) / (1 + e*sin(latc)))^(e/2)
D = B (1 - e^2)^0.5 / (cos(latc) * ( 1 - e^2*sin^2(latc))^0.5)
if D < 1 to avoid problems with computation of F make D^2 = 1
F = D + (D^2 - 1)^0.5 * SIGN(latc)
H = F*(to)^B
G = (F - 1/F) / 2
gammao = asin(sin(alphac) / D)
lonO = lonc - (asin(G*tan(gamma0))) / B
vc = 0
In general: uc = (A / B) atan2((Dsq - 1)^0.5 , cos (alphac) ) * SIGN(latc) (see Gn7-2 implementation notes in preface for atan2 convention)
but for the special cases where alphac = 90 degrees (e.g. Hungary, Switzerland) then
uc = A*(lonc - lonO)

Forward case: To compute (E,N) from a given (lat,lon) :

t = tan(pi/4 - lat/2) / ((1 - e sin (lat)) / (1 + e sin (lat)))^(e/2)
Q = H / t^B
S = (Q - 1 / Q) / 2
T = (Q + 1 / Q) / 2
V = sin(B (lon - lonO))
U = (- V cos(gammao) + S sin(gammao)) / T
v = A ln((1 - U) / (1 + U)) / 2 B

In general:
u = (A atan2((S cos(gammao) + V sin(gammao)) , cos(B (lon - lonO))) / B) - (ABS(uc) . SIGN(latc))

but when alphac = pi/2 rad
if lon = lonc, u = 0
else u = (A atan((S cos(gammao) + V sin(gammao)) / cos(B (lon - lonO))) / B) - (ABS(uc) . SIGN(latc) . SIGN(lonc – lon))

The rectified skew co-ordinates are then derived from:
E = v cos(gammac) + u sin(gammac) + Ec
N = u cos(gammac) - v sin(gammac) + Nc

Reverse case: Compute (lat,lon) from a given (E,N) :

v’ = (E - Ec) cos(gammac) - (N - Nc) sin(gammac)
u’ = (N - Nc) cos(gammac) + (E - Ec) sin(gammac) + (ABS(uc) . SIGN(latc))

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(gammac) + S’ sin(gammac)) / T’
t’ = (H / ((1 + U’) / (1 - U’))^0.5)^(1 / B)

chi = pi / 2 - 2 atan(t’)

lat = chi + sin(2chi).( e^2 / 2 + 5*e^4 / 24 + e^6 / 12 + 13*e^8 / 360) + sin(4*chi).( 7*e^4 /48 + 29*e^6 / 240 + 811*e8 / 11520) + sin(6chi).( 7*e^6 / 120 + 81*e8 / 1120) + sin(8chi).(4279 e^8 / 161280)

lon= lonO - atan2 ((S’ cos(gammao) , V’ sin(gammao)) / cos(B*u’ / A)) / B

Example:

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 = 0.081472981and e^2 = 0.006637847

Latitude Projection Centre fc = 4°00'00"N = 0.069813170 rad
Longitude Projection Centre lc = 115°00'00"E = 2.007128640 rad
Azimuth of central line ac = 53°18'56.9537" = 0.930536611 rad
Rectified to skew gc= 53°07'48.3685" = 0.927295218 rad
Scale factor ko= 0.99984
Easting at projection centre Ec = 590476.87 m
Northing at projection centre Nc = 442857.65 m

Forward calculation for:
Latitude lat = 5°23'14.1129"N = 0.094025313 rad
Longitude lon = 115°48'19.8196"E = 2.021187362 rad

B = 1.003303209 F = 1.072121256
A =6376278.686 H = 1.000002991
to = 0.932946976 g0 = 0.927295218
D = 1.002425787 lon0 = 1.914373469
D2 =1.004857458
uc =738096.09 vc =0.00

t =0.910700729 Q =1.098398182
S =0.093990763 T = 1.004407419
V =0.106961709 U = 0.010967247
v =-69702.787 u = 163238.163

Then Easting E = 679245.73 m
Northing N = 596562.78 m

Reverse calculations for same easting and northing first gives :
v’ = -69702.787 u’ = 901334.257
Q’ = 1.011028053
S’ = 0.010967907 T’ = 1.000060146
V’ = 0.141349378 U’ = 0.093578324
t’ = 0.910700729 c = 0.093404829

Then Latitude = 5°23'14.113"N
Longitude = 115°48'19.820"E

Method
Parameters:

Parameter Name	Parameter Code	Sign reversal	Parameter Description
Latitude of projection centre	8811	No	For an oblique projection, this is the latitude of the point at which the azimuth of the central line is defined.
Longitude of projection centre	8812	No	For an oblique projection, this is the longitude of the point at which the azimuth of the central line is defined.
Azimuth at projection centre	8813	No	The azimuthal direction (north zero, east of north being positive) of the great circle which is the centre line of an oblique projection. The azimuth is given at the projection centre.
Angle from Rectified to Skew Grid	8814	No	The angle at the natural origin of an oblique projection through which the natural coordinate reference system is rotated to make the projection north axis parallel with true north.
Scale factor at projection centre	8815	No	The factor by which the map grid is reduced or enlarged during the projection process, defined by its value at the projection center.
Easting at projection centre	8816	No	The easting value assigned to the projection centre.
Northing at projection centre	8817	No	The northing value assigned to the projection centre.

Meta Data

Information Source:

EPSG guidance note #7-2, http://www.epsg.org

Data Source:

EPSG

Revision Date:

August 29, 2018

Change ID:

[1999.811] [2004.43] [2004.6] [2007.044] [2010.058] [2017.024]

Alias:

Alias	Naming System	Remarks
Rectified Skew Orthomorphic (RSO)	EPSG alias	This same name is also used for variant A.
Oblique Mercator	EPSG alias	This was the method name used prior to October 2010.

Wildcard	Example
% or *	Any wildcard characters
? or _	One wildcard character
[ - ]	[ - ] Range e.g. N[a-c] will return all geodetic objects that contains a string starting with the letter 'N' or 'n' followed by any of the letters contained within the range specified within the square bracket, e.g. letters a, b and c.
\|	Or e.g. E[a\|e]M will return all geodetic objects that contains a string starting with the letter E(e), ends with the letter M(m) and also contain either the letter a or letter e.