EPSG Dataset :: v12.054

Laborde Oblique Mercator

Coordinate Operation Method Details [VALID]

Name:

Laborde Oblique Mercator

Code:

9813

Operation is Reversible:

Yes

Formula:

Note : these formulas have been transcribed from IGN Document NT/G 74. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

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

B = {1+[e^2 cos^4(phiC)]/(1– e^2)}^0.5
phiS = asin[sin(phiC) / B]
R = a kC {(1–e^2)^0.5 / [1–e^2 sin^2(phiC)]}
C = ln[tan(pi/4+phiS /2)] – B. ln{tan(pi/4+phiC /2) ([1 – e sin(phiC)]/[1+e sin(phiC)])^(e/2)}

Forward case: To compute (E,N) from a given (lat,lon)
L = B.(lon–lonC)
q = C + B . ln{tan(pi/4+lat/2) ([1–e sin(lat)] / [1+e sin(lat)])^(e/2)}
P = 2.atan[e ^q] – pi/2 where e is the base of natural logarithms
U = cos(P).cos(L).cos(phiS) + sin(P).sin(phiS)
V = cos(P).cos(L).sin(phiS) - sin(P).cos(phiS)
W = cos(P).sin(L)
d = (U^2+V^2)^0.5
if d <> 0 then L' = 2.atan(V/(U+d)) and P' = atan(W/d)
if d = 0 then L' = 0 and P' = sign(W).pi/2
H = –L' + i.ln(tan(pi/4+P'/2)) where i^2 = –1
G = (1 – cos(2.alphaC) + i.sin(2.alphaC))/12
E = FE + R . IMAGINARY(H + G.H^3)
N = FN + R . REAL(H + G.H^3)

Reverse case: To compute (lat, lon) from a given (E,N):
G = (1–cos(2.alphaC) + i.sin(2.alphaC))/12 where i^2 = –1
To solve for Latitude and Longitude, a re-iterative solution is required, where the first two elements are
H0 = (N–FN)/R + i.(E–FE)/R ie k = 0
H1 = H0/(H0 + G.H0^3), i.e. k = 1,
and in subsequent reiterations, k increments by 1
Hk+1 = (H0+2.G.Hk^3)/(3.G.Hk^2+1)
Re-iterate until ABSOLUTE(REAL([H0-Hk-G.Hk^3)])) < 1E-11

L' = –1.REAL(Hk)
P' = 2.atan{ e ^[IMAGINARY(Hk)]} – pi/2 where e is the base of natural logarithms.
U' = cos(P').cos(L').cos(phiS) + cos(P').sin(L').sin(phiS)
V' = sin(P')
W' = cos(P').cos(L').sin(phiS) – cos(P').sin(L').cos(phiS)
d = (U'^2+ V'^2)^0.5
if d <> 0 then L = 2 atan[V'/( U'+d)] and P = atan(W'/d)
if d = 0 then L = 0 and P = SIGN(W') . pi/2
lon = lonC + (L/B)

q' = {ln[tan(pi/4+P/2)] – C}/B
The final solution for latitude requires a second re-iterative process, where the first element is
lat'(0) = 2.atan(e ^q') – pi/2 where e is the base of natural logarithms.
And the subsequent elements are
lat'(k) = 2.atan{({1+e.sin[lat(k-1)]} / {1–e.sin[lat(k-1)]})^(e/2).e ^q'} – pi/2 for K =1 ?
Iterate until ABSOLUTE(lat(k)-lat(k-1)) < 1E-11
lat = lat(k)

Example:

See information source.

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.
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.
False easting	8806	No	Since the natural origin may be at or near the centre of the projection and under normal coordinate circumstances would thus give rise to negative coordinates over parts of the mapped area, this origin is usually given false coordinates which are large enough to avoid this inconvenience. The False Easting, FE, is the value assigned to the abscissa (east or west) axis of the projection grid at the natural origin.
False northing	8807	No	Since the natural origin may be at or near the centre of the projection and under normal coordinate circumstances would thus give rise to negative coordinates over parts of the mapped area, this origin is usually given false coordinates which are large enough to avoid this inconvenience. The False Northing, FN, is the value assigned to the ordinate (north or south) axis of the projection grid at the natural origin.

Meta Data

Information Source:

"La nouvelle projection du Service Geographique de Madagascar"; J. Laborde; 1928. Also IGN Paris technical note NT/G 74.

Data Source:

EPSG

Revision Date:

November 2, 2010

Change ID:

[1997.613] [2006.96] [2007.04] [2010.058]

Alias:

Alias	Naming System	Remarks
Laborde Madagascar	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.