Mercator (1SP)

Name Mercator
EPSG Code 9804
GeoTIFF Code CT_Mercator (7)
OGC WKT Mercator_1SP
Supported By EPSG, GeoTIFF, PROJ.4, OGC WKT

Projection Parameters

Name EPSG # GeoTIFF ID OGC WKT Units Notes
Latitude of natural origin 1 NatOriginLat latitude_of_origin Angular
Longitude of natural origin 2 NatOriginLong central_meridian Angular
Scale factor at natural origin 5 ScaleAtNatOrigin scale_factor Unitless
False Easting 6 FalseEasting false_easting Linear
False Northing 7 FalseNorthing false_northing Linear


I believe the formula (EPSG) notes are exactly the same for Mercator (1SP) and Mercator (2SP). There is only one projection in GeoTIFF for straight Mercator.

PROJ.4 Organization

PROJ.4 does not support a latitude of natural origin other than the equator.
  +proj=merc  +lon_0=Longitude of natural origin
              +k_0=Scale factor at natural origin 
              +x_0=False Easting
              +y_0=False Northing
A more common formulation for Mercator is to drop the +k_0, and instead to provide a latitude of true scale using the +lat_ts parameter, which is the latitude at which the scale is 1.

EPSG Notes


The Mercator projection is a special case of the Lambert Conic Conformal projection with the equator as the single standard parallel. All other parallels of latitude are straight lines and the meridians are also straight lines at right angles to the equator, equally spaced. It is little used for land mapping purposes but is in universal use for navigation charts and is the basis for the transverse and oblique forms of the Mercator. As well as being conformal, it has the particular property that straight lines drawn on it are lines of constant bearing. Thus navigators may derive their course from the angle the straight course line makes with the meridians.

In the few cases in which the Mercator projection is used for terrestrial applications or land mapping, such as in Indonesia prior to the introduction of the Universal Transverse Mercator, a scale factor may be applied to the projection. This has the same effect as choosing two standard parallels on which the true scale is maintained at equal north and south latitudes either side of the equator.

The formulas to derive projected Easting and Northing coordinates are:

For the two standard parallel case, k0 is first calculated from

k0 = cos*1/(1 - e2sin2*1)1/2 
where *1 is the absolute value of the first standard parallel (i.e. positive).  

Then, for both one and two standard parallel cases, 

	E = FE + ak0(* - *0)              
	N = FN + ak0 logn{tan(*/4 + */2)[(1 - esin*) / (1 + esin*)]e/2 } 
           		where symbols are as listed above and logarithms are natural.
The reverse formulas to derive latitude and longitude from E and N values are:
	* = * + (e2/2 + 5e4/24 + e6/12 + 13e8/360) sin(2*) 
		+ (7e4/48 + 29e6/240 + 811e8/11520) sin(4*)
		+ (7e6/120 +  81e8/1120) sin(6*)  + (4279e8/161280) sin(8*)

where 		* = */2 - 2 arctan t
		t = B (FN-N)/(ak0)
		B = base of the natural logarithm, 2.7182818...
		and  for the 2 SP Case, k0 is calculated as for the forward transformation 
	* =  ((E - FE)/ak0)  + *0","For Projected Coordinate System Makassar / NEIEZ

Ellipsoid   Bessel 1841  a = 6377397.155 m   1/f = 299.15281
then e = 0.08169683

Latitude Natural Origin         00o00'00""N  = 0.0000000 rad
Longitude Natural Origin    110o00'00""E  = 1.91986218 rad
Scale factor ko                  0.997
False Eastings FE              3900000.00 m
False Northings FN              900000.00 m

Forward calculation for: 
Latitude            3o00'00.00""S   = -0.05235988 rad
Longitude     120o00'00.00""E   =  2.09439510 rad
Easting  E   =      5009726.58 m
Northing N  =        569150.82 m

Reverse calculation for same easting and northing first gives :
t    = 1.0534121
chi = -0.0520110

Then Latitude     =   3o00'00.000""S
         Longitude  = 120o00'00.000""E