Mercator (2SP)

Projection Parameters

Notes

The difference between 1SP and 2SP formulations is that the initial parameters are different: in 1SP you start from the longitude of natural origin and the scale factor, while in 2SP you start from the longitude of natural origin and the latitude of one of the standard parallels (the other is symmetrical in relation to the equator). So, the formulas of 2SP are just those for 1SP after calculating the scale factor from the absolute value of the latitude of the standard parallels and the "e" parameter of the datum.

PROJ.4 Organization

  +proj=merc  +lat_ts=Latitude of first standard parallel
              +lon_0=Longitude of natural origin
              +x_0=False Easting
              +y_0=False Northing

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 
above.
 
	* =  ((E - FE)/ak0)  + *0","For Projected Coordinate System Pulkovo 1942 / Mercator Caspian Sea

Parameters:
Ellipsoid  Krassowski 1940   a = 6378245.00m   1/f = 298.300
then e = 0.08181333 and e^2 = 0.00669342

Latitude first SP                             42o00'00""N = 0.73303829 rad
Longitude Natural Origin                51o00'00""E = 0.89011792 rad
False Eastings FE                          0.00 m
False Northings (at equator) FN     0.00 m
  
then natural origin at latitude of 0oN has scale factor k0=  0.74426089

Forward calculation for: 
Latitude        53o00'00.00""N = 0.9250245 rad
Longitude     53o00'00.00""E  = 0.9250245 rad

gives Easting E   =     165704.29 m 
          Northing N =   5171848.07 m

Reverse calculation for same easting and northing first gives :
t = 0.33639129    chi = 0.92179596

Then Latitude     =  53o00'00.000""N
          Longitude  =  53o00'00.000""E"