Oblique Stereographic

Projection Parameters

Notes

PROJ.4 Organization

  +proj=sterea +lat_0=Latitude of natural origin 
               +lon_0=Longitude of natural origin
               +k_0=Scale factor at natural origin 
               +x_0=False Easting
               +y_0=False Northing

EPSG Notes

The Stereographic projection may be imagined to be a projection of the earth's surface onto a plane in contact with the earth at a single tangent point from the opposite end of the diameter through that tangent point.

This projection is best known in its polar form and is frequently used for mapping polar areas where it complements the Universal Transverse Mercator used for lower latitudes. Its spherical form has also been widely used by the US Geological Survey for planetary mapping and the mapping at small scale of continental hydrocarbon provinces. In its transverse or oblique ellipsoidal forms it is useful for mapping limited areas centred on the point where the plane of the projection is regarded as tangential to the ellipsoid., e.g. the Netherlands. The tangent point is the origin of the projected coordinate system and the meridian through it is regarded as the central meridian. In order to reduce the scale error at the extremities of the projection area it is usual to introduce a scale factor of less than unity at the origin such that a unitary scale factor applies on a near circle centred at the origin and some distance from it.

The coordinate transformation from geographical to projected coordinates is executed via the distance and azimuth of the point from the centre point or origin. For a sphere the formulas are relatively simple. For the ellipsoid the parameters defining the conformal sphere at the tangent point as origin are first derived. The conformal latitudes and longitudes are substituted for the geodetic latitudes and longitudes of the spherical formulas for the origin and the point .

Oblique and Equatorial Stereographic Formula

* Given the geodetic origin of the projection at the tangent point (*0, *0), the parameters defining the conformal sphere are:

	R= *( *0,*0)
     	n=  *[(1+e2 cos4*0)/(1-e2)]
	c=  (n+sin*0) (1-sin*0)/[(n-sin*0) (1+sin(*0)]

where:	sin*0 = (w1-1)/(w1+1)
	w1 = (S1.S2e)n
	S1 = (1+sin*0)/(1-sin*0)
	S2 = (1-e sin*0)/(1+e sin*0)

	*0 = sin-1[(w-1)/(w+1)]

	where S1 and S2 are as above and  w = c (S1S2e)n

 	*0  = *0

	* = sin-1[(w-1)/(w+1)]

where 	w = c (S1S2e)n
	S1 = (1+sin*)/(1-sin*)
	S2 = (1-e.sin*)/(1+e.sin*)

and 	* = n( * - *0 ) + *0 
 
Then	B = [1+sin* sin*0 + cos*cos*0cos(* -*0 )]

and	N = FN + 2 R k0 [sin* cos*0 - cos*sin*0cos(* -*0 )] / B

	E = FE + 2 R k0 cos* sin(* -*0 ) / B

The parameters of the conformal sphere and conformal latitude and longitude at the origin are computed as above. Then for any point with Stereographic grid coordinates (E,N) :

	* = *0 + 2 tan-1[{(N-FN)-(E-FE) tan (j/2)} / (2 Rk0)]

	* = j + 2 i + *0

where	g = 2 Rk0 tan (*/4 - *0/ 2 )
	h = 4 Rk0 tan *0 + g
	i = tan-1 [(E-FE) / {h+(N-FN)}]
	j = tan-1 [(E-FE) / (g-(N-FN)] - i

Geodetic longitude 	* = (* -*0 ) / n +  *0

Isometric latitude	* = 0.5 ln [(1+ sin*) / { c (1-  sin*)}] / n

First approximation 	*1 = 2 tan-1 e*  - * / 2  where e=base of natural logarithms.

			*i = isometric latitude at *i

where			*i= ln[{tan(*i/2+* / 4}  {(1-e sin*i)/(1+e sin*i)}e/2]
 
Then iterate		*i+1 = *i - ( *i - * ) cos *i ( 1 -e2 sin2*i) / (1 - e2)

until the change in 	*   is sufficiently small.

If the projection is the equatorial case, *0 and *0 will be zero degrees and the formulas are simplified as a result,but the above formulae remain valid.

For the polar version, *0 and *0 will be 90 degrees and the formulae become indeterminate.See below for formulae for the polar case.

For Stereographic projections centred on points in the southern hemisphere, including the south Polar Stereographic, the signs of E, N, *0, *, must be reversed to be used in the equations and * will be negative anyway as a southerly latitude. For Projected Coordinate System RD / Netherlands New

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

Latitude Natural Origin      52o09'22.178""N  = 0.910296727 rad
Longitude Natural Origin     5o23'15.500""E  =  0.094032038 rad
Scale factor k0                 0.9999079
False Eastings FE             155000.00 m
False Northings FN           463000.00 m

Forward calculation for: 

Latitude    53oN = 0.925024504 rad
Longitude   6oE = 0.104719755 rad

first gives the conformal sphere constants:

rho0 = 6374588.71    nu0 = 6390710.613
R = 6382644.571    n = 1.000475857    c  = 1.007576465

where S1 = 8.509582274  S2 = 0.878790173  w1 = 8.428769183
sin chi0 = 0.787883237

w   = 8.492629457   chi0 = 0.909684757      D0 = d0 

for the point  chi  = 0.924394997    D = 0.104724841

hence B = 1.999870665    N = 557057.739    E = 196105.283

reverse calculation for the same Easting and Northing first gives:

g = 4379954.188    h = 37197327.96   i = 0.001102255   j = 0.008488122

then  D = 0.10472467  Longitude = 0.104719584 rad =  6 deg E

chi  = 0.924394767    psi = 1.089495123
phi1 = 0.921804948       psi1 = 1.084170164
phi2 = 0.925031162       psi2 = 1.089506925
phi3 = 0.925024504       psi3 = 1.089495505
phi4 = 0.925024504

Then Latitude      = 53o00'00.000""N
          Longitude   =   6o00'00.000