EPSG guidance note #7-2, http://www.epsg.org
2018-08-29
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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 52°09'22.178"N = 0.910296727 rad
Longitude Natural Origin 5°23'15.500"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 53°N = 0.925024504 rad
Longitude 6°E = 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 = 53°00'00.000"N
Longitude = 6°00'00.000"E
urn:ogc:def:method:EPSG::9809
Oblique Stereographic
This is not the same as the projection method of the same name in USGS Professional Paper no. 1395, "Map Projections - A Working Manual" by John P. Snyder.
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.
Given the geodetic origin of the projection at the tangent point (lat0, lon0), the parameters defining the conformal sphere are:
R= sqrt( rho0 * nu0)
n= {1 + [e^2 * cos^4(latC) / (1 - e^2)]}^0.5
c= [(n+sin(lat0)) (1-sin(chi0))]/[(n-sin(lat0)) (1+sin(chi0))]
where:
sin(chi0) = (w1-1)/(w1+1)
w1 = (S1.(S2)^e)^n
S1 = (1+sin(lat0))/(1-sin(lat0))
S2 = (1-e sin(lat0))/(1+e sin(lat0))
The conformal latitude and longitude (chi0,lambda0) of the origin are then computed from :
chi0 = asin[(w2-1)/(w2+1)]
where S1 and S2 are as above and w2 = c (S1(S2)^e)^n
lambda0 = lon0
For any point with geodetic coordinates (lat, lon) the equivalent conformal latitude and longitude (chi, lambda) are computed from
lambda = n(lon-lambda0) + lambda0
chi = asin[(w-1)/(w+1)]
where w = c (Sa (Sb)^e)^n
Sa = (1+sin(lat))/(1-sin(lat))
Sb = (1-e.sin(lat))/(1+e.sin(lat))
Then B = [1+sin(chi) sin(chi0) + cos(chi) cos(chi0) cos(lambda-lambda0)]
N = FN + 2 R k0 [sin(chi) cos(chi0) - cos(chi) sin(chi0) cos(lambda-lambda0)] / B
E = FE + 2 R k0 cos(chi) sin(lambda-lambda0) / B
The reverse formulae to compute the geodetic coordinates from the grid coordinates involves computing the conformal values, then the isometric latitude and finally the geodetic values.
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) :
chi = chi0 + 2 atan[{(N-FN)-(E-FE) tan (j/2)} / (2 R k0)]
lambda = j + 2 i + lambda0
where g = 2 R k0 tan(pi/4 - chi0/2)
h = 4 R k0 tan(chi0) + g
i = atan2[(E-FE) , {h+(N-FN)}]
j = atan2[(E-FE) , (g-(N-FN)] - i
(see GN7-2 implementation notes in preface for atan2 convention)
Geodetic longitude lon = (lambda-lambda0 ) / n + lambda0
Isometric latitude psi = 0.5 ln [(1+ sin(chi)) / { c (1- sin(chi))}] / n
First approximation lat1 = 2 atan(e^psi) - pi/2 where e=base of natural logarithms.
psii = isometric latitude at lati
where psii= ln[{tan(lati/2 + pi/4} {(1-e sin(lati))/(1+e sin(lati))}^(e/2)]
Then iterate lat(i+1) = lati - ( psii - psi ) cos(lati) (1 -e^2 sin^2(lati)) / (1 - e^2)
until the change in lat is sufficiently small.
For Oblique Stereographic projections centred on points in the southern hemisphere, the signs of E, N, lon0, lon, must be reversed to be used in the equations and lat will be negative anyway as a southerly latitude.
An alternative approach is given by Snyder, where, instead of defining a single conformal sphere at the origin point, the conformal latitude at each point on the ellipsoid is computed. The conformal longitude is then always equivalent to the geodetic longitude. This approach is a valid alternative to the above, but gives slightly different results away from the origin point. It is therefore considered by EPSG to be a different coordinate operation method to that described above.