NAD83 / Alberta 10-TM (Forest)

For Projected Coordinate System OSGB 1936 / British National Grid

Parameters:
Ellipsoid  Airy 1830  a = 6377563.396 m  1/f = 299.32496
then e'^2 = 0.00671534 and e^2 = 0.00667054

Latitude Natural Origin         49°00'00"N   = 0.85521133 rad
Longitude Natural Origin        2°00'00"W  = -0.03490659 rad
Scale factor ko                     0.9996013                                                                                              False Eastings FE                 400000.00 m
False Northings FN              -100000.00 m

Forward calculation for: 
Latitude       50°30'00.00"N  = 0.88139127 rad
Longitude    00°30'00.00"E  = 0.00872665 rad
A  = 0.02775415       C = 0.00271699
T =  1.47160434       M = 5596050.46
M0 = 5429228.60     nu  = 6390266.03

Then Easting E =        577274.99 m
          Northing N =       69740.50 m

Reverse calculations for same easting and northing first gives :
e1 =    0.00167322      mu1 = 0.87939562
M1 = 5599036.80        nu1 = 6390275.88
lat1  = 0.88185987      D = 0.02775243
rho1 =6372980.21       C1 =  0.00271391
T1 = 1.47441726

Then Latitude       = 50°30'00.000"N
         Longitude    = 00°30'00.000"E

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.

The formulas to derive the projected Easting and Northing coordinates are in the form of a series as follows:

Easting, E =  FE + k0*nu[A + (1 - T + C)A^3/6 + (5 - 18T + T^2 + 72C - 58e'sq)A^5/120]	

Northing, N =  FN + k0{M - M0 + nu*tan(lat)[A^2/2 + (5 - T + 9C + 4C^2)A^4/24 + (61 - 58T + T^2 + 600C - 330e'sq)A^6/720]} 
where T = tan^2(lat)                nu = a /(1 - esq*sin^2(lat))^0.5
C = esq*cos^2(lat)/(1 - esq)
A = (lon - lon0)cos(lat), with lon and lon0 in radians.
M = a[(1 - esq/4 - 3e^4/64 - 5e^6/256 -....)lat - (3esq/8 + 3e^4/32 + 45e^6/1024+....)sin(2*lat) + (15e^4/256 + 45e^6/1024 +.....)sin(4*lat) - (35e^6/3072 + ....)sin(6*lat) + .....]
with lat in radians and M0 for lat0, the latitude of the origin, derived in the same way.

The reverse formulas to convert Easting and Northing projected coordinates to latitude and longitude are:

lat = lat1 - (nu1*tan(lat1)/rho1)[D^2/2 - (5 + 3*T1 + 10*C1 - 4*C1^2 - 9*e'^2)D^4/24 + (61 + 90*T1 + 298*C1 + 45*T1^2 - 252*e'^2 - 3*C1^2)D^6/720]
lon = lon0 + [D - (1 + 2*T1 + C1)D^3/6 + (5 - 2*C1 + 28*T1 - 3*C1^2 + 8*e'^2 + 24*T1^2)D^5/120] / cos(lat1)
where lat1 may be found as for the Cassini projection from:

lat1 = mu1 + ((3*e1)/2 - 27*e1^3/32 +.....)sin(2*mu1) + (21*e1^2/16 -55*e1^4/32 + ....)sin(4*mu1)+ (151*e1^3/96 +.....)sin(6*mu1) + (1097*e1^4/512 - ....)sin(8*mu1) + ......
and where
nu1 = a /(1 - esq*sin^2(lat1))^0.5
rho1 = a(1 - esq)/(1 - esq*sin^2(lat1))^1.5
e1 = [1- (1 - esq)^0.5]/[1 + (1 - esq)^0.5]
mu1 = M1/[a(1 - esq/4 - 3e^4/64 - 5e^6/256 - ....)]
M1 = M0 + (N - FN)/k0
T1 = tan^2(lat1)
C1 = e'^2*cos^2(lat1)
D = (E - FE)/(nu1*k0), with nu1 = nu for lat1

For areas south of the equator the value of latitude lat will be negative and the formulas above, to compute the E and N,  will automatically result in the correct values. Note that the false northings of the origin, if the equator, will need to be large to avoid negative northings and for the UTM projection is in fact 10,000,000m. Alternatively, as in the case of Argentina's Transverse Mercator (Gauss-Kruger) zones, the origin is at the south pole with a northings of zero. However each zone central meridian takes a false easting of 500000m prefixed by an identifying zone number. This ensures that instead of points in 
different zones having the same eastings, every point in the country, irrespective of its projection zone, will have a unique set of projected system coordinates. Strict application of the above formulas, with south latitudes negative, will result in the derivation of the correct Eastings and Northings. 

Similarly, in applying the reverse formulas to determine a latitude south of the equator, a negative sign for lat results from a negative lat1 which in turn results from a negative M1.