EPSG guidance note #7-2, http://www.epsg.org
2012-02-13
true
false
true
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 of natural origin (LatO) = 49°00'00"N = 0.85521133 rad
Longitude of natural origin (LonO) = 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
Constants of the projection:
n = 0.00167322
B = 6366914.609
h1 = 0.0008347452
h2 = 0.0000007554
h3 = 1.18487E-09
h4 = 2.40864E-12
QO = 0.9787671618
?O0 = 0.8518980373
?O1 = 0.0008273732
?O2 = -0.0000001986
?O3 = -1.0918E-09
?O4 = 1.2218E-12
Mo = 5429228.602
Q = 1.0191767215
? = 0.8781064142
?0 = 0.0278629616
?0 = 0.8785743280
?1 = -0.0000086229
?1 = 0.0008215669
?2 = -0.0000000786
?2 = -0.0000002768
?3 = 1.05551E-10
?3 = -1.01855E-09
?4 = 3.97791E-13
?4 = 1.67447E-12
? = 0.0278542603
? = 0.8793956171
Then
Easting E = 577274.99 metres
Northing N = 69740.50 metres
Reverse calculation for same easting and northing first gives:
h1' = 0.0008347455
h2' = 0.0000000586
h3' = 1.65563E-10
h4' = 2.13692E-13
Then
?' = 0.87939562
?' = 0.0278542603
?1' = 0.0008213109
?1' = -0.0000086953
?2' = -0.0000000217
?2' = -0.0000000061
?3' = -1.41881E-10
?3' = 1.486E-11
?4' = 1.49609E-13
?4' = 3.50657E-14
?0' = 0.8785743280
?0' = 0.0278629616
?' = 0.8781064142
Q' = 1.0191767215
Q" 1st iteration = 1.0243166838
Q" 2nd iteration = 1.0243306667
Q" 3rd iteration = 1.0243307046
Q" 4th iteration = 1.0243307047
Then
Latitude (Lat) = 50°30'00.000"N
Longitude (Lon) = 00°30'00.000"E
http://www.opengis.net/def/method/EPSG/0/9807
Transverse Mercator
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.
For the calculation of easting and northing from latitude and longitude, first calculate constants for the projection:
n = f / (2-f)
B = [a/(1+n)] (1 + n^2/4 + n^4/64)
h1 = n/2 ? (2/3)n^2 + (5/16)n^3 + (41/180)n^4
h2 = (13/48)n^2 ? (3/5)n^3 + (557/1440)n^4
h3 = (61/240)n^3 ? (103/140)n^4
h4 = (49561/161280)n^4
Then the meridional arc distance from equator to the projection origin (Mo) is computed from:
If LatO = 0 then Mo = 0
else if LatO ? 90°N ? ?/2 radians
Mo = B (?/2)
else if LatO ? 90°S ? -?/2 radians
Mo = B (-?/2)
else
Qo = asinh(tan LatO) ? [e atanh(e sin LatO)]
?o = atan(sinh Qo)
?O0 = asin (sin ?o)
Note: The previous two steps are taken from the generic calculation flow given below for latitude Lat, but here for LatO may be simplified to ?O0 = ?o = atan(sinh Qo).
?O1 = h1 sin(2?Oo)
?O2 = h2 sin(4?Oo)
?O3 = h3 sin(6?Oo)
?O4 = h4 sin(8?Oo)
?O = ?O0+ ?O1+ ?O2+ ?O3+ ?O4
Mo = B ?O
end
Note: if the projection grid origin is very close to the pole (within 2" or 50m), the tangent function in the equation for Qo is unstable and may fail. Mo may instead be calculated as:
Mo = a[(1 ? e^2/4 ? 3e^4/64 ? 5e^6/256 ?....)LatO ? (3e^2/8 + 3e^4/32 + 45e^6/1024+....)sin2LatO
+ (15e^4/256 + 45e^6/1024 +.....)sin4LatO ? (35e^6/3072 + ....)sin6LatO + .....]
with LatO in radians.
Then
Q = asinh(tan Lat) ? [e atanh(e sin Lat)]
? = atan(sinh Q)
?0 = atanh [cos ? sin(Lon ? LonO)]
?0 = asin (sin ? cosh ?0)
?1 = h1 sin(2?0) cosh(2?0)
?1 = h1 cos(2?0) sinh(2?0)
?2 = h2 sin(4?0) cosh(4?0)
?2 = h2 cos(4?0) sinh(4?0)
?3 = h3 sin(6?0) cosh(6?0)
?3 = h3 cos(6?0) sinh(6?0)
?4 = h4 sin(8?0) cosh(8?0)
?4 = h4 cos(8?0) sinh(8?0)
? = ?0 + ?1 + ?2 + ?3 + ?4
? = ?0 + ?1 + ?2 + ?3 + ?4
and
Easting, E = FE + ko B ?
Northing, N = FN + ko (B ? ? Mo)
For the reverse formulas to convert Easting and Northing projected coordinates to latitude and longitude first calculate constants of the projection where n is as for the forward conversion, as are B and Mo:
h1' = n/2 ? (2/3)n^2 + (37/96)n^3 ? (1/360)n^4
h2' = (1/48)n^2 + (1/15)n^3 ? (437/1440)n^4
h3' = (17/480)n^3 ? (37/840)n^4
h4' = (4397/161280)n^4
Then
?' = (E ? FE) / (B ko)
?' = [(N ? FN) + ko Mo] / (B ko)
?1' = h1' sin(2?') cosh(2?')
?1' = h1' cos(2?') sinh(2?')
?2' = h2' sin(4?') cosh(4?')
?2' = h2' cos(4?') sinh(4?')
?3' = h3' sin(6?') cosh(6?')
?3' = h3' cos(6?') sinh(6?')
?4' = h4' sin(8?') cosh(8?')
?4' = h4' cos(8?') sinh(8?')
?0' = ?' ? (?1' + ?2' + ?3' + ?4')
?0' = ?' ? (?1' + ?2' + ?3' + ?4')
?' = asin(sin ?0' / cosh ?0')
Q' = asinh(tan ?')
Q" = Q' + [e atanh(e tanh Q')] = Q' + [e atanh(e tanh Q")] which should be iterated until the change in Q" is insignificant. Then
Lat = atan(sinh Q")
Lon = LonO + asin(tanh(?0') / cos ?')
EPSG guidance note number 7.
1999-09-09
false
The latitude of the point from which the values of both the geographical coordinates on the ellipsoid and the grid coordinates on the projection are deemed to increment or decrement for computational purposes. Alternatively it may be considered as the latitude of the point which in the absence of application of false coordinates has grid coordinates of (0,0).
http://www.opengis.net/def/parameter/EPSG/0/8801
Latitude of natural origin
Abbeviated as "CM".
Abbreviation for "Central Meridian".
EPSG guidance note number 7.
2002-06-22
false
The longitude of the point from which the values of both the geographical coordinates on the ellipsoid and the grid coordinates on the projection are deemed to increment or decrement for computational purposes. Alternatively it may be considered as the longitude of the point which in the absence of application of false coordinates has grid coordinates of (0,0). Sometimes known as "central meridian (CM)".
http://www.opengis.net/def/parameter/EPSG/0/8802
Longitude of natural origin
EPSG guidance note number 7.
1999-09-09
false
The factor by which the map grid is reduced or enlarged during the projection process, defined by its value at the natural origin.
http://www.opengis.net/def/parameter/EPSG/0/8805
Scale factor at natural origin
This alias applies only in the case of projection methods which have an axis positive west, e.g. Transverse Mercator (South Orientated).
EPSG guidance note number 7.
2002-07-31
false
Since the natural origin may be at or near the centre of the projection and under normal coordinate circumstances would thus give rise to negative coordinates over parts of the mapped area, this origin is usually given false coordinates which are large enough to avoid this inconvenience. The False Easting, FE, is the value assigned to the abscissa (east or west) axis of the projection grid at the natural origin.
http://www.opengis.net/def/parameter/EPSG/0/8806
False easting
This alias applies only in the case of projection methods which have an axis positive south, e.g. Transverse Mercator (South Orientated).
EPSG guidance note number 7.
2002-07-31
false
Since the natural origin may be at or near the centre of the projection and under normal coordinate circumstances would thus give rise to negative coordinates over parts of the mapped area, this origin is usually given false coordinates which are large enough to avoid this inconvenience. The False Northing, FN, is the value assigned to the ordinate (north or south) axis of the projection grid at the natural origin.
http://www.opengis.net/def/parameter/EPSG/0/8807
False northing