Modified Azimuthal Equidistant
GML Report Print View

Modified Azimuthal Equidistant Open
Coordinate Operation Method Details [VALID]
Name: Modified Azimuthal Equidistant
Code: 9832
Operation is Reversible: Yes
Formula: 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.

First calculate a constant for the projection:
nu_O = a /(1 – e^2 sin^2(latO))^0.5

Then the forward conversion from latitude and longitude is given by:
nu = a /(1 – e^2 sin^2(lat))^0.5
psi = atan [(1 – e^2) tan(lat) + e^2 * nu_O * sin(latO) / (nu * cos(lat))]
alpha = atan2{sin (lon – lonO) , [cos(latO) * tan(psi) – sin(latO) * cos (lon – lonO)]} (see implementation notes in GN7-2 preface for atan2 convention)
G = e sin(latO) / (1 – e^2)^0.5
H = e cos(latO) * cos(alpha) / (1 – e^2)^0.5
Then
if sin(alpha)) = 0, s = asin (cos(latO) * sin(psi) – sin(latO) * cos(psi)) * SIGN(cos(alpha))
else s = asin [sin (lon – lonO) * cos(psi) / sin(alpha))

and in either case
c = nu_O * s {[1 – s^2 * H^2 (1 – H^2) /6] + [(s^3/8)GH(1-2H^2)] + (s^4/120)[H^2(4-7H^2) – 3G^2(1-7H^2)] – [(s^5/48)GH]}

Then
E = FE + * sin(alpha)
N = FN + * cos(alpha)

For the reverse conversion from easting and northing to latitude and longitude:
c' = [(E FE)^2 + (N – FN)^2]^0.5
alpha' = atan2 [(E – FE) , (N – FN)]
A = e^2 * cos^2(latO) * cos^2(alpha') / (1 – e^2)
B 3e^2 * (1-A) * sin(latO) * cos(latO) * cos(alpha') / (1 – e^2)
D = c'nu_O
J = D – [A (1 + AD^3 / 6] – [B (1 + 3A) D^4 / 24]
K = 1 – (* J^2 / 2) – (B *J^3 / 6)
psi' = asin (sin(latO) cos(J) + cos(latO) sin(J) cos(alpha'))

Then
lat = atan [(1 – e^2 * K sin(latO) / sin(psi')) * tan(psi') / (1 – e^2)]
lon = lonO + asin (sin(alpha') * sin(J) / cos(psi'))
Example: See information source or EPSG Guidance Note 7.
Method
Parameters:
Parameter Name Parameter Code Sign reversal Parameter Description
Latitude of natural origin 8801 No 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).
Longitude of natural origin 8802 No 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)".
False easting 8806 No 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.
False northing 8807 No 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.