EPSG Dataset :: v12.003

Equidistant Conic

Coordinate Operation Method Details [VALID]

Name:

Equidistant Conic

Code:

1119

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.

The equidistant conic projection is neither conformal nor equal area, but a compromise between the two. It is most frequently used for very small scale maps in atlases based on a sphere. EPSG carries only the ellipsoidal development in which the meridional arc distance between the parallels of latitude reflects that on the ellipsoid.

The formulas to convert geodetic latitude and longitude (φ, λ) to easting (E) and northing (N) are:

Easting, E = EF + r sin θ
Northing, N = NF + rF – r cos θ

where
M = a [(1 – e^2/4 – 3e^4/64 – 5e^6/256 - …)φ – (3e^2/8 + 3e^4/32 + 45e^6/1024 + …)sin 2φ + (15e^4/256 + 45e^6/1024 + …)sin 4φ – (35e^6/3072 + …)sin 6φ + …]
with φ in radians, and with MF, M1 and M2 for φF (latitude of false origin), φ1 (latitude of 1st standard parallel) and φ2 (latitude of 2nd standard parallel) derived in the same way;

m = cos φ / (1 – e^2 sin2 φ)^0.5 with φ in radians, and m1 and m2 for φ1 and φ2 derived in the same way;
n = a (m1 – m2) / (M2 – M1)
G = (m1 / n) + (M1 / a)
rF = a G – MF

r = a G – M
θ = n (λ – λF)

For the reverse calculation:

G and rF are constants for the projection calculated as for the forward calculation above. Then:

r' = ± {(E – EF)^2 + [rF – (N – NF)]^2 }^0.5, taking the sign of n

If n is positive (northern hemisphere case of the projection),
θ' = atan2 {(E – EF) , [rF – (N – NF)]}
but if n is negative (southern hemisphere case of the projection), the signs of both arguments of the atan2 function must be reversed:
θ' = atan2 {– (E – EF), – [rF – (N – NF)]}

M = a G – r'
μ = M / [a (1 – e^2/4 – 3e^4/64 – 5e^6/256 - …)]

Latitude, φ = μ + (3e1/2 – 27e1^3/32 + …)sin 2μ + (21e1^2/16 – 55e1^4/32 + …)sin 4μ + (151e1^3/96 - …)sin 6μ + (1097e1^4/512 - …)sin 8μ
with μ in radians

where e1 = [1 – (1 – e^2)^0.5] / [1 + (1 – e^2)^0.5]

Longitude, λ = λF + (θ' / n)

Example:

Example (Northern Hemisphere):

Parameters:
Ellipsoid: Clarke 1866 a = 6378206.4 metres and b = 6356583.8 metres
from which: 1/f =294.9786982, e = 0.08227185, e2 =0.00676866 and e1 = 0.001697916

Latitude of false origin = 23°00'00"N
Longitude of false origin = 96°00'00"W
Latitude of 1st standard parallel = 29°30'00"N
Latitude of 2nd standard parallel = 45°30'00"N
Easting at false origin = 0.00 metre
Northing at false origin = 0.00 metre

Forward calculation for:
Latitude = 35°00'00.00"N
Longitude = 75°00'00.00"W

first gives :
φF = 0.401425728 rad
λF = -1.675516082 rad
φ1 = 0.514872129 rad
φ2 = 0.794124810 rad
M1 = 3264511.20 m
m1 = 0.871070821
M2 = 5040295.01 m
m2 = 0.702119143
MF = 2544389.75 m
n = 0.606835507
G = 1.947254290
rF = 9875600.03 m

Then :
φ = 0.610865238 rad
λ = -1.308996939 rad
M = 3874395.26 m
m = 0.820065623
r = 8545594.52 m
θ = 0.222416830 rad

Then
Easting = 1885051.86 m
Northing = 1540507.64 m

Reverse calculation for same easting and northing gives:
r' = 8545594.52 m
θ' = 0.222416830 rad
M = 3874395.26
μ = 0.608473702

Then
Latitude = 35°00'00.000"N
Longitude = 75°00'00.000"W

Method
Parameters:

Parameter Name	Parameter Code	Sign reversal	Parameter Description
Latitude of false origin	8821	No	The latitude of the point which is not the natural origin and at which grid coordinate values false easting and false northing are defined.
Longitude of false origin	8822	No	The longitude of the point which is not the natural origin and at which grid coordinate values false easting and false northing are defined.
Latitude of 1st standard parallel	8823	No	For a conic projection with two standard parallels, this is the latitude of one of the parallels of intersection of the cone with the ellipsoid. It is normally but not necessarily that nearest to the pole. Scale is true along this parallel.
Latitude of 2nd standard parallel	8824	No	For a conic projection with two standard parallels, this is the latitude of one of the parallels at which the cone intersects with the ellipsoid. It is normally but not necessarily that nearest to the equator. Scale is true along this parallel.
Easting at false origin	8826	No	The easting value assigned to the false origin.
Northing at false origin	8827	No	The northing value assigned to the false origin.

Meta Data

Remarks:

This is the ellipsoidal form of the projection.

Information Source:

US Geological Survey Professional Paper 1395; "Map Projections - A Working Manual"; J. Snyder.

Data Source:

EPSG

Revision Date:

June 29, 2023

Change ID:

[2023.004]

Wildcard	Example
% or *	Any wildcard characters
? or _	One wildcard character
[ - ]	[ - ] Range e.g. N[a-c] will return all geodetic objects that contains a string starting with the letter 'N' or 'n' followed by any of the letters contained within the range specified within the square bracket, e.g. letters a, b and c.
\|	Or e.g. E[a\|e]M will return all geodetic objects that contains a string starting with the letter E(e), ends with the letter M(m) and also contain either the letter a or letter e.