EPSG Dataset :: v12.054

Orthographic

Coordinate Operation Method Details [VALID]

Name:

Orthographic

Code:

9840

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 Orthographic Projection forward conversion from 2D geographic coordinates latitude and longitude (lat, lon) and the origin on the ellipsoid (latO, lonO) is given by:

E = FE + nu cos(lat) sin (lon – lonO)
N = FN + nu [sin(lat) cos(latO) – cos(lat) sin(latO) cos (lon – lonO)] + e^2 (nuO sin(latO) – nu sin(lat)) cos(latO)

where
nu is the prime vertical radius of curvature at latitude lat; nu = a /(1 – e^2 sin^2(lat))^0.5,
nuO is the prime vertical radius of curvature at latitude of origin latO; nuO = a /(1 – e^2 sin^2(latO)^0.5,
e is the eccentricity of the ellipsoid and e^2 = (a^2 – b^2)/a^2 = 2f – f^2
a and b are the ellipsoidal semi-major and semi-minor axes,
1/f is the inverse flattening, and
the latitude and longitude of the projection origin are latO and lonO.

These formulas are similar to those for the orthographic case of the vertical perspective (method code 9839) except that, for the Orthographic Projection given here, h = 0 and the term (nu + h) reduces to nu. The projection origin is at the topocentric system origin latO, lonO with false origin coordinates FE and FN.

For the reverse formulas for latitude and longitude corresponding to a given Easting (E) and Northing (N), iteration is required as the prime vertical radius (nu) is a function of latitude.

Begin by seeding the iteration with the center of projection (or some better guess):
lat = latO
lon = lonO

Enter the iteration here with the (next) best estimates of lat and lon. Then solve for the radii of curvature in the prime vertical (nu) and meridian (rho):
nu = a / (1 – e^2 sin^2(lat))^0.5
rho = a (1 – e^2) / (1 – e^2 sin^2(lat))^1.5

Compute test values of E and N (E' and N') using the forward equations:
E' = FE + nu cos(lat) sin(lon – lonO)
N' = FN + nu [sin(lat) cos(latO) – cos(lat) sin(latO) cos(lon – lonO)] + e^2 (nuO sin(latO) – nu sin(lat) ) cos(latO)

Partially differentiate the forward equations to solve for the elements of the Jacobian matrix:
J11 = dE/dlat = – rho sin(lat) sin(lon – lonO)
J12 = dE/dlon = nu cos(lat) cos(lon – lonO)
J21 = dN/dlat = rho [cos(lat) cos(latO) + sin(lat) sin(latO) cos(lon – lonO)]
J22 = dN/dlon = nu sin(latO) cos(lat) sin (lon – lonO)

Solve for the determinant of the Jacobian:
D = J11 J22 – J12 J21

Solve the northerly and easterly differences this iteration:
dE = E – E'
dN= N – N'

Adjust the latitude and longitude for the next iteration by inverting the Jacobian and multiplying by the differences:
lat = lat + (J22 dE – J12 dN) / D
lon = lon + (–J21 dE + J11 dN) / D

Return to the entry point with new estimates of latitude and longitude and iterate until the change in lat and lon is not significant.

Example:

See EPSG Guidance Note 7-2.

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.

Meta Data

Remarks:

If the natural origin of the projection is at the topocentric origin, this is a special case of the Vertical Perspective (orthographic case) (method code 9839) in which the ellipsoid height of all mapped points is zero (h = 0).

Information Source:

EPSG guidance note #7-2, http://www.epsg.org

Data Source:

EPSG

Revision Date:

November 1, 2007

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.