EPSG Dataset :: v12.003

Lambert Conic Conformal (2SP Michigan)

Coordinate Operation Method Details [VALID]

Name:

Lambert Conic Conformal (2SP Michigan)

Code:

1051

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.

To derive the projected Easting and Northing coordinates of a point with geographical coordinates (lat,lon) the formulas for the one standard parallel case are:

E = EF + r sin(theta)
N = NF + rF - r cos(theta)
where
m = cos(lat)/(1 - e^2 sin^2(lat))^0.5 for m1, lat1, and m2, lat2 where lat1 and lat2 are the latitudes of the two standard parallels.
t = tan(pi/4 - lat/2)/[(1 - e sin(lat))/(1 + e sin(lat))]^(e/2) for t1, t2, tF and t using lat1, lat2, latF and lat respectively.
n = (loge(m1) - loge(m2))/(loge(t1) - loge(t2))
F = m1/(n t1^n)
r = a K F t^n for rF and r, where rF is the radius of the parallel of latitude of the false origin and K is the ellipsoid scaling factor.
theta = n(lon - lon0)

The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:

lat = pi/2 - 2atan{t'[(1 - esin(lat))/(1 + esin(lat))]^(e/2)}
lon = theta'/n +lon0
where
r' = +/-[(E - EF)^2 + {rF - (N - NF)}^2]^0.5 , taking the sign of n
t' = (r'/(aKF))^(1/n)
theta' = atan2 [(E- EF),(rF - (N- NF))] (see GN7-2 implementation notes in preface for atan2 convention)
and n, F, and rF are derived as for the forward calculation.

Note that the formula for lat requires iteration. First calculate t' and then a trial value for lat* using
lat = π/2-2atan(t'). Then use the full equation for lat substituting the trial value into the right hand side of the equation. Thus derive a new value for lat. Iterate the process until lat does not change significantly. The solution should quickly converge, in 3 or 4 iterations.

Example:

For Projected Coordinate System NAD27 / Michigan Central

Parameters:
Ellipsoid Clarke 1866, a = 6378206.400 metres = 20925832.16 US survey feet
1/f = 294.97870
then e = 0.08227185 and e^2 = 0.00676866

First Standard Parallel = 44°11'00"N = 0.771144641 rad
Second Standard Parallel = 45°42'00"N = 0.797615468 rad
Latitude False Origin = 43°19'00"N = 0.756018454 rad
Longitude False Origin = 84°20'00"W = -1.471894336 rad
Easting at false origin = 2000000.00 US survey feet
Northing at false origin = 0.00 US survey feet
Ellipsoid scaling factor = 1.0000382

Forward calculation for:
Latitude = 43°45'00.00"N = 0.763581548 rad
Longitude = 83°10'00.00"W = -1.451532161 rad

first gives :
m1 = 0.718295175 m2 = 0.699629151
t = 0.429057680 tF = 0.433541026
t1 = 0.424588396 t2 = 0.409053868
n = 0.706407410 F = 1.862317735
r = 21436775.51 rF = 21594768.40
theta = 0.014383991

Then Easting X = 2308335.75 US survey feet
Northing Y = 160210.48 US survey feet

Reverse calculation for same easting and northing first gives:
theta' = 0.014383991 r' = 21436775.51
t' = 0.429057680

Then Latitude = 43°45'00.000"N
Longitude = 83°10'00.000"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.
Ellipsoid scaling factor	1038	No	Ratio by which the ellipsoid is enlarged so that survey observations are reduced to a surface above the ellipsoid surface.

Meta Data

Information Source:

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

Data Source:

EPSG

Revision Date:

August 29, 2018

Change ID:

[2013.02] [2017.024]

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.