EPSG Dataset :: v12.054

Krovak Modified (North Orientated)

Coordinate Operation Method Details [VALID]

Name:

Krovak Modified (North Orientated)

Code:

1043

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.

From the defining parameters the following constants for the projection may be calculated :

A = a(1 - e^2)^0.5 / [1 - e^2 sin^2.(latC)]
B = {1 + [e^2 * cos^4(latC) / (1 - e^2)]}^0.5
gammao = asin[sin(latC) / B]
to = tan(pi/4 + gammao/2).[(1 + e sin(latC)) / (1 - e sin(latC))]^(e.B/2) / [tan(pi/4 + latC/2)]^B
n = sin(latp)
ro = kp.A / tan(latp)

To derive the projected Southing and Westing coordinates of a point with geographical coordinates (lat, lon) the formulas for the Krovak are:

U = 2(atan{to.tan^B(lat/2 + pi/4) / [(1 + e sin(lat)) / (1 - e sin(lat))]^[e.B/2]} - pi/4)
V = B(lonO - lon) where lonO and lon must both be referenced to the same prime meridian.
T = asin[cos(alphaC).sin(U) + sin(alphaC).cos(U). cos(V)]
D = asin[cos(U).sin(V)/cos(T)]
theta = n.D
r = ro.tan^n(pi/4 + latp/2) / tan^n(T/2 + pi/4)
Xp = r.cos(theta)
Yp = r.sin(theta)
Xr = Xp – Xo
Yr = Yp – Yo
dX = C1 + C3.Xr – C4.Yr – 2.C6.Xr.Yr + C5.(Xr^2 – Yr^2) + C7.Xr.(Xr^2 – 3.Yr^2) – C8.Yr.(3.Xr^2 – Yr^2) + 4.C9.Xr.Yr.(Xr^2 – Yr^2) + C10.(Xr^4 + Yr^4 – 6.Xr^2.Yr^2)
dY = C2 + C3.Yr + C4.Xr + 2.C5.Xr.Yr + C6.(Xr^2 – Yr^2) + C8.Xr.(Xr^2 – 3.Yr^2)+ C7.Yr.(3.Xr^2 – Yr^2) - 4.C10.Xr.Yr.(Xr^2 – Yr^2) + C9.(Xr^4 + Yr^4 – 6.Xr^2.Yr^2)
Southing = FN + Xp – dX
Westing = FE + Yp – dY
Easting = -(Westing)
Northing = -(Southing)

The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:
Southing = -(Northing)
Westing = -(Easting)

Xr' = (Southing - FN) – Xo
Yr' = (Westing - FE) – Yo
dX' = C1 + C3.Xr' – C4.Yr' – 2.C6.Xr'.Yr' + C5.(Xr'^2 – Yr'^2) + C7.Xr'.(Xr'^2 – 3.Yr'^2) – C8.Yr'.(3.Xr'^2 – Yr'^2) + 4.C9.Xr'.Yr'.(Xr'^2 – Yr'^2) + C10.(Xr'^4 + Yr'^4 – 6.Xr'^2.Yr'^2)
dY' = C2 + C3.Yr' + C4.Xr' + 2.C5.Xr'.Yr' + C6.(Xr'^2 – Yr'^2) + C8.Xr'.(Xr'^2 – 3.Yr'^2)
+ C7.Yr'.(3.Xr'^2 – Yr'^2) - 4.C10.Xr'.Yr'.(Xr'^2 – Yr'^2) + C9.(Xr'^4 + Yr'^4 – 6.Xr'^2.Yr'^2)
Xp' = (Southing - FN) + dX'
Yp' = (Westing - FE) + dY'
r' = [(Yp')^2 + (Xp')^2]^(1/2)
theta' = atan2[Yp' , Xp'] (see GN7-2 implementation notes in preface for atan2 convention)
D' = theta' / sin(latp)
T' = 2{atan[((ro / r')^(1/n)).tan(pi/4 + latp/2)] - pi/4}
U' = asin[cos(alphaC).sin(T') - sin(alphaC).cos(T').cos(D')]
V' = asin(cos(T').sin(D') / cos(U'))

Then latitude lat is found by iteration using U' as the value for lat(j-1) in the first iteration:
lat(j) = 2*(atan{tO^(-1/B) tan^(1/B).(U'/2 + pi/4).[(1 + e sin(lat(j-1)) / (1 - e sin(lat(j-1))]^(e/2)} - pi/4)

Then
lon = lonO - V' / B where lon is referenced to the same prime meridian as lonO.

Example:

For Projected Coordinate Reference System: S-JTSK (Ferro) / Modified Krovak

Parameters:
Ellipsoid Bessel 1841 a = 6377397.155m 1/f = 299.15281
then e = 0.081696831 e^2 = 0.006674372

Latitude of projection centre = 49°30'00"N = 0.863937979 rad
Longitude of Origin = 42°30'00"E of Ferro = 0.741764932 rad
Co-latitude of cone axis = 30°17'17.30311" = 0.528627763 rad
Latitude of pseudo standard parallel = 78°30'00"N = 1.370083463 rad
Scale factor on pseudo Standard Parallel (ko) = 0.9999
False Easting = 0.00 m
False Northing = 0.00 m
Ordinate 1 of evaluation point Xo = 1089000.00 m
Ordinate 2 of evaluation point Yo = 654000.00 m
C1 = 2.946529277E-02
C2 = 2.515965696E-02
C3 = 1.193845912E-07
C4 = -4.668270147E-07
C5 = 9.233980362E-12
C6 = 1.523735715E-12
C7 = 1.696780024E-18
C8 = 4.408314235E-18
C9 = -8.331083518E-24
C10 = -3.689471323E-24

Calculated projection constants:
A = 6380703.611
B = 1.000597498
gammao = 0.863239103
to = 1.003419164
n = 0.979924705
ro = 1298039.005

Forward calculation for:
Latitude = 50°12'32.442"N = 0.876312568 rad
Longitude = 34°30'59.1790"E of Ferro = 0.602425500 rad

Then the forward calculation first gives

U = 0.875596951
V = 0.139422687
T = 1.386275051
D = 0.506554627
theta = 0.496385393
r = 1194731.002
Xp = 1050538.631
Yp = 568990.995
Xr = -38461.369
Yr = -85009.005
dX = -0.077
dY = 0.088
Southing = 6050538.71 m
Westing = 5568990.91 m
and then
Easting X = -5568990.91 m
Northing Y = -6050538.71 m

Reverse calculation for the same Easting and Northing:
Southing = 6050538.71 m
Westing = 5568990.91 m
Xr' = -38461.292
Yr' = -85009.093
dX' = -0.077
dY' = 0.088
Xp' = 1050538.631
Yp' = 568990.995
r' = 1194731.002
theta' = 0.496385393
D' = 0.506554627
T' = 1.386275051
U' = 0.875596951
V' = 0.139422687
lat(iteration 1) = 0.876310603
lat(iteration 2) = 0.876312562
lat(iteration 3) = 0.876312568

Latitude = 0.876312568 rad = 50°12'32.442"N
Longitude = 0.602425500 rad = 34°30'59.179"E of Ferro.

Method
Parameters:

Parameter Name	Parameter Code	Sign reversal	Parameter Description
Latitude of projection centre	8811	No	For an oblique projection, this is the latitude of the point at which the azimuth of the central line is defined.
Longitude of origin	8833	No	For polar aspect azimuthal projections, the meridian along which the northing axis increments and also across which parallels of latitude increment towards the north pole.
Co-latitude of cone axis	1036	No	The rotation applied to spherical coordinates for the oblique projection, measured on the conformal sphere in the plane of the meridian of origin.
Latitude of pseudo standard parallel	8818	No	Latitude of the parallel on which the conic or cylindrical projection is based. This latitude is not geographic, but is defined on the conformal sphere AFTER its rotation to obtain the oblique aspect of the projection.
Scale factor on pseudo standard parallel	8819	No	The factor by which the map grid is reduced or enlarged during the projection process, defined by its value at the pseudo-standard parallel.
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.
Ordinate 1 of evaluation point	8617	No	The value of the first ordinate value of the evaluation point.
Ordinate 2 of evaluation point	8618	No	The value of the second ordinate of the evaluation point.
C1	1026	No	Coefficient used in polynomial transformations.
C2	1027	No	Coefficient used in polynomial transformations.
C3	1028	No	Coefficient used in polynomial transformations.
C4	1029	No	Coefficient used in polynomial transformations.
C5	1030	No	Coefficient used in polynomial transformations.
C6	1031	No	Coefficient used in polynomial transformations.
C7	1032	No	Coefficient used in polynomial transformations.
C8	1033	No	Coefficient used in polynomial transformations.
C9	1034	No	Coefficient used in polynomial transformations.
C10	1035	No	Coefficient used in polynomial transformations.

Meta Data

Remarks:

Incorporates a polynomial transformation which is defined to be exact and for practical purposes is considered to be a map projection.

Information Source:

Land Survey Office (ZU), Prague. www.cuzk.cz/zu

Data Source:

EPSG

Revision Date:

August 29, 2018

Change ID:

[2010.071] [2010.1] [2012.011] [2017.037] [2017.024]

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