# EPSG:1043

## Krovak Modified (North Orientated)

### Attributes

Data source: EPSG

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

Revision date: 2018-08-29

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

### 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.```