Geodetical Mapping - Krovak System

Doc. Dr. Vladimír Homola, Ph.D.

Krovak mapping is Gaussian equi-angular conic projection in the skew position. The Bessel ellipsoid is projected into the plain using the reference sphere. In Czechoslovakia, the Krovak mapping was established in 1922 for cadastral maps; later it was used for definitive military mapping. Since 1968, the sheets of Basic Map System of CZ are built in this projection.

Fig. 1: Krovak projection

The calculation of the projection is exceedingly difficult and is described in steps A to D.

The Bessel ellipsoid (parameters: a = 6377.397 [km], b = 6356.079
[km], f = 1/299.15, M_{0} = 6359.3926 [km], N_{0} =
6402.0861 [km]) is conformly displayed on the sphere with radius
R_{0} = sqrt (M_{0} x N_{0}) = 6380.7036 [km];
this ellipsoid touches the sphere in only one point. By this way, it
is preserved the length of parallel with the geographical latitude
j_{0} = 49.5
°; this parallel is displayed as
the parallel of sphere with the sphere latitude U_{0} = 49.459957
°. The ratio of length preservation is

a = sin (
f_{0}) / sin(U_{0}) = 1.0005975.

The geographical longitudes l on ellipsoid
correspond with the spherical longitudes V on sphere; there is V =
a x l. The
geographical latitudes are on principle related to Ferro meridian ( =
17° 40' West of Greenwich). The
basic meridian is l_{0} =
42.5° [Ferro degrees]. For all points of grid,
j and l must be
transformed to U and V.

The U coordinate is given like U = ( b^{2} / a^{2} ) x tg
j, or in this way:

DU" = 9985.859795 x
D
j + 2.402875 x
D
j^{2} - 0.011658 x
D
j^{3}
+ ...

where
D
j =
j -
j_{0}
[in sec x 10^{-4}]; then U = U_{0} +
DU [in
°]. In place
of V coordinate, we search
DV =
a x
(l_{0} -
l).

The very open cone with
l' =
352.98456° is put onto the sphere. The top
of cone is about 131 [km] above the town Tallin. The constructive
pole of cone is on j_{P} =
59.7575 °,
l_{P} =
42.5°.
The meridian l_{P} =
l_{0} is about 15 [km] from the end of
the former Czechoslovak Republic east border (today's Carpathian
Ukraine). To the coordinate
j_{P}
(l_{P}) corresponds the constructive
coordinate U_{P} = 59.711860°
(Z_{P} = 30.288140°) and
DV = 0°.

The touch constructive parallel
F_{0}
= 78.5° with radius
r_{0} = 1298.039 [km]
conducts approximately along the middle of the mapped area (former
Czechoslovak Republic). For all points of the grid, the spherical
coordinates must be transformed to construction coordinates:
U®
D,
DV
®
L. The coordinate
D
can be calculated from Z = 90
° - U
according to cosinus rule, the coordinate
L
according to sinus rule:

cos
L = cos Z x cos Z_{P} + sin Z x
sin Z_{P} x cos
DV

sin
L = sin Z x sin
DV / sin
D

On account of decrease of the maximum length distortion (in place of 24 [cm/km] - 14 [cm/km] only), the radius of sphere is contracted by ratio 1.0001; new radius R = 6380.0655 [km]. For all points of grid, the polar coordinates can be calculated by transformation of construction coordinates according to Gauss conic mapping:

D_{0} =
90° -
F_{0}

l' =
L x cos
D_{0}

r = R x tg
D_{0} x (tg
(D/2) x cotg
(D_{0}/2)) ^ (cos
D_{0})

The final step is the conversion of polar coordinates to rectangular coordinates (with exchange X and Y coordinate):

Y =
r x sin
l'

X =
r x cos
l'

The example for the point
j =
49° 53' 29.1" and
l_{Fe} =
32° 6' 10.5", i.e.
l_{Gr} =
14° 26' 10.5":

U = 49.850832° | DV = 10.403297° |

D = 11.51291° | L = 35.68706° |

l' = 39.97063° | r = 1299.478 [km] |

X = 1064.852 [km] | Y = 744.804 [km] |