Approximation of the Krovák coordinates in the Czech Republic by Lambert Conformal Conic projection for GIS applications*
Gábor Timár1,**, Petr Urban2
1Dept. of Geophysics, Eötvös University of Budapest, Hungary
2ARCDATA Ltd., Praha, Czech Republic
The Krovák projection, an oblique conformal conic (OCC) one, is unique among the world’s national grids. Many GIS softwares simply don’t contain it and not only the Krovák projection but also the OCC is missing in them. Therefore in these packages, the Krovák grid coordinates cannot be ’taught’ with exact accuracy.
However, an approximation
is given in this paper, appropriate for all GIS packages and for some GPS
receivers. This substitute grid is based on the Lambert Conformal Conic (LCC)
projection, whose parameters are also given here. The difference of the
standard Krovák and the approximating LCC grid values are averagely 30 meters,
maximum 82 meters, throughout the Czech Republic. This moderate accuracy
enables to use the substitute grid for most GIS and some topographic purposes.
The approximation can be made with significantly lower errors for Slovakia.
After the WWI and the
dissolve of the Habsburg Empire, the newly formed Czechoslovak Republic built
up its own geodetic and cartographic system. The geodetic base, containing old
Austrian and Hungarian basepoints, re-equalized on the Bessel ellipsoid, is the
S-JTSK, while for the cartographic system the Krovák projection has been
introduced (for details, see eg. Mugnier, 2000).
The Krovák system has been
defined to provide minimum distortion throughout the uniqually shaped country;
this is a conformal conic projection in oblique position (Oblique Conformal
Conic, OCC; Kuska, 1960). There is no other national grid in the world based on
OCC projection. Beacuse of this uniquality, the GIS packages either know the
exact Krovák projection itself or don’t know any parametrizable OCC projection.
In the latter case, there is no possibility to define the Krovák system exactly
for those software. In this paper we propose an approximate solution with an
error of 3-82 meters, which is not precise enough for geodetic or even
cartographic purposes but appropriate for some non-precision GIS applications.
The exact Krovák is a
double projection: first from the S-JTSK datum of the Bessel1841 ellipsoid to
the Gauss-sphere then from the sphere to the cone.
The parameters of the
first (from ellipsoid to sphere) projection:
The normal parallel:
Φn = 49° 30’ (on
the ellipsoid)
φn = 49° 27’ 35.8463” (on the
Gauss-sphere)
The constants of the ellipsoid-->sphere projection:
n = 1.00059749835949
k = 1.00341916389791
The radius of the Gauss-sphere:
RGauss = 6380065.5402 meters.
The parameters of the Oblique
Conformal Conic (from sphere to cone) projection:
Coordinates of the projection centre (the pseudo-pole):
Φc = 59° 45’ 27”
Λc = 24° 50’ (from Greenwich)
Latitude of the pseudo standard-parallel:
φps = 78°30’
Starting point of the projection:
Φ0 = 49° 30’
Λ0 = 24° 50’ (from Greenwich)
Scale factor at the starting point: 0.9999
For the equations of this
projection, see e.g. Hojovec (1987). Note that all meridians here were given
relative to Greenwich, albeit the prime meridian of the original datum is the
Ferro one. The Ferro-Greenwich shift used here is 17° 40’.
As we mentioned in the Introducion, this
projection usually cannot be parametrized in GIS softwares. To find a
substitute projection with an accuracy of a few meters, we assumed that a
parameter set can be found for the LCC projection, suitable for our needs. The
coordinates of crossing points of all integer and half parallels and meridian
in the Czech Republic (with an additional crossing point near Cheb) were taken
into account. The Krovák coordinates of these points were calculated with the
algorithm of Kuska (1960). Besides, the LCC coordinates of them were also
computed with the „substituting” LCC projection (the equations of Snyder, 1987,
were used), and the best fit were detected with the following parameter set:
Projection centre:
Φc = 59° 50’ 28.30704”
Λc = 24° 50’ 01.80636” (from Greenwich)
The two standard parallels:
φs1 = 47° 52’ 12”
φs2 = 49° 34’ 12”
False Easting = 0 meters
False Northing = 0 meters.
Using these parameters, the maximum error of the approximation (the distance between the real Krovák and the substituting LCC coordinates) was 82 meters in the Czech Republic, occuring at the western extremities, around Cheb. In the Praha region the error is around 18 meters, while the average error at the investigated points is 30 meters. The best fit (error under 4 meter) is in the Znojmo region.
The signs of the coordinates are reversed in this approximation (the LCC projection is NE-directed) compared to the SW-directed Krovák grid.
The approximate LCC projection is not accurate
enough for any precise usage but for some GIS applications, where the pixel
size is larger than 100 meter. The LCC projection is common enough to be built
in every GIS packages.
A further possible usage of this new projection is
to parametrize the Magellan GPS receivers to get approximating Krovák
coordinates. Most receivers allows to parametrize only transverse cylindric
projection (e.g. Gauss-Krüger) but in the Magellans it is possible to set LCC,
thus the newly presented grid, too.
For the correct usage of the approximating LCC
grids – and the original Krovák grid itself –, the parameter set of the S-JTSK
datum (the shift parameters between the WGS84 and the S-JTSK) should be also
set, as follows:
dX = 589 meters;
dY = 76 meters;
dZ = 480 meters;
transformation direction is from S-JTSK to WGS84 (DMA, 1990). For GPS settings, the
ellipsoid shape difference parameters:
da = 740 meters;
df = 1e-5 (in scientific format).
Also note, that a similar but significantly better approach can be made for Slovakia as the area of that country is nearer to the original centre of the Krovák projection.
Defense Mapping Agency (1990): Datums, Ellipsoids, Grids and Grid
Reference Systems. DMA Technical Manual 8358.1. Fairfax, Virginia, USA
Hojovec, Vladimír (1987): Kartografie. Geodetický a
kartografický podnik, Praha
Kuska, František (1960): Matematická Kartografia. Slovenské Vydateľstvo Technickej Literatúry, Bratislava, 388 p.
Mugnier, Clifford J. (2000): Grids & Datums – the Czech Republic. Photogramm. Eng. & Rem . Sens. 66: 30-31.
Snyder, John P. (1987): Map projections - a working manual. USGS Prof. Paper 1395: 1-261
* Reference: Timár, G., Urban, P. (2003): Aproximace Křovákova zobrazení pro území České Republiky Lambertovým konformním kuželovým zobrazením pro potřeby GIS. ArcRevue [Praha] 12(2): 24-25.
** Corresponding author, e-mail: timar@ludens.elte.hu