Approximation of the Křovák projection with the Lambert Conformal Conic projection in the GIS and GPS applications on the territory of Slovakia
Gábor Timár1,*, Martin Danišík2
1Dept. of Geophysics, Eötvös University of Budapest, Hungary
2Institute of Geology and Palaeontology, University of Tübingen, Germany
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 implemented 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 substitute LCC grid values are averagely 6 meters, maximum 12 meters, throughout Slovakia. This moderate accuracy enables to use the substitute grid for most GIS and some topographic purposes. The approximation is not valid for the Czech Republic, where a similar LCC grid can be defined only with significantly higher errors.
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 system, 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 e.g. Mugnier, 2000).
The Krovák system has been defined to provide minimum distortion throughout the uniquely 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. Because of this uniqueness, 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 that software. In this paper we propose an substitute solution with 6-12 meter accuracy, which is not precise enough for geodetic purposes but appropriate for GIS and non-precision GPS practice.
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 (ellipsoid » 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 (sphere » cone) projection:
Coordinates of the projection centre (the pseudo-pole):
Φc = 59° 45’ 27”
Λc = 24° 50’ (E from Greenwich)
Latitude of the pseudo standard-parallel:
φps = 78°30’
Starting point of the projection:
Φ0 = 49° 30’
Λ0 = 24° 50’ (E from Greenwich)
Scale factor at the starting point: 0.9999
For the equations of this projection, see e.g. Kuska (1960). 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 Introduction, the Krovák 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 meridians in Slovakia (with the additional crossing points of the 47° parallel and the meridians of 18° and 18° 30’) 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:
Φc = 59° 50’ 0.5712”
Λc = 24° 50’ (E from Greenwich)
The two standard parallels:
φs1 = 47° 36’ 18”
φs2 = 49° 13’ 30”
False Easting = 4.7 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) is 12 meters in Slovakia, occurring at the eastern, north-eastern extremities. In the Bratislava region, the error is around 11 meters, while the average error at the investigated points is 6.3 meters. The best fit (error under 1 meter) is at the upper Hron valley, around Brezno.
The signs of the coordinates are reversed in this approximation (the LCC projection is NE-directed) compared to the SW-directed Krovák grid.
The substitute LCC projection is precise enough for most raster-based GIS application, where the pixel size is larger than 10 meter or for any GIS application where the precision claim doesn’t exceed 10-12 meters. 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 allow to parametrize only transverse cylindric projections (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 substitute LCC grid – as well as 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 set, as follows:
dX = 589 meters;
dY = 76 meters;
dZ = 480 meters;
transformation direction: S-JTSK » WGS84 (DMA, 1990). For GPS settings, the ellipsoid shape difference parameters:
da = 740 meters;
df = 1e-5 (in scientific format).
Also note, that the presented approximating grid is not valid for the Czech Republic with similar accuracy. While the Czech region is quite far from the starting point of the original projection (in Zakarpatie), the real central line is far from the direction of the parallels. Therefore a similar LCC approximation can be found for the Czech Republic only with substantially less accuracy (approximately 60 meters).
Defense Mapping Agency (1990): Datums, Ellipsoids, Grids and Grid Reference Systems. DMA Technical Manual 8358.1. Fairfax, Virginia, USA
Kuska, František (1960): Matematická Kartografia. Slovenské Vydavateľstvo 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
Timár, G., Danišík, M. (2003): Approximation of the Křovák projection with the Lambert Conformal Conic projection in the GIS and GPS applications on the territory of Slovakia (in Slovakian with English summary). Kartogaficke listy [Bratislava] 11: 100-102.