Conversion between Hungarian and Austrian map projection systems is
presented in this paper. The conversion may be performed in two steps: at
first any kind of map projection systems should be transformed into WGS-84
ellipsoidal co-ordinates in one country, and then from WGS-84 ellipsoidal
co-ordinates should be transformed into the desired system of  the other
country. An algorithm and a computer program have  been developed to carry out
this transformation

Varga, József

Völgyesi, Lajos

English

Periodica Polytechnica (Budapest University of Technology and Economics)

PERIODICA POLYTECHNICA SER. CIV. ENG. VOL. 46, NO. 2, PP. 231–238(2002)GPS AS THE DEVICE OF JUNCTION OF TRIANGULATIONNETWORKSLajos VÖLGYESI and József VARGADepartment of Geodesy and SurveyingBudapest University of Technology and EconomiesH–1521 Budapest, HungaryReceived: Jan. 25, 2002AbstractConversion between Hungarian and Austrian map projection systems is presented in this paper. Theconversion may be performed in two steps: at first any kind of map projection systems should betransformed into WGS-84 ellipsoidal co-ordinates in one country, and then from WGS-84 ellipsoidalco-ordinates should be transformed into the desired system of the other country. An algorithm and acomputer program have been developed to carry out this transformation.Keywords: Hungarian and Austrian map projection systems, transformation between map projectionsystems.1. IntroductionMap projection systems of large scale maps and their reference surfaces, as wellas their triangulation networks usually differ with countries – moreover, they maybe different inside the same state. A contiguous triangulation network can onlybe considered as uniform according to map projections if the whole network wasadjusted at the same time. Geodetic co-ordinates of triangulation stations will bechanged even if the network had been readjusted, e.g. with new conditions.Exact conversions between two map projection systems with closed math-ematical formulae can be made in cases only when both projection systems havethe same reference surface and points of the same triangulation network comingfrom the same adjustment are represented in both projection systems. When anyof the above mentioned requirements have not been met the conversion can be per-formed only using certain common points that have co-ordinates in both projectionsystems (HAZAY , 1964; VARGA, 1981, 1982, 1986). In such cases the accuracyof transformed co-ordinates depends on the reliability of triangulation networksand the position and number of the selected common points. Slightly different co-ordinates will result after the conversion process when other common points hadbeen chosen. If there is no exact conversion using closed mathematical formulaebetween two map projection systems, the transformation can be performed only byHelmert’s transformation or polynomials up to the maximum degree five (Manualfor the Application of Unified National Projection, 1975). Applying these methods232 L. VÖLGYESI and J. VARGAwe can eliminate the distortions of projection and the discrepancies of triangulationnetworks at the same process making a single plain transformation.More precise and secure conversion can be made using the so-calledmixedmethod. In this method the transformation can be performed in two steps: first thedistortions of projection and then the discrepancies of triangulation networks can beeliminated. In the first step we suppose that the two map projection systems havethe same reference surface and the same triangulation network, and we performthe computation by theco-ordinate method using closed mathematical formulae(VARGA, 1986). So in the first step we get approximate plane co-ordinates in thesecond projection system. Then in the second step we perform a transformationby polynomials using common points. The common points for determining thecoefficients of this transformation polynomials should be the points which have boththe previously computed approximate values and the original plane co-ordinates inthe second projection system. We can use transformation polynomials having lowerdegrees in the second step of transformation to eliminate the discrepancies of thedifferent triangulation networks, against the case if we make the conversion in onlyone step using series.2. Conversion between Hungary and AustriaThere are some difficulties in the case of conversion between map projection systemof neighboring countries – when the reference surfaces of the applied map projectionsystems are the same and the triangulation networks are connected to them. This isthe case between the Hungarian and Austrian map projection systems that we haveinvestigated. Conversion between Hungarian and Austrian map projection systemscannot be computed byco-ordinate method using closed mathematical formulaebecause the position and orientation of reference surfaces are slightly different, andthe triangulation networks had been adjusted one by one – although there is theBessel’s ellipsoid as a reference surface of projection systems which is applied inHungary and Austria too, and there are some common points of different triangula-tion networks. So the conversion between the two countries can be performed onlyby transformation polynomials using common points.Map projection systems of neighboring countries can be expanded only for afew tens of kilometers range from the common border because common points canalways be found only in this region. GPS is the most powerful tool for fixing com-mon points anywhere, because determining ofX , Y , Z spatial geocentric Cartesian,WGS-84, or UTM co-ordinates of points of triangulation network by GPS, we cancreate such system of common points which are very suitable for conversion of mapprojection system between the neighboring countries.Such conversions between countries are necessary only if somebody wantsto use its own special map projection system in the neighbor country and does notwant to use e.g. the simpler UTM projection system. But in this case a projectionsystem of a country can be expanded into a direction only not too far from theGPS AS THE DEVICE OF JUNCTION 233common border.Having enough common points determined by GPS affords the chance to makecomputer program for the conversion between map projection systems of Hungaryand Austria. So it is all the same whether we transform co-ordinates betweenmap projection systems of Hungary and Austria with different reference surfaces(Bessel’s ellipsoid in Austria, and Bessel’s, Krassovky’s or IUGG-67 ellipsoids inHungary) and different meridians of origin (prime meridian of Ferro for Austriaand prime meridian of Greenwich for Hungary).3. Practical SolutionThe conversion logic between the different map projection systems can be seen inFig. 1. Transformation paths – and their directions – between different systemsare pictured by arrows. It can be seen that it is possible to convert between bothWGS-84↔ Unified National Projection and WGS-84↔ Gauss-Krüger systemsonly through other intermediate systems. E.g. if a conversion between WGS andEOV systems is needed, then WGS-84 co-ordinates first have to be converted intothe new Gaussian sphere (NGS) and then into a so-called auxiliary system (AUX)and finally they should be converted from this AUX system into EOV co-ordinates;or e.g. if a conversion between GAK and WGS systems is needed, then Gauss-Krüger co-ordinates first have to be converted into an auxiliary system (AUX) andthen into the new Gaussian sphere (NGS) and finally they should be converted fromthe new Gaussian sphere into the WGS-84 ellipsoid.  HER HKRSZTKSTVTNGAKEOVOGS NGSBES KRA IUGNGSWGSUTMAUXGWEWGEEVESGSXYZ ABEAGKAW  Fig. 1. Conversion logic between the different map projection systemsIf any two systems inFig. 1 are connected through a hexagonal block, thenbetween these two systems only an approximately accurate conversion could be234 L. VÖLGYESI and J. VARGAmade by transformation polynomials. InFig. 1 the two-letter abbreviations inhexagonal blocks show which data files containing transformation polynomials haveto be used to convert between the two neighboring systems. If any two systemsin Fig. 1 are connected by a continuous line, then an exact conversion by the co-ordinate method, i.e. through closed mathematical formulae can be made.Since application of the correct methods of conversion between a multitude ofmap projection systems may cause problems even for experts, a program packagehas been developed by which conversions can be made between Hungarian andAustrian map projection systems and their reference co-ordinates in all combina-tions, the usage of which is simple even for users having no deep knowledge in mapprojections.Conversions between co-ordinatesVTN = System without projectionBES = Hungarian Bessel’s EllipsoidalSZT = Budapest Stereographic ProjectionKST = Military Stereographic ProjectionHER = North Cylindrical SystemHKR = Middle Cylindrical SystemABE = Austrian Bessel’s EllipsoidalAGK = Austrian Gauss-Krüger ProjectionIUG = IUGG-67 EllipsoidalEOV = Unified National ProjectionKRA = Krassovsky’s EllipsoidalGAK = Hungarian Gauss-Krüger ProjectionWGS = WGS-84 Ellipsoidal /GPS/XYZ = Spatial Cartesian Geocentric /GPS/UTM = Universal Transverse Mercatorare performed by the conversion program in the area of Hungary and Austria in 212combinations as enlisted inTable1. South cylindrical projection system (HER) andBudapest city stereographic projection (VST) are not to be found on the above listbecause the regions where these two Hungarian map projection systems are usedare not neighboring to Austria and using these two systems there is no practicalneed to make conversions between Hungary and Austria.This table informs us about the possibility and accuracy of conversions verysimply.Double lines in this table separate map projection systems belonging to dif-ferent reference surfaces. (By reference surface the ellipsoid is meant, though thefact should be acknowledged that the approximating /Gaussian/ sphere serves alsoas a reference surface for those map projection systems where a double projectionis applied and an intermediate sphere is the reference surface at the second step ofthe projection to get co-ordinates on a plane. Co-ordinates on this approximatingsphere have no practical role for users.)GPS AS THE DEVICE OF JUNCTION 235Plus ‘+’ signs at the intersection fields of rows and columns indicate thatan exact conversion between the two map projection systems is possible usingclosed mathematical formulas found in reference works of (HAZAY , 1964) and(VARGA, 1981, 1986) for transformation. In this case the accuracy of transformedco-ordinates is the same as the accuracy of co-ordinates to be transformed.Cross ‘×’ signs of this table indicate the impossibility of transformation be-tween the two map projection systems with closed mathematical formulae, and theconversion – according to rules found in (Manual for. . ., 1975) – is performedusing polynomials as of a finite (maximum five) degree with limited accuracy(VÖLGYESI et al, 1994, 1996; VÖLGYESI, 1997).Minus ‘−’ signs in the table are reminders of the fact that an identical (trans-formation into itself) conversion has no meaning except of the Gauss-Krüger andUTM projection systems where the need of conversion between different zonesfrequently arises. Hence a ‘!+!’ sign indicates that it is possible to make exactconversions between different zones of the Gauss-Krüger and UTM map projectionsystems.4. Accuracy of ConversionsIt was mentioned previously that it is possible to convert through closed mathemat-ical formulae between certain map projection systems. A conclusion could havebeen drawn as a result of our test computations that in these cases the accuracyof transformed plane co-ordinates is equal to the accuracy of initial co-ordinates(1 mm or 0.0001′′). These conversions are referred to inTable1 with ‘+’ and ‘!+!’signs or these systems are connected by continuous lines (arrows) inFig.1.In all other cases when the transformation path between any two systems goesthrough a hexagonal block (or blocks), the accuracy of transformed co-ordinatesdepends, on the one hand, how accurately the control networks of these systemsfit into each other; and on the other, how successful the determination of transfor-mation polynomial coefficients was. It follows also from these facts that no matterhow accurately these transformation polynomial coefficients were determined, ifthe triangulation networks of these two systems do not fit into each other accurately– since there were measurement, adjustment and other errors during their estab-lishment – then certainly no conversion of unlimited accuracy can be performed(in other terms, accuracy level conversions between two map projection systemsare defined by the determination errors or discrepancies of these control networks).This fact, of course, does not mean that one should not be very careful when themethod of transformation is selected or – when the polynomial method is applied– the coefficients are determined.The objective of our first tests was to decide which one of the two methods:Helmert transformation or polynomial method is more advantageous to be used.We arrived at the result that although the Helmert transformation is computationallysimpler, its accuracy in the majority of cases does not even approximate the accuracy236 L. VÖLGYESI and J. VARGAprovided by the polynomial method. Since a simple programming can be a motiveonly for software ‘beginners’, we voted firmly for the use of polynomial method.When the polynomial method is chosen, the next important step is to determinethe optimal degree of the polynomial. By considering a simple way of reasoningone could arrive at the conclusion that the higher the degree of the polynomialthe higher the accuracy of map projection conversions. On the contrary, it couldbe proved by our tests that the maximum accuracy was resulted by applying fivedegree polynomials. No matter whether the degree was decreased or increased,the accuracy of transformed co-ordinates was lessened alike (more considerably bydecreasing, less considerably by increasing).Coefficients of transformation polynomials are based on co-ordinates of com-mon pointsyi , xi and y′i , x′i in systemsI and I I , respectively. Thenyi , xi co-ordinates in systemI are transformed into co-ordinatest y′i , t x′i in systemI I byusing these coefficients and finally the standard error characteristic to conversion,µ =√√√√√n∑i=1(t y′i − y′i)2 +n∑i=1(t x ′i − x ′i )2n(1)will be determined.For information it could be mentioned that, for example, between the Bu-dapest Stereographic and the EOV systems the standard error is±0.252 m fromthe expression (1) for the complete area of Hungary when 134 common points areused and the same figures are±0.037 m and±0.217 m between EOV and WGS-84,and EOV and Gauss-Krüger systems by using 34 and 50 common points, respec-tively. Between Austrian Gauss-Krüger and WGS-84 systems the standard erroris ±0.152 m for the complete area of Austria when 57 common points are used todetermine the coefficients of transformational polynomials.Our experience has shown that although the accuracy can be somewhat in-creased by raising the number of common points within the polynomial methodbut the accuracy of conversion cannot be increased beyond a certain limit evenwith this method since there is a difference between the two triangulation networks.In certain cases, however, an improvement could be gained when transformationpolynomial coefficients are not determined for the complete area of the countrybut common points are given and transformation polynomial coefficients are de-termined only for a smaller region. In such cases conversions, of course, must notbe made outside the sub-area where the coefficients of transformation polynomialswere determined, and the junction of these regions is not a simple problem.5. SummaryConversion between Hungarian and Austrian map projection systems is performedin two steps: first any kind of map projection systems should be transformed intoGPS AS THE DEVICE OF JUNCTION 237Table 1. Possible transformations between Hungary and AustriaVTN BES SZT KST HER HKRABE AGK IUG EOV KRA GAK WGS XYZ UTMVTN − × × × × × × × × × × × × × ×BES × − + + + + × × × × × × × × ×SZT × + − + + + × × × × × × × × ×KST × + + − + + × × × × × × × × ×HER × + + + − + × × × × × × × × ×HKR × + + + + − × × × × × × × × ×ABE × × × × × × − + × × × × × × ×AGK × × × × × × + !+! × × × × × × ×IUG × × × × × × × × − + × × × × ×EOV × × × × × × × × + − × × × × ×KRA × × × × × × × × × × − + × × ×GAK × × × × × × × × × × + !+! × × ×WGS × × × × × × × × × × × × − + +XYZ × × × × × × × × × × × × + − +UTM × × × × × × × × × × × × + + !+!WGS-84 ellipsoidal co-ordinates in one country, and then from WGS-84 ellip-soidal co-ordinates should be transformed into the desired system for the othercountry. A precise and secure conversion can be made using the so-calledmixedmethod: first the distortions of projection, and then the discrepancies of triangula-tion networks can be eliminated performing a transformation by polynomials usingcommon points. Using our method and software for the given common points, thetransformation between Austrian and Hungarian map projection systems can beperformed with a few centimeters accuracy for a few ten kilometers range of thecommon border.AcknowledgementsOur investigations were supported by theHungarian National Research Fund (OTKA), con-tract No. T-030177; and theGeodesy and Geodynamics Research Group of the HungarianAcademy of Sciences.References[1] HAZAY,I., Map Projections, Tankönyvkiadó, Budapest, 1964, (in Hungarian).[2] Manual for the Application of Unified National Projection, MÉM OFTH, Budapest, 1975, (inHungarian).238 L. VÖLGYESI and J. VARGA[3] VARGA, J.,New Methods of Conversion between our Projection Systems, Budapest, TechnicalDoctoral Thesis, 1981, (in Hungarian).[4] VARGA, J., Conversion between the Unified National Projection (EOV) and between our FormerProjections,Geodézia és Kartográfia, No. 2, 1982, (in Hungarian).[5] VARGA, J.,Control Networks I, (Map projections). Tankönyvkiadó, Budapest, 1986, (in Hun-garian).[6] VÖLGYESI, L. – TÓTH, GY. – VARGA, J., Transformation between Hungarian Map ProjectionSystems,Geodézia és Kartográfia, 46 No. 5–6 (1994), (in Hungarian).[7] VÖLGYESI, L. – TÓTH, GY. – VARGA, J., Conversion between Hungarian Map ProjectionSystems,Periodica Polytechnica Civil Eng., 40 No. 1 (1996), pp. 73–83.[8] VÖLGYESI, L., Transformation of Hungarian Unified National and Gauss-Krüger ProjectionSystem into WGS-84.Reports on Geodesy, Warsaw University of Technology, 27 No. 4 (1997),pp. 281–294.

GPS AS THE DEVICE OF JUNCTION OF TRIANGULATION NETWORKS

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GPS AS THE DEVICE OF JUNCTION OF TRIANGULATION NETWORKS

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