CONVERSIONS BETWEEN HUNGARIAN MAP PROJECTION SYSTEMS

CONVERSIONS BETWEEN HUNGARIAN MAP PROJECTION SYSTEMS Lajos VÖLGYESI, Gula TÓTH, József VARGA Department of Geodes Techncal Unverst of Budapest H-11 Budapest, Hungar Abstract When dfferent map projecton sstems are appled smultaneousl over a gven area the need of converson permanentl arses n the overlappng areas of these sstems. Conversons, however, can not alwas be made b closed mathematcal formulas and n these cases t frequentl rases serous problems to appl a correct transformaton method. Hence such algorthm and program package was developed for all combnaton converson between Hungaran map projecton sstems and ther reference surfaces whch should not cause dffcultes even for users havng no deeper knowledge n map projectons. Kewords: map projecton, converson between map projectons. 1. INTRODUCTION A multtude of map projecton sstems was resulted n Hungar because there were man subsequent (and mostl justfed) changes n geodetc reference sstems. Three dfferent stereographc map projecton sstems were used for geodetc purposes and three conformal tangent clndrcal sstems were requred as well. Two 6 o zones of Gauss-Krüger and UTM projecton cover over the area of the countr thus more than one sstem s used even wthn a sngle knd of projecton. Besdes, the Unfed Natonal Projecton sstem (EOV) was ntroduced n the whole area of Hungar as well. The former Hungaran Gaussan sphere, tangent to the Bessel ellpsod s the common reference surface of Hungaran stereographc and conformal tangent clndrcal sstems, the new Hungaran Gaussan sphere, tangent to the IUGG-67 ellpsod, s the reference for EOV sstem and the Krassovsk ellpsod s the reference surface for Gauss-Krüger projectons n our countr. Furthermore, WGS-8

ellpsodal or geocentrc Cartesan co-ordnates result from GPS measurements more recentl and even t s requred n nternatonal relatons to use the UTM sstem n more recent tmes. Ths pcture s complcated further on b the fact that besdes the above mentoned sstems also mltar stereographc, n the area of Budapest ct stereographc and n some vllages of the countr even sstem wthout projecton are used. When dfferent map projecton sstems are used smultaneousl over the same area the need of converson frequentl arses nsde overlappng areas. Crcumstances are the same when there are dfferent zones wthn a sngle projecton (e.g. Gauss-Krüger or UTM projectons), then nsde the perpher of neghbourng zones co-ordnates are to be converted usuall. More generall speakng: when the projecton sstem of our maps dffer from that of our control ponts avalable, our measurement results have to be transformed nto the projecton sstem of our map that the could be represented on t. Conversons ma take place ether b the so called co-ordnate method (wth closed mathematcal epressons) or through transformaton equatons (polnomals), whch were provded b usng so called common ponts that have known co-ordnates n both sstems. It s possble to make eact conversons wth closed mathematcal epressons n cases onl when both projecton sstems has the same reference surface and ponts of the same trangulaton network comng from the same adjustment are represented n both projecton sstems. It s snce f a pont belongng to a dfferent trangulaton network s converted from one sstem nto the other then transformed co-ordnates wll not ft sutabl nto ponts of the trangulaton network presented on the projecton plane n queston. It s true because one should consder dfferences that ma arse from the dfferent poston and orentaton of the two trangulaton networks and also base etenson networks and angle observatons are qute dfferent. A refnement of an trangulaton network wth recent measurements or wth a readjustment alters co-ordnates of horzontal control ponts wth respect to the reference surface and hence co-ordnate on the projecton plane as well. The effects are the same when some parameters of the reference surface are modfed even n that case when otherwse our trangulaton network remans the same. An re-orentaton of the network (a rotaton of the reference surface) does not hnder eact converson. When the co-ordnate method s appled, converson ma be made b rgorous mathematcal epressons found n some reference works enlsted. In each such case when an of the above mentoned requrements has not been met the converson s not possble b closed mathematcal formulas. The converson therefore can be performed onl b transformaton equatons, whch were deduced as polnomals from so called common ponts that have co-ordnates n both projecton sstems. In ths case mamum fve-order conformal polnomals can be appled dependng on the number of common ponts. For eample, the connecton between, co-ordnates of the projecton sstem I. and,,, co-ordnates of the projecton sstem II. s establshed b the

' = A A A 10 18 0 ' = B B B 10 18 0 1 11 11 19 19 1 1 0 0 1 1 1 1 1 6 6 1 1 7 7 16 16 8 8 17 17 9 + 9 + + + (1) polnomals. Coeffcents A 0 A 0 and B 0 B 0 (altogether coeffcents) can be determned b usng common ponts sutabl through an adjustment process. In such a case slghtl dfferent co-ordnates wll be resulted after the converson process dependng on the poston and number of selected common ponts and the appled method.. COMPUTER SOFTWARE DEVELOPMENT Snce t ma cause problems even for eperts to appl correct methods of converson between a multtude of map projecton sstems so we worked out such a program package b whch conversons can be made between Hungaran map projecton sstems and ther reference co-ordnates n all combnaton, the usage of whch can cause no problem even for users havng no deeper knowledge n map projectons. Converson between co-ordnates VTN = Sstem wthout projecton BES = Bessel's Ellpsodal SZT = Budapest Stereographc Projecton KST = Mltar Stereographc Projecton HER = North Clndrcal Sstem HKR = Mddle Clndrcal Sstem HDR = South Clndrcal Sstem VST = Stereographc Sstem of the Ct Budapest IUG = IUGG-67 Ellpsodal EOV = Unfed Natonal Projecton KRA = Krassovsk's Ellpsodal GAK = Gauss-Krüger Projecton WGS = WGS-8 Ellpsodal XYZ = Spatal Cartesan Geocentrc /GPS/ UTM = Unversal Transverse Mercator are performed b the converson program n the area of Hungar n 1 combnatons as t s enlsted n Table 1.

Table 1 VTN BES SZT KST HER HKR HDR VST IUG EOV KRA GAK WGS XYZ UTM VTN - ( ) BES - + + + + + SZT + - + + + + KST + + - + + + HER + + + - + (+) ( ) HKR + + + + - + ( ) HDR + + + (+) + - ( ) VST ( ) ( ) ( ) ( ) - IUG - + EOV + - KRA - + GAK +!+! WGS - + + XYZ + - + UTM + +!+! Ths table conves us nformaton on the possblt and accurac of conversons ver smpl. Double lnes n ths table separate map projecton sstems belongng to dfferent reference surfaces. (B reference surface the ellpsod s meant, though the fact should be acknowledged that the appromatng /Gaussan/ sphere serves also as a reference surface for those map projecton sstems where a double projecton s appled and an ntermedate sphere s the reference surface at the second step of the projecton to get co-ordnates on a plane or on a plane developable surface. Co-ordnates on ths appromatng sphere have no practcal role for users.) Plus " + " sgns at the ntersecton felds of rows and columns ndcate that an eact converson between the two map projecton sstem s possble usng closed mathematcal formulas found n reference works of (HAZAY, 196), (VARGA, 1981, 1986) for transformaton. In ths case the accurac of transformed co-ordnates s the same as the accurac of co-ordnates to be transformed. Cross " " sgns of ths table ndcate the mpossblt of transformaton between the two map projecton sstems wth closed mathematcal formulas and the converson accordng to rules found n (RULES FOR MAP PROJECTION'S USE, 197) s performed usng e.g. polnomals as n Eq. (1) of a fnte (mamum fve) degree. In these cases theoretcall there s onl a possblt of converson wth lmted accurac (e.g. the accurac of converted plane co-ordnates s generall about onl ±10 cm ±0 cm). Parenthetc plus " (+) " and cross " ( ) " sgns remnd us of the fact that a converson s possble and t can be done b our program but there s no practcal need ecept of

scentfcal reasons to make t. (E.g. between map projecton sstems wth no overlappng areas or f the are not ver close to each other there can be no practcal need to make converson). Mnus " " sgns n the table are remnders of the fact that an dentcal (transformaton nto tself) converson has no meanng ecept of the Gauss-Krüger and UTM projecton sstems where the need of converson between dfferent zones frequentl arses. Hence a "!+! " sgn ndcates that t s possble to make eact conversons between dfferent zones of the Gauss-Krüger and UTM map projecton sstems. It has to be noted that onl an appromate converson usng common ponts s possble from the Stereographc projecton sstem of the ct Budapest nto some other (e.g. nto Budapest Stereographc) projecton sstems that have even the same reference surface (Bessel ellpsod) because the trangulaton networks are dfferent. Snce our recent nformaton shows that there are some vllages not onl n the southern part of Transdanuba but also n the countr Szabolcs-Szatmár-Bereg that have maps wthout projecton, hence converson between North Clndrcal Sstem (HER) and Sstem Wthout Projecton (VTN) s allowed and the sgn " " appears nstead of "( )" sgn n the correspondng feld of the table. The logcal frame of our map projecton converson software can be grasped n Fg. 1 and Fg.. Module 1 Vetpol.Ee Common ponts eov_szt.pol eov_vst.pol eov_wgs.pol szt_vtn.pol gak_eov.pol gak_szt.pol gak_wgs.pol doc. Vetulet.Ee work.dat Vetrajz.Ee Ponts to be converted Olvas.Ee Vet.Ee out.dat Module átszámított Converted ponts Fg. 1

Our software has two man parts: a module whch elds coeffcents of transformaton polnomals and another module whch performs actual conversons. Broken lnes surround these two modules n Fg. 1. Module 1 computes coeffcents of transformaton polnomals n equaton (1) n those cases t s mpossble to convert between the two sstems through the co-ordnate method, that s through closed mathematcal epressons. Program Vetpol.Ee makes t possble to calculate the coeffcents of polnomals when some common ponts are adequatel gven. Program Vetpol creates bnar fles eov_szt.pol, eov_vst.pol, eov_wgs.pol, szt_vtn.pol, gak_eov.pol, gak_szt.pol, and gak_wgs.pol contanng coeffcents of transformaton polnomals, whch are requred for conversons between EOV-Budapest Stereographc, EOV Ct Stereographc, EOV WGS-8, Budapest Stereographc Wthout Projecton, Gauss-Krüger EOV, a Gauss-Krüger Budapest Stereographc and Gauss-Krüger WGS-8. Program Vetpol determnes the degree of transformaton polnomals automatcall as a functon of the number of common ponts. If there are 1 or more common ponts then all the (namel ) coeffcents of a fve-degree polnomal n the epresson (1) can be determned. When the number of common ponts les between 1 and 0 then the degree of polnomals s, f the number of common ponts s between 10 and 1 then the degree s, and then the number of common ponts s between 6 and 9 the degree of polnomals requred for transformaton s. At least 6 common ponts are necessar to compute coeffcents of the polnomals, however an effort should be made to use as man common ponts as possble to determne these polnomal coeffcents. If the number n of common ponts s such as 7 n 9, 11 n 1, 16 n 0 or n 1, than the number of equatons s greater than t s necessar (the problem s over determned), hence the most relable values of unknown polnomal coeffcents are determned through adjustment b program Vetpol. Program Vetrajz.Ee s also a member of Module 1 b whch the geometrcal arrangement of common ponts can be dsplaed on screen to check the evenness of our pont dstrbuton. Actual conversons can be made b Module (Fg. 1). Three mportant programs can be found n ths module: nput-output organzer program of the converson software, namel Vetulet.Ee, man converson program Vet.Ee and Olvas.Ee s an utlt program to read and prnt output fles. Co-ordnates of ponts to be converted can be nputted from both keboard and dsk fles b the program Vetulet.Ee. A bult-n specal edtor helps to handle co-ordnates from the keboard or to transfer them nto a work fle work.dat n the requred format. Ths specal edtor also serves to check nputted co-ordnates on a hgh level and therefore t s practcall mpossble to read errorneous co-ordnates. Co-ordnates from dsk fles wll also 6

pass through the above strct trouble shootng process and the wll be transferred nto a work fle work.dat as well. HER HKR HDR VST UTM EV KST SZT GS ES GE EOV EW SVR SV VTN RGG GAK UGG UGG BES KRA IUG WGS GW XYZ Fg. Co-ordnates n the work fle work.dat are transformed nto the requred sstem b the man converson program Vet.Ee. The operaton of ths man program and the converson logc between the 1 dfferent map projecton sstems can be overvewed on Fg.. Transformaton paths and ther drectons between dfferent sstems are pctured b arrows. It can be seen that n most cases t s possble to convert between two arbtrar sstems onl through other ntermedate sstems (e.g. f a converson between UTM and EOV sstems s needed then UTM co-ordnates frst have to be converted nto WGS-8 ellpsod, then nto the new Gaussan sphere and then nto a so-called aular sstem and fnall the should be converted from ths SVR sstem nto EOV). If an two sstems are connected n Fg. b a contnuous lne then an eact converson b the co-ordnate method,.e. through closed mathematcal epressons can be made; when the path, however, passes through an heagonal block then the two sstems, ponted b arrows, onl a appromatel accurate converson could be made b transformaton plonomals. Twoletter abbrevatons n heagonal blocks show whch bnar data fle, contanng transformaton polnomals, have to be used to convert between the two neghbourng sstems (ther meanng n accord wth Fg. 1 s: ES = eov_szt.pol, EV = eov_vst.pol, EW = eov_wgs.pol, SV = szt_vtn.pol, GE = gak_eov.pol, GS = gak_szt.pol, GW = 7

gak_wgs.pol). When there s more than one path possble between an two sstems, the path s chosen along whch converson s more accurate. Transformed co-ordnates n dfferent formats are passed nto out.dat and trf.dat fles b program Vet.Ee. Olvas.Ee s a utlt program that serves to dspla (read) and prnt output fles. The content of the output fle out.dat can be eamned b ths program on the screen and t can also be prnted optonall.. SOFTWARE TESTING AND TESTS OF ACCURACY It was mentoned prevousl that t s possble to convert through closed mathematcal epressons between certan map projecton sstems. A concluson could have been drawn as a result of our test computatons that n these cases the accurac of computed plane coordnates s 1 mm and of geodetc co-ordnates s 0.0001". These conversons are referred to n Table 1 wth " + ", " (+) ", and "!+! " sgns or these sstems are connected b contnuous lnes (arrows) n Fg.. In all other cases when the transformaton path between an two sstems passes through an heagonal block (or blocks), the accurac of transformed co-ordnates depends, on one sde, how accuratel the control networks of these sstems ft nto each other; and on the other sde, how successful was the determnaton of transformaton polnomal coeffcents. It follows also from these facts that no matter how accuratel these transformaton polnomal coeffcents was determned, f the trangulaton networks of these two sstems do not ft nto each other accuratel snce there were measurement, adjustment and other errors durng ther establshment then certanl no converson of unlmted accurac can be performed (n other terms, onl such an accurate converson between two map projecton sstems s possble that the accurac allowed b the determnaton errors or dscrepances of these control networks). Ths fact, of course does not mean that ones should not be ver careful when the method of transformaton s selected or when the polnomal method s appled the coeffcents n Equaton (1) are determned. Our frst tests amed at the queston to decde whch one of the two methods: Helmert transformaton or polnomal method s more advantageous to be used. We arrved at the result that although the Helmert transformaton s computatonall more smple ts accurac n the majort of cases does not even appromate the accurac provded b the polnomal method. Snce a smple programmng can be a motve for onl software "begnners" therefore we took our stand frml on the sde of the use of polnomal method. When the polnomal method s chosen the net mportant queston s to determne the optmal degree of the polnomal. B consderng a smple wa of reasonng one could 8

arrve at the concluson that the hgher the degree of the polnomal the hgher the accurac of map projecton conversons wll be. On the contrar, t could be proved b our tests that the mamum accurac was resulted b applng fve degree polnomals. No matter whether the degree was decreased or ncreased, the accurac of transformed co-ordnates was lessened alke (more consderabl b decreasng, less consderabl b ncreasng). It s true, reall, that mnmum 1 common ponts are requred to determne coeffcents of a fve degree polnomal, but our eperences revealed the fact that the accurac of conversons can be ncreased further on b usng a consderabl greater amount of common ponts and the most probable values of these unknown polnomal coeffcents are determned through an adjustment. A documentaton fle, provded b the program Vetpol, conves some nformaton characterstc to the accurac of conversons b the polnomal method. Coeffcents of transformaton polnomals are frst provded b the program Vetpol based on co-ordnates of common ponts, and ', ' n sstems I and II, respectvel. Then, coordnates n sstem I are transformed nto co-ordnates t ', t ' n sstem II b usng these coeffcents and fnall the standard error characterstc to converson, n n ( t ' ') + ( t ' ') = 1 = 1 µ = n () wll be determned. For our gudance t could be mentoned that for eample, between the Budapest Stereographc and the EOV sstems the standard error s ±0. m from the epresson () for the complete area of Hungar when 1 common ponts are used and the same fgures are ±0.00 m, ±0.07 m and ±0.17 m between Budapest Ct Stereographc an EOV, EOV and WGS-8, and EOV and Gauss-Krüger sstems b usng, and 0 common ponts respectvel. Our eperences showed the fact that although the accurac can somewhat be ncreased b ncreasng the number of common ponts wthn the polnomal method but the accurac of converson can not be ncreased beond a certan lmt even wth ths method snce there s a dfference between the two trangulaton networks. In certan cases, however, an mprovement could be ganed when transformaton polnomal coeffcents are not determned for the complete area of the countr but for onl smaller sub-areas common ponts are gven and transformaton polnomal coeffcent are determned b program Vetpol. In such cases conversons, of course, must not be made outsde the sub-area where the coeffcents of transformaton polnomals were determned b program Vetpol. 9

It s worth of note that also heghts of ponts can be handled, when necessar, b the software. For eample when XYZ geocentrc co-ordnates, determned from GPS, are to be transformed nto an other sstem then besdes the transformed, projecton coordnates or ϕ, λ ellpsodal (geodetc) co-ordnates, also the h = H + N heghts above the WGS-8 ellpsod wll be resulted, where N denotes geod-ellpsod dstance.e. geod undulaton above WGS-8 ellpsod and H s the heght above geod (heght above sea level). So f the geod-ellpsod dstance n a certan pont s known there s also a possblt to determne heghts of practcal value b the GPS technque. Fnall we would lke to menton that b our software wth certan modfcatons one s able to convert between other map projecton sstems as well that are used n other countres. REFERENCES 1. Haza I. (196): Map projectons. Tankönvkadó, Budapest (n Hungaran). Rules for the Applcaton of Unfed Natonal Projecton (197). MÉM OFTH, Budapest (n Hungaran). VARGA J. (1981): New Methods of Converson Between our Projecton Sstems. Budapest, Techncal doctoral dssertaton (n Hungaran).. VARGA J. (198): Converson between the Unfed Natonal Projecton (EOV) and between our Former Projectons. Geodéza és Kartográfa No. (n Hungaran). VARGA J. (1986): Control Networks I. (Map projectons). Tankönvkadó, Budapest (n Hungaran) * * * Völges L, Tóth G, Varga J. (1996) Converson between Hungaran Map Projecton Sstems. Perodca Poltechnca Cv.Eng., Vo1.0, Nr.1, pp. 7-8. Dr. Lajos VÖLGYESI, Department of Geodes and Surveng, Budapest Unverst of Technolog and Economcs, H-11 Budapest, Hungar, Műegetem rkp.. Web: http://sc.fgt.bme.hu/volges E-mal: volges@ek.bme.hu 10