M athem ati cal El em ents of Ori enteeri ng M aps

. Mathematical elements of maps, in addition to the methods of presenting content with cartographic symbols and the criteria of cartographic generalization, are the most important elements of modern ground plan view maps. At the same time, on maps for orienteering disciplines, which are the most internationally unified maps in the world, the mathematical basis is not defined in very detailed specifications. In this paper, we describe the background and reasons for the apparent insignificance of mathematical content in orienteering maps, and on the example of the analysis of selected maps of Croatia and Slovenia, we determine in which period and to what extent mathematical elements were present on the maps.


Mathematical Elements of Maps
According to the still officially valid definition, a map is a symbolised representation of geographical reality, representing selected features or characteristics, resulting from the creative effort ofits author's execution of choices, and is designed for use when spatial relationships are ofprimary relevance (ICA 2003). To the extent that it should ensure the presentation of geographical reality and the corresponding mutual relations of the displayed objects, it must satisfy the appropriate positional correctness. Unambiguous positional definition of a geographical object can only be given both in nature and on a map with the help of mathematically defined coordinate systems (Ormeling, Kraak 2003;Robinson et al 1995). These differ from each other due to the diversity ofthe shape ofthe Earth (body of a mathematically indescribable shape) and the ground plan map (plane), while the mutual relations between them are determined by mapping the position of the object in nature on the map. The description of the location of the object on the map in the chosen plane coordinate system thus depends on determining the shape and size of the physical surface of the Earth, choosing a conditional mathematically describable (curved) surface, projecting this surface onto a plane and reducing it to the scale of the map ( Figure 1). All these are called mathematical elements of the map (Peterca 2001).

Orienteering Maps
Sports orienteering is a set ofsports disciplines that are managed and developed on an international scale by the International Orienteering Federation (hereinafter: the IOF). The disciplines of sports orienteering are foot orienteering, ski orienteering, mountain bike orienteering and trail orienteering. In the first three disciplines, the goal is for the competitor to complete the course oflegs between the control points in the shortest possible time, by navigating aided only by a map and compass, in the first case on foot, in the second on skis and in the third on a mountain bike. In the case oftrail orienteering, the goal is to recognize the control point correctly shown on the map on the ground with the help ofa compass. As with any sport, it is also necessary to ensure equal conditions for all competitors. Most of the definitions ofthe IOF regarding the map refer to the oldest and most widespread discipline, foot orienteering, but they also apply sensibly to other discipline.
From the competitors' point of view, a legible and accurate map is necessary for a qualified choice ofroute, enabling navigation along the route chosen to suit their navigational skills and physical abilities. However, skill in route choice and map reading loses all meaning ifthe map is not a good representation ofthe ground -ifit is of poor legibility, inaccurate or out-of-date. In the ideal case no competitor should gain an advantage or suffer a disadvantage because offaults on the map (IOF 2022).
The development of maps for orienteering always follows the needs ofthe sport development itselfand the progress of map production technology. In the initial period of orienteering, at the end of the 19th century, official state topographic maps available at the time were used, initially at smaller scales, e.g. 1:100,000 (Figure 2), while later more detailed maps at larger scales were available. After several decades, development led to the production of the maps specifically intended for orienteering. Today's modern maps for orienteering have significantly improved accuracy and precision while the most important is that they are much more readable (Figure 2). Special orienteering maps have never been made by cartographers, but by enthusiastic amateurs, mostly competitors in orienteering. Therefore, the primary criterion that map makers follow has always been suitability for the orienteering events, not theoretical cartographic principles. The main problem of the first specially made orienteering maps were that the cartographic symbols used on them differed greatly between regions and especially countries (Zentai 2007). In order to ensure equal competition conditions for all competitors, unification was necessary. Thus, in 1969 the IOF published the first version of the international specifications for orienteering maps -ISOM 1969, describing on two pages 45 map symbols in four colours (black, blue, brown, yellow). The specifications were mandatory for all international competitions and were intended to be used to create orienteering maps of all the varied terrains world-wide that are suitable for orienteering. In 1972, green was added as the fifth colour to show the runnability of the terrain and the set of signs was expanded. In 1975 the first comprehensive specifications brochure was published − ISOM 1975 (Figure 3), which, in addition to the description of cartographic symbols, also contained specifications regarding scale, contour interval, positional accuracy, level of generalization, content description and printing method. In the following years, the specifications were quickly adopted in orijentacijskog sporta su orijentacijsko trčanje, skijaška orijentacija, orijentacija brdskim biciklima i precizna orijentacija. U prvim trima granama cilj je da natjecatelj samo uz navigacijske vještine i uz pomoć kompasa i karte u što kraćem vremenu obiđe rutu između kontrolnih točaka, u jednom slučaju pješice, u drugom na skijama, a u trećoj na brdskom biciklu. U slučaju precizne orijentacije cilj je uz pomoć kompasa na terenu ispravno prepoznati kontrolnu točku prikazanu na karti. Kao i u svakom sportu, u granama sportske orijentacije potrebno je osigurati jednake uvjete za sve natjecatelje. Većina odredbi IOF-a vezanih uz kartu odnose se na najstariju i najrašireniju granu orijentacijskog sporta -orijentacijsko trčanje, ali se primjenjuju i na druge grane. Natjecatelju je u orijentacijskom trčanju neophodna čitka i dovoljno precizna karta za pouzdan izbor rute odabrane prema natjecateljevoj sposobnosti orijentacije i njegovoj fizičkoj spremnosti te praćenje terena uzduž rute. Naravno, sposobnost pravilnog odabira rute i čitanja karte gubi svaki smisao ako karta ne prikazuje pravilno teren, ako je nečitka, netočna ili zastarjela. Dobra karta je takva da se pri njenom korištenju niti jedan natjecatelj ne smije izgubiti ili dobiti vrijeme na stazi nauštrb neadekvatnog prikaza na karti (IOF 2022).
Izrada karata za orijentacijsko trčanje uvijek prati potrebe razvoja sportske grane i napredak tehnologije izrade karata. U početnom su razdoblju razvoja, krajem 19. stoljeća, korištene tada dostupne službene državne topografske karte, isprva u manjim mjerilima, npr. 1:100 000, a kasnije su bile dostupne i detaljnije u većim mjerilima. Nakon nekoliko desetljeća razvoja pokazala se potreba izrade posebnih karata za orijentacijsko trčanje. Današnje moderne orijentacijske karte imaju značajno bolju točnost, preciznost, a pogotovo je poboljšana njihova čitljivost (slika 2). most countries and started to be used for national and local competitions as well. Over the years, the sport itself has changed somewhat, new disciplines have been added, and above all, map production procedures and technologies were developed. For this reason, the specifications for the production oforienteering maps were also updated. The specifications from 1975 were followed by specifications in 1982, 1990and 2000(Zentai 2001). Today's, the latest version of the specifications from 2017 (ISOM 2017, Figure 3), prescribes the use of around 100 different cartographic symbols. For maps intended for other sport orienteering disciplines (mountain bike orienteering, ski orienteering and sprint maps, which are also used for trail orienteering), there are separate additional specifications, the basis of which is ISOM.
Due to all of the above, we can state with great certainty that maps for foot orienteering and other sports orienteering disciplines (hereinafter referred to as orienteering maps) are today the most internationally unified, i.e. standardized, maps on a global scale. The closest to such a large international level of standardization would be nautical charts, but their content is generally less complex and even between countries, deviations in colours are permissible.

Definition of Mathematical Elements in the Specifications of Orienteering Maps
As long as official national topographical maps were used for orienteering, mathematical elements were the default. The first maps were certainly completely default in terms ofcontent, but later they were quite likely used for additions and changes to the content, and most likely did not interfere with the mathematical elements. However, when the production of special orienteering maps began in the 1950s, a decision had to be made regarding the mathematical elements ofthe map, too. The first goal ofthe unification and description oforienteering maps was primarily the unification of the use of symbols to display objects and phenomena, therefore the ISOM 1969 specification contains only two pages of symbols (Zentai 2022). However, one ofthe signs, parallel black vertical lines, is described as "Grid Lines (Magnetic North)". In the first, more extensive ISOM 1975 specification, we already find descriptions that also define properties related to the mathematical elements of the map. Possible map scales of 1:20,000 or 1:15,000 and an equidistance of5 m are defined. In addition, it is stated: "North lines must be black 0.1 to 0.15 mm lines pointing to magnetic north. Their spacing on the map should represent 500 m on the ground..." Ofcourse, only the map, oriented to magnetic north allows the competitor, when using a compass, to directly transfer the bearing from the map to nature without taking into account magnetic declination, and in some cases also meridian convergence. We assume that the direction of the north lines towards magnetic north began to be used even before the first version of ISOM, in the 50s, when orienteering maps were first purposely produced, especially in areas of the world where the value ofmagnetic declination is significant. If we assume that when using a compass for navigation on the terrain, directions are determined with an accuracy of up to 2 degrees (Petrovič 2005), we can assume that with a declination value exceeding 2 degrees, the lag of the orientation of the map between orientation on the terrain and direction tracking at longer distances is already known. Unfortunately, research on the orientation of the first purpose-built orienteering maps is not available, and the number ofpreserved maps from that period is also not very large. It is even more difficult to determine whether magnetic north lines may have been additionally added to the existing topographic maps. Among others, Zentai (2007) dealt with the analysis ofthe first maps used for orienteering, but it is very difficult to find maps that were used in competitions. Partly because the competitors had to return them after the competition due to restricted access, but of course the paper was often too damaged during the first long competitions and was not properly preserved. We can assume, however, that course planners avoided adding additional direction lines to existing maps with a map grid, as the duality of vertical lines would have further worsened the readability of the maps and introduced confusion among the competitors.
ISOM 1982 prescribes only 1:15,000 as an acceptable map scale, but in addition to the foreseen equidistance of 5 m, it also allows a half smaller one, 2.5 m for flat areas. In addition to the basic 1:15,000, ISOM 2000 also allows a scale of 1:10,000, and later, for new disciplines, scales of1:7500, 1:5000, 1:4000 and 1:3000 are also used.
3. Definicija matematičkih elemenata u specifikacijama orijentacijskih karata Sve dok su se za orijentacijsko trčanje koristile službene državne topografske karte, matematički su elementi bili njima definirani. Prve su karte,svakako, i sadržajno bile u potpunosti preuzete, no kasnije su, vrlo Slika 3. Prikaz naslovnice prve knjižice iz 1 975. i aktualne knjižice međunarodnih specifikacija za orijentacijske karte (ISOM). Fig. 3 The cover of the first booklet from 1 975 and the current booklet of International Specifications for Orienteering Maps (ISOM). system, without any global or projection mathematical basis. The accuracy ofdistances and directions on a map often depended on the accuracy offield measurements with a compass and counting footsteps. Even with the first maps made on the basis of geolocated source data (topographic maps), the orienteering map makers often used a local coordinate system that was not shown anywhere on the map. With the beginnings of computerized map production, various CAD and graphic programs were used for the first orienteering maps, mainly OCAD and Adobe Illustrator, where the data had to be described in coordinates. Although the maps were generally already made on the basis of existing topographical maps with additional field checking, the data were imported in the digital environment by digitizing or vectorizing scanned field sketches in local coordinates of the graphical interface. Only by preparing the basis ofthe map in the computer environment based on the combination of various source data in digital form, for example topographic maps, orthophoto, vector data oftopographic bases, the creation oforienteering maps in the computer environment is introduced in the actual projection coordinates ofthe sources.
Of all the mathematical elements on orienteering maps, in accordance with the specifications of the IOF, only the map scale, equidistance and the mandatory orientation ofthe sides ofthe map and the north lines in the direction ofmagnetic north are prescribed. The required positional accuracy is described very loosely, both horizontal and vertical, only appropriate relative ratios are prescribed, in which competitors must not notice discrepancies. The absolute accuracy of orienteering maps as well as their georeferencing are unnecessary for the competitor, but they are recommended when using the map in connection with GNSS, but only during production, media monitoring of the competitor or for subsequent analysis.
In the article, we try to analyse how orienteering maps in Croatia and Slovenia followed these requirements and specifications.
Orienteering in Croatia developed from team mountaineering competitions, in which orientation was also an important (but not the only) element of knowledge. These were organized in the entire area of the former SFRY from the 1950s onwards. The Mountaineering Association ofYugoslavia was present in 1959 at the preparatory meeting for the establishment of the International Orienteering Federation (IOF), but two positioned with sufficient accuracy to ensure that a competitor using compass and pacing will perceive no discrepancy between map and ground. In general, ifthe distance between neighbouring features deviates less than 5% this will satisfy accuracy requirements.
Absolute height accuracy is of little significance on an orienteering map. On the other hand, it is important that the map shows as correctly as possible the relative height difference between neighbouring features. Accurate representation of shape is of great importance for the orienteer, because a correct, detailed and sometimes exaggerated picture of the land form is an essential precondition for map reading. However, the inclusion of a lot of small detail must not disguise the overall shapes. Drawing accuracy is of primary importance to any map user because it is closely connected with the reliability ofthe final map.
Grafička točnost (npr. jednakost debljina crta znakova, jednakost veličina znakova) je stupanj do kojeg se slijede odobrene specifikacije. Točnost crtanja je od primarne važnosti za svakog korisnika karte jer je usko povezana s pouzdanošću konačne karte." Od specifikacija ISOM 2000 dalje dodatno se navodi: "Apsolutna položajna točnost karte (koordinatna točnost) također je važna ako se orijentacijska karta koristi sa sustavom za pozicioniranje ili zajedno sa skupovima geografskih podataka iz drugih izvora. U takvim slučajevima također mora biti moguće transformirati kartu u dobro poznati geografski sustav." U ISOM 2017 dodane su rečenice: "Čitljivost je uvijek važnija od apsolutne točnosti, pa kartograf može pomicati pojedine objekte na karti ako to omogućuje bolju čitljivost karte." Konačno, u najnovijim specifikacijama ISOM 2017 postoji i ova dodatna odredba: "Georeferenciranje karte znači da je karta koordinatno postavljena u odabrani years later, when the IOF was officially established, it was not among its founding members. Despite this, already in 1963, the first individual orienteering competition based on the Scandinavian model (without additional tasks) was organized in Croatia. In the following years, these competitions were supplemented by the still existing team mountain orienteering competitions, which also contained fewer additional tasks. At the beginning of the 1970s, there was a growing resistance to individual orienteering competitions in the mountaineering community, as they did not encourage other traditional mountaineering skills, neglected the important principle of cooperation and solidarity between team members, and encouraged "rambling" on mountain trails and mountains, which is dangerous and against the mountaineering codex. These views slowed down the development of orienteering for a few years, but at the end of the 1970s it prevailed as a completely unique sport, separated from traditional mountain orienteering. In 1981, the first orienteering club in Croatia, OK Maksimir, was founded. In 1982, the Mountaineering Association of Yugoslavia also joined the IOF (OKV2 2022). The Croatian Orienteering Association, founded on 22 December 2005 in Zagreb is today the umbrella organization for orienteering sports in Croatia, as well as the representative body ofCroatia in the IOF (HOO 2022).
The expansion of development and the increasing number of contacts with foreign competitions, and through them also with special orienteering maps, brought the desire to prepare orienteering maps in accordance with the IOF specifications in Croatia as well. The production ofthe first orienteering maps for sports orienteering was undertaken by experienced competitors, who knew best the characteristics and needs ofthe sports and maps, as elsewhere in the world. Based on the available sources, there may be a bit ofa dilemma as to which was the first orienteering map in Croatia, since around the same time, several authors started making maps in different areas. According to OKV1 2022 and OK2 2022, in 1980, Zlatko Smerke made the orienteering map of Trakošćan, the first five coloured orienteering maps based on IOF specifications in Croatia (Figure 4).
In the 1990s, the creation oforienteering maps with the help of computer programs gave a new impetus to the production of orienteering maps, which simplified the restoration ofcontent and preparation for printing, while improving the graphic quality at the same time (Frangeš et al. 2002). Maps were increasingly produced by professional draftsmen from various parts ofEurope. The demanding and interesting terrains, also in connection with vacationing on the Adriatic, have attracted many competitors from all over the world to competitions in Croatia for years, and many maps of different areas have been produced for the needs ofthe competitions. In addition to maps for foot orienteering, quite a few maps for mountain bike orienteering and maps for trail orienteering have been produced. Unfortunately, there is no uniform list of orienteering maps produced in the territory of Croatia. The maps are in the owned and kept by individual clubs, and we can only assume the total number ofmaps produced.

Development Orienteering and Orienteering Maps in Slovenia
The development oforienteering in Slovenia started a little later than in Croatia, but generally followed a similar pattern (OZS 2015). The first individual orienteering competition in Slovenia was only held in 1974, but in the second half of the 70s and further into the 80s, some Slovenian beginners in orienteering took part in already established competitions in Croatia. What was happening regarding the creation and development of orienteering maps is better known. In 1982, on the slopes of Mrzlica above Trbovlje, around the Podmeja village, Roman Sladič made a three-colour map for orienteering, which, due to the number of colours used, did not fully follow the ISOM ofthe time, but was made as especially for orienteering. The first five-colour map according to ISOM, the map ofRožnik in Ljubljana was made in 1987 by Serbian orienteers Mile Stevanović and Slobodan Radovanović, while the part covering Tivoli was added by one of the pioneers of sports orienteering in Slovenia, Bojan Jevševar ( Figure 5).
The first computer-made maps were made by Daniel Lebar, Boris Bauman and Dušan Petrovič at the end of 1992 and at the beginning of1993 (Petrovič 2005) using OCAD 3, then still a DOS program. Later map quality increased with the addition oforthophotos, GNSS devices, laser distance and angle measurements, and in the last decade mainly by using data from the laser scanning of Slovenia, which was available for the entire area of Slovenia in 2015. By the middle of2022, a total ofaround 900 orienteering maps had been produced in Slovenia, 410 in accordance with the basic ISOM, 377 maps for sprints (these are also used for trail orienteering), 25 for mountain bike orienteering, 12 for ski orienteering and an additional 69 school orienteering maps (OZS 2022). These are produced on a larger scale and are not internationally unified, only a proposed set of cartographic symbols is prepared.
In the analysis, we were interested in the extent to which various mathematical elements were used in the produced orienteering maps in Croatia and Slovenia in a certain period, both those that are prescribed and mandatory by the specifications (otherwise only for the use of maps for international competitions) and those that are only recommended in the specifications. The first mandatory provision concerns the orientation of the map; since the very first specifications from 1969, the vertical lines on the map should point in the direction of magnetic north. In 1990, the specification on the north lines was supplemented by the fact that the magnetic north lines must be parallel to the trimmed sides ofthe map paper. The second provision refers to the map scale, which is also prescribed, but in certain cases deviation is possible. This is especially true in cases ofhighly detailed terrains, where with an excessive degree of generalization, the possibility ofshowing details on the terrain would be lost, and therefore the mapmakers, in agreement with the organizers, sometimes decide on a larger scale. However, in the case of the oldest maps, made only on the basis ofimprecise field measurements with a compass and step counting, the correctness of determining the actual scale ofthe map is questionable.
Positional accuracy is mentioned very loosely in the specifications, but the competitor should not notice positional deviations. In the analysis, we therefore looked for possible significant positional displacements of the content ofthe analysed maps in relation to the comparative source data.
The last mentioned and recommended feature is the georeferencing ofthe map in the selected projection coordinate system. In addition to helping mapmakers to combine different source data, this is important for users in subsequent analysis with the help ofGNSS tracks or in following athletes equipped with trackers during the actual competitions, for example through the media.
U specifikacijama se vrlo slobodno spominje pozicijska točnost, tek toliko da natjecatelj ne bi trebao niti We checked all four mentioned properties on the selected orienteering maps of Croatia and Slovenia. For the area of Croatia, we only had available maps in printed form or their scanned version intended for the competitor. From this we were able to determine the first three properties, but not the georeferencing, since the coordinates are never indicated on the printed map, even if the digital map is georeferenced in the selected projection coordinate system. For the maps ofSlovenia, we had access to data in digital format for all maps produced after 1993, from which, in addition to the first three properties, we could also determine whether the map was georeferenced in a selected global or national coordinate system.
For maps available only in printed form or in the form of a raster image, we compared them with the content of the topographic map 1:25,000 (TK 25), namely with the raster files of sheets from the period before the year 2000. All these sheets cover the area of Slovenia and Croatia. They contain a rectangular 1 km grid of the Gauss-Krueger map projections, calculated from ellipsoidal coordinates determined on the Bessel 1841 ellipsoid. The map grid is directed towards the projection north, but the meridian convergence value is given for each sheet. The positional accuracy of the TK 25 is estimated to be up to 15 m (Peterca et al. 1974), depending on the type ofthe elements displayed.
The analysis process was carried out by importing the raster image of each orienteering map into the file in which we had the georeferenced TK 25 raster sheets and aligning it in the direction ofthe vertical lines that indicate north. We then attached the imported orienteering map to TK 25 at one ofthe well-recognizable points and checked the direction deviation at another recognizable point as far away as possible. With this, we determined a possible rotation ofthe orienteering map, considering meridian convergence and magnetic declination. We also compared the distance between the same recognizable points and thus checked the adequacy ofthe map scale. Furthermore, by reviewing the entire content, we looked for positional deviations of recognizable objects and thus evaluated the positional accuracy. For maps available in digital vector format, we imported TK 25 as a background. In cases when the orienteering map was not georeferenced, we simply aligned it with the background, and then checked the direction, possible scale deviation and maximum positional displacements in the same way.
In the analysis, we have included some breakaway orienteering maps for the area ofCroatia ( Figure 6). The Ponikva map from 1984 belongs to the group ofthe first Croatian orienteering maps, made mainly on the basis of field measurements with a compass and by counting steps. The map ofSolarske Šume, made only three years later, was already made on the basis of aerial photographs from 1977. Both maps were still made entirely analogically, by drawing with ink on plastic foils. The Jurakovo map is made according to the specifications for mountain bike orienteering maps in 2004 with an upgrade ofthe previous orienteering map, originally made in 1982 and reworked in digital format and supplemented in 2002. The Dolje map is also made according to the specifications for mountain bike orienteering maps with an upgrade of the older orienteering map by Swedish cartographers from 1984. The Sestvetski Kraljevac map from 2021, unlike the previous ones, is one of the latest maps, made on the basis of all available topographical and other sources with the help of computer software tools and used in an international competition, the South-Eastern Europe Orienteering Championship.
For the sake of a more complete review of the state of orienteering maps and easier accessibility we included several maps for the area ofSlovenia in the analysis. In addition to the two oldest, three-colour maps of Podmeja from 1982 and the five-colour map of Rožnik from 1987, we have included two more from the era of analogue map production, the Udin Boršt map from 1989, which was an example ofa superbly produced orienteering map ofthat time, and the Črni vrh map from 1991, which was made in the area of a very unreliable topographical base, an outdated TTN 10 sheet. In the same area in Črni Vrh, in 1999, a map of Predgriže was produced, this time already in the OCAD computer SW, but it was analysed in the same way as the analogue maps with the aim ofmaking the comparison as appropriate as possible (Figure 7). For the rest ofthe maps of Slovenia, available in digital format, we only checked the possible rotation from the descriptive properties of the file in relation to the projection coordinate grid and checked the geolocation.

Rezultati analize i rasprava
Rezultati usporedbe svih deset karata prikazani su u tablici 1. Analiza primjerenosti mjerila karte pokazuje da mjerilo značajno odstupa od nazivnog samo za karte Dolje (5% preveliko, 1:9500 umjesto 1:10 000) i Podmeja (7,5% premalo). Za sve je ostale razmatrane karte detektirana razlika manja od 12,5 m / 1 km, što je grafička točnost usporedne karte TK 25. S obzirom na to da se radi o dvjema najstarijim analiziranim kartama, izrađenima na understood. Experienced competitors might be able to notice such an error when estimating the distance on the field, and mostly on the terrains where there are not many other details, and it is really necessary to estimate the length by counting steps at longer distances. Similar deviations would be expected in some other older, analogously produced orienteering maps, such as Ponikve or Črni Vrh, but apparently the authors of the maps used existing topographic maps as a basis for adopting the basic map skeleton. In the case ofthe Dolje map, one might expect that the scale error was eliminated during the conversion into digital form, but apparently the authors only digitized the existing map.
The analysis ofthe orientation ofthe discussed maps shows that, in most orienteering maps, the north lines are aligned with the map grid ofthe comparative TK 25, i.e. the projection north. Perceived differences of less than 0.5°are smaller than the accuracy threshold ofthe angle measurement on the analogue map. It could be concluded from this that the mentioned maps are not directed towards magnetic north in accordance with the specifications. However, additional thinking shows that this is not necessarily the case. The value of the magnetic declination in the area ofCroatia and Slovenia has only in recent decades reached a value that is perceptible when measuring angles. Table 2 shows the magnetic declination values for Zagreb and Ljubljana for the period from 1980 to 2020 in ten-year intervals.
In addition to the declination, the deviation between the magnetic and projection north must also include the meridian convergence value, which for the Zagreb area in the TK 25 map projection is around 1°, and for the Ljubljana area around −0.5°. Thus, we can see that in the 80s, the possible shift of the north lines on the orienteering maps in relation to the rectangular grid ofthe cartographic projection used would not have reached the accuracy of the angle measurements on the map, and for the following two decades the accuracy of determining the direction in nature with a compass, as used in orienteering. That's why the authors of the orienteering maps didn't even bother with possible map rotation, which appears only in the last two decades. From the analysis, this can be seen on the Sesvetski Kraljevac map from 2021, where the measured offset is 4°. With Slovenian maps in digital form, the use ofrotation appears in individual maps produced after 2005, but at the same time, in many maps produced in recent years, the rotation is not implemented.
The relative positional accuracy, which must not exceed the value noticeable to the competitor when used in the field, is difficult to consistently evaluate with established methods and ways of assessing the quality of maps (Jobst 2006). What competitors notice on the field as inaccuracy depends on very different factors, such as the density of objects, their diversity, height relationships, visibility on the terrain, etc. Therefore, instead of an exact analysis, we visually checked the actual maximum detected absolute deviations ofthe presentations of elements on the map. Since we used TK 25 as the background for comparison, we can only evaluate deviations greater than 15 m as reliable. In any case, before the evaluation, we corrected the detected scale error in the previously mentioned maps. Perceived deviations of up to 50 m were detected only on the map of Ponikve (Figure 8), and even then, mainly in the reliefelements, which are difficult to determine in position.
The last mathematical element oforienteering maps defined by the specifications is their georeferencing. As expected, the first digitally produced maps were still in the local coordinate system with an origin point in the map area. The first georeferenced maps appeared at the end ofthe 90s, but not because ofa different use ofcartographic resources, but because this was made possible by the then new version of the OCAD software tool, which is almost exclusively used for the production of orienteering maps in Croatia and Slovenia. Since the mentioned period of the end of the 90s, all the maps produced have been georeferenced, but due to the change of national coordinate systems in Croatia and Slovenia, their coordinate base is not the same everywhere. It will certainly make sense in the future to transform the coordinates of all georeferenced maps into the current national coordinate systems of both countries. Cartographic software tools enable such transformation, but the user must understand and know the state ofthe national coordinate systems in order to perform the appropriate conversion in the tool. Also, some digitally produced maps from the first period were subsequently georeferenced, but often with a not very reliable method ofalignment to a selected number of recognized identical points on the map and background.

Conclusion
As a rule, mathematical elements represent the most exact and unequivocally definable property of maps (Field 2018). This is not the case for orienteering maps, despite their highest level ofstandardization on a global scale. There are several reasons for this. First of all, orienteering maps are not made by cartographic experts, but by the competitors themselves, who often have no knowledge of the geodetic basis and mathematical elements ofthe map. Despite all the technological Slika 7. Slovenske orijentacijske karte uključene u analizu, uz karte sa slike 5 (vlasnici OK Tivoli i OK Azimut). progress in all disciplines and forms of orienteering sports, orienteering maps are still used in analogue form, printed on paper, and when using them in the field, competitors can only use a compass to determine the direction and count steps for determining distances. Any coordinates are completely irrelevant for them, reading global coordinates from the map could even be problematic, because competitors could help themselves by using GNSS devices, which are prohibited by the rules. Also, that's the reason why the ISOM prescribes as mandatory mathematical elements only the map scale and its orientation towards magnetic north. Thus, orienteering maps are the perfect example ofhow user needs are decisive on maps. These maps are also so different from the usual topographic and thematic maps that only the users themselves can create highquality ones.
Despite the fact that the ISOM rules are binding only for international competitions, due to the adaptation of the competitors to the maps used in international competitions, they are also used in the vast majority on maps intended for national or local use. The analysis of the selected orienteering maps of Croatia and Slovenia thus showed that deviations from the specifications were present only in some ofthe earliest maps. Orientation of orienteering maps towards magnetic north became necessary in our area only after the year 2000, so understandably maps produced before this period are not rotated. However, we have detected orientational inconsistency even in some later produced orienteering maps, such as the analysed upgraded maps for mountain bike orienteering in Croatia or many ofthe digitally produced maps in the last two decades in Slovenia. The positional accuracy of all analysed maps is generally good considering the requirements and needs of navigation in the field, but for a reliable assessment, a much more extensive analysis ofa carefully planned sample of objects or even the entire content should be performed. However, the problem of absolute accuracy, together with the problem of inadequate or imprecise georeferencing, is becoming increasingly apparent in other uses of orienteering maps − in subsequent analysis or realtime competitor tracking media shows, where significant deviations from the position ofthe competitor occur temelju nepoznatih izvora s terenskim mjerenjem pomoću brojanja koraka i kompasa, razlika je razumljiva. Takvu bi grešku kod procjene udaljenosti na terenu možda mogli primijetiti iskusni natjecatelji, a i to uglavnom na terenima gdje nema puno drugih detalja pa je stvarno potrebno procjenjivanje duljine brojanjem koraka na većim udaljenostima. Slična bi se odstupanja očekivala i kod nekih drugih, starijih, analogno izrađenih orijentacijskih karata, poput Ponikva ili Črni vrh, no vjerojatno su autori karata koristili postojeće topografske karte kao osnovu za usvajanje osnovnog kostura karte. U slučaju karte Dolje moglo bi se očekivati da je pogreška mjerila otklonjena prilikom pretvorbe u digitalni oblik, no očito su autori samo digitalizirali postojeću kartu. Analiza orijentacije razmatranih karata pokazuje da su kod većine orijentacijskih karata sjeverne linije usklađene s pravokutnom mrežom usporedne TK 25, odnosno projekcijskim sjeverom. Razlike manje od 0,5°su ispod praga točnosti mjerenja kuta na analognoj karti. Iz toga bi se moglo zaključiti da navedene karte nisu usmjerene prema magnetskom sjeveru u skladu sa specifikacijama. Međutim, promišljanje pokazuje da to nije nužno tako. Vrijednost magnetske deklinacije na području Hrvatske i Slovenije tek je u posljednjim desetljećima dosegla vrijednost uočljivu pri mjerenju kutova. Tablica 2 prikazuje vrijednosti magnetske deklinacije za Zagreb i Ljubljanu za razdoblje od 1980. do 2020. godine u desetogodišnjim intervalima.
Relativnu položajnu točnost, koja ne smije prelaziti vrijednost koju natjecatelj primjećuje pri uporabi na terenu, teško je dosljedno vrednovati utvrđenim metodama i načinima ocjenjivanja kvalitete karata (Jobst 2006  due to the limited absolute accuracy ofthe orienteering map, but also limited precision determined by GNSS. Perhaps in the future we can also expect a technological upgrade oforienteering sports in the area ofmap use. Due to their precision and up-to-datedness, orienteering maps are not only useful for orienteering sports competitions. Maps on digital paper, monitoring a competitor or anyone else on the field with augmented reality methods, using multimedia glasses for technique training and more will certainly require unambiguously defined mathematical elements of the map, which will enable the synthesis of various data, sensors and displays. The performed analysis will thus provide map owners with a cursory assessment of the condition of their maps and the necessary upgrades for various possible future forms ofuse.