Optimum deformasyonlu kartografik projeksiyonların Türkiye için araştırılması

Abstract

Kıtaların şekil ve büyüklüklerini veya yeryüzündeki herhangi iki noktanın birbirine göre olan durumunu deformasyonsuz olarak harita düzlemine aktaran herhangi bir projeksiyon türü sözkonusu değildir. Bir kartograf hazırlanacak harita için uygun projeksiyon seçimi ile karşılaştığında, haritanın hangi amaç için kullanılacağını, hangi özelliklerin arandığım ve diğer istekleri gözönünde bulundurması gerekir. Ancak bu düşüncelerden sonra, projeksiyonu yapılan bölge için bu koşullardan bazılarını yerine getiren en uygun projeksiyon yöntemi seçilir. Harita projeksiyonları, (bu çalışmada optimum deformasyonlu haritaların elde edilmesi araştırmaları, yani; projeksiyon denklemlerinin bulunması ve haritaların çizilme işi "harita projeksiyonu" olarak adlandırılacaktır) alan koruma, açı koruma veya ortalayıcı olma gibi farklı özelliklere sahiptirler. Bu farklı özellikler haritaların çeşitli kullanım amaçlan için az veya çok daha uygun olmaları sonucunu doğurur. Yeryuvarı yüzeyinin herhangi bir projeksiyon yöntemine göre bir düzleme veya başka bir yüzeye aktarılması istenirse, haritadaki uzunluk, açı, alan v.b.elemanlar asıllarına göre deformasyona uğrarlar. Haritada projeksiyonu sözkonusu olan yüzeyin değişik elemanlarının deformasyonlarının bilinmesi, kullanılan projeksiyon türünün ve dolayısıyla haritanın değerlendirilmesi ve haritadan aranan büyüklüklerin doğru değerlerinin bulunması açısından önemlidir. Bu çalışma beş bölümden oluşmaktadır. Giriş bölümünde harita projeksiyonu ve önemi hakkında genel bilgiler verilmiştir. İkinci bölümde, dünyanın şekli, referans yüzeyleri ve temel kavramlar konusunda açıklayıcı detaylı bilgiler verilmiştir. Üçüncü bölümde, projeksiyonların değişik türleri, (konik projeksiyon, silindirik projeksiyon ve düzlem projeksiyon) ele alınarak bu projeksiyonların genel özellikleri açıklanmış ve değişik türleri için, projeksiyon denklemleri ve deformasyon katsayıları verilmiştir. Dördüncü bölümde, optimum deformasyonlu harita projeksiyonları konusu ele alınarak, nokta-uzunluk deformasyon kriterleri ve ortalama uzunluk deformasyon kriterleri verilmiş ve optimum deformasyonlu harita projeksiyonlarının araştırılmasından, seçilen deformasyon kriterlerinden birini minimum yapan projeksiyon parametrelerinin varyasyonlar hesabından yararlanılarak elde edilebileceği gösterilmiştir. Beşinci bölümde, Türkiye'nin içinde bulunduğu paralel daire kuşağı için, değişik projeksiyon türlerine göre en uygun ortalama uzunluk deformasyon kriterleri ve diferansiyel anlamdaki deformasyon katsayıları hesaplanmış, bunlara bağlı olarak çizilen deformasyon elipsleri verilmiştir. Sonuç ve öneriler bölümünde ise, ortalama uzunluk deformasyon kriterlerinin ve hesaplanan diferansiyel anlamdaki deformasyon değerlerinin karşılaştırılmasından, ortalama uzunluk deformasyon kriterlerinden birinin minimum olması yöntemlerden biri olarak ortaya çıkmış, böylece elde edilen bir kritere göre Türkiye için en uygun projeksiyon önerilmiştir.

The most advantageous projections are those which most minimize distortions, i.e., ideal projections and the best projections. In practice the number of most advantageous projections available to cartographers is very modest. The best include Chebyshev projections ( as the best of the conformal projections ) and a number of others that differ in scope, and whose use is dictated by the conditions in the area to be mapped : the best Postel projection ( azimuthal ), the Lambert projection ( equal - area ), the Tissot projection ( conic equal - area ), the conic projections of Singer and Kavrajskij, and a variant of the best of the Eulerian projections No ideal projection is known. Although cartographers have a high opinion of Chebyshev projections and recommend their extensive use ( Kavrajskij, Urmayev ), they also deny any need to search for the most advantageous projections, which they see as an exercise of value only to establish which of the known projections is nearest to an ideal projection (Kavrajskij ). The conclusion is that there are now two alternatives in cartography, namely either to accept or reject the need to search for most advantageous projections. Although one can, of course, get by without most advantageous projections in practice, it should be the object of cartography, as of any science, to find the best and most rigorously founded solution of the question, and then to incorporate it in practical instructions We therefore reject the negative approach, and consider that the question cannot be finally answered until the most advantageous projections are available for practical use. Moreover, practical considerations should influence the formulation of the problem by indicating the utility of the various projections, since their multiplicity of figures will not give rise to a unique solution. This is quite understandable, since the problem is, in terms of analysis, one of approximating a function of two variables, for which there can be no unique formulation even at the stage of detailed formulation. Since, however, this latter circumstance does not prevent intensive development of the theory of the approximation of a function, we conclude that the problem of the most advantageous projections should be investigated in its entirety in the light oh the theory of the approximation of functions. Projections of a sphere onto a plane should be considered for the practical purposes of cartography, but it is also possible in wider formulation of the problem to investigate the depiction of one regular arbitrary surface one another, especially because, first, it is very simple to proceed from this to the case of depiction of a XI sphere on a plane and, second, the given case, for all its frequency and apparent simplicity, does not involve any significant simplifications when compared with the general case of depiction of arbitrary surfaces. In representation of a domain S of a regular surface 5, ds2 = L2 (u,v)\du2 +dv2] on a domain D of a surface iS2 dS2 = Ll(x,y)[dx2+dy2] effected by the functions x = x(u,v), y = y(u,v) In construction of the most advantageous representations of the given domain S of the surface Sx on some domain D of the surface 52, the above function must be sought with the condition that the scale of representation m = m (u, v) in the domain S should be close to unity. How then are we to understand to closeness of m to unity, or how are we to determine the above function in the domain S so that (m-1) or İn m should tend most advantageously to zero in S? Unambiguous response to this question is not possible because of the variety of concepts of distance in metric spaces. We shall consider two possibilities leading to the best and to the mean square approximations, and shall deal in this connection with the most advantageous representations of the minimax type and the most advantageous representations of the variational type. Although it may be accepted that a procedure has been developed mathematical formulation of the problem remains unclear for ideal representations of the minimax type. This study is devoted to ideal and best projections of the variational type. Before considering these projections, let us examine the measures of line distortions that will be needed subsequently : these measures have been excellently dealt with by Kavrajskij. When considering measures of the distortion of lengths around a given point of the representations (Sl on S2 ) along a given direction the following are usually selected d = m-l 4=1-- m d2 =lnm xn d,=~{m2-\) Each of these quantities may readily be expressed in terms of any of the other quantities, and they differ only in small quantities of the second and higher orders. When the territories involved are not very large it is almost a matter of indifference which of these quantities is taken as the measure of length distortion at a given point in a given direction when calculating the most advantageous projections ; at least this is true for territory on the earth's surface. The above quantities will most simplify the calculations may therefore be used. Turning now the general measure of length distortion in all directions at a given point of a representation, let us consider the following measures of distortion. Airy Airy - Kavrajskij ^=|[(lna)2+(ln6)2] Jordan In I Where a is the azimuth of a direction on the surface Sx represented on the surface S2. Jordan - Kavrajskij, 2x In v In addition to these, other measures of lengths in all directions at a given point of a representation have been proposed Fiorini 4=(ai>-l)2+£-l)2 b The distortion measure initially adopted by Airy in his investigations. This measure has been the subject of thorough investigation by A. K. Malovichko ; we owe its general expression to A. Klingach Xlll pl(a.b-l) + p2£-l) b P1+P2 Where px and p2 are the weights of distortions of angles ( forms ) and areas at a given point of a projection. Finally, it should be noted that all of the measures of distortion of lengths in all directions at a given point are mean square distortion. The measures of the distortion of lengths within the entire territory to be depicted, i.e., to criteria of the utility of projections. From the point of view of criterion of the minimax type or, to be more precise, the Chebyshev criterion ( corresponding to uniform or best approximations of the functions ) the most advantageous projections is that for which the ratio of the greatest value of the scale m^ to the least value mmin is minimum within the limits of the region to be depicted. An ideal projection of the minimax type is therefore a rrr projection of S cz S, on Da.S2 for which m = - deviates least from unity within ds the limits of the region depicted ; or, since b < m < a, for which the extreme scales a and b deviate least from unity. From the point of view of a criterion of the variational type ( corresponding in the great majority of instances to power mean square approximations-in terms of the theory of approximation of functions ) the most advantageous projection of an assigned region Sc^on to another surface S2 is that for which one of the following functions E assumes the least value : Airy E2A-\\e\ds Airy - Kavrajskij ss Jordan 1 lit E1J=^~\(,\im-\fda)dS S2n s 0 Jordan - Kavrajskij xiv E2x= - l(l(hım)2da)dS Sin s Each of these functions expresses one or other of the criteria of the utility of projections. We shall apply the term ideal projections of some given region 5" c Sx on to a surface S2 to projections satisfying one of the criteria indicated, ascribing them to a definite type minimax or variational. But whereas there is as yet only one condition for determination of the projection corresponding to Chebyshev's criterion for ideal projections of the minimax type, ideal representations of the variational type are ambiguously defined because of the existence of a number of criteria, all of which are logically based, whose use for the same region SczS^ represented on S2 will obviously lead to different results. There is no basis for according preference to any one of these criteria. We may now formulate the general problems whose solution would permit the construction of ideal projections. A. Solution of the following mathematical problem is required to obtain ideal projections of the minimax type : it is required to find in S c Sx the functions for which the extreme scales a and b. B. To construct ideal projections of the variational type we must solve variational problems involving minimization of the functions E already indicated. Where as the first problem A is one not previously raised in mathematics, but one which must first be solved if attempts to construct ideal projections of the minimax type are to be successful, the procedure for solution of variational problems of type B is known in principle, but great difficulties are encountered in its implementation.