数学在地图学中的应用（MathematicalapplicationinCartography）

数学在地图学中的应用(Mathematical application inCartography)Mathematical applicat ion in Cartog raphyZhong Yexun 1,2 Hu Baoqing 1 Joe Junjun 3(1, Guangxi Normal University， Key Laboratory of Ministxy of education and the evolution of resource environment in Beibu Gulf la by Col lege of resources and environment science lb Nanning, 530001 ; 2, the Guangxi Bureau of Surveying a nd mapping, Nanning, 530023; 3, WuhanUniversity Institute of Surveying and mapping, Wuhan, 430079)Abstract: the map object is in tho geographical space of natural phenomena and social economic phenomenon, the representation of objects bet ween the qua ntita tive re la tions and spatial forms in the objective, quantitative relations and spatial form as the research object of mathematics, and closely related to cartography. This paper discusses the topology and function theory, algebra, geometry, calculus, graph theory, set theory, application of probability theory and mathematical statistics, fractai geometry and fuzzy mathematics in cartography, mathematical cartography application of various mathematical tools and methods makes a brief introduction. That mathematics is widely used in cartography, mathematics plays an important role in promoting the development of cartography.Keywords: mathematics; algebra; topology; calculus; set theory; Cartography; app1icationMathematics is a discipline that studies quantitertive relations and spatisl forms in the real world 1 Map representation and object is reflected in lhe geographical space of natural phenomena and social economic phenomcnon, which is the earth atmosphere, hydrosphere, biosphere, 1ithosphere and pedosphere interaction within the area of things 2. The space between entities to learn the quantitative relations and spatial forms of objective existence, determines there is a close relation between mathematics and cartographyAccording to the mathematical application in cartography, respectively discusses the topology and function theory, geometry, calculus etc. 1 topology and function theoryMap projection is the mathematical foundation of a map. Map projection is the function of estab 1ishing points on a plane (represented by plane rectangular coordinates or polar coordinates) and points on the earths surface (represented by atitude and longi tude)(1)Different ones, which determine different specific map projections, 3.A map project ion transfonnation is defined as a topological transformation between two two-dimensional fields If the curve coordinate with the surface of the earth is cut open, the two-dimensional field, then, a special case of 4 map projection and inverse projection transform is. The so-called topological transformotion is a neither tearing nor kneading, but allows the expansion and bonding of the graph transform 5. The two maps of South America, shown in Figure 1, represent the meaning of the topological transformation intuitively.According to the mathematical definition of the network topology, can coordinate mathematical definition of the 6 nctwork, the network, road network and other notwork map derived in cartography.Depende nt variables are fun cl ions of in depe ndent variables 7(1) in the formula, x, y are changed by given values, and X, y is a function. Get ting Xs Fl and getting Ys F2 are two d i fferentuncli ons. Correspondenee, mappi ng, and transformation are synonyms for function 8Map symbols are the language of maps. A map symbol is essentially a plane image of a drawing object under three topological maps. These three topologies are: 3D space X to earth ellipsoid, S mapping f: X = S,The mapping of ellipsoid S to cartographic cognitive structure G: S YProject source: supported by the National Natural Science Foundation of China (4087125040661005); special education support plan for the new century of the Ministry of Education (NCET-06-0760) Guangxi Natural Science SectionAcademic fund key project (0832021Z)Author: bell Yexun (1939-), male, Professor, research direction: Cartography theory E-mai1:gxzyxun163. comThe mapping of Y and Y to 2D planar Q: Y = Z = Z. Set X as the drawing object within the drawing area A, thenThe project!on of the el 1 ipsoid, the knowledge of X and f (x) for the figure, exists in the cognitive structure of the mapper in conceptual form Y. Map symbols Mappist. cn according to the attribute of thematic maps of selected x, X and QGF through the subjective assuranee operation (x) corresponding to 9Figure 1 the shape of South America in two different projectionsFigForm, of, South, America, in, Two, Different, Projection2 GeometryThe famous five set (the parallel axiom) geometry system deduce, called Euclidean geometry. Perspective azimuth project!on is the traditional method of map projection using Euclidean geometry. According to the viewpoint and perspective azimuthal projection, the earth center dis tan ce, and can be divided into positive projection (point at infinily), exterrml projection (located in the view sphere at a finite distance), spherical projection (the viewpoint on the earth，s surface and center of project ion (view) in the center of the earth).The double azimuth projection created by Li Guozao, a Chinese scholar, belongs to projection 10 established by geometric method.Projection de format i on is in evitable in map projectio n In the document 11, the author gives a proof of image by geometric method.Map application often area. The geometry area calculation in calculation method, grid method, parallei line, transit n etwork method equivalent calculation met hod, are based on the basic principle of 12.3 calculusThe application of calculus in cartography is common in.The basic formula of map projection, for a basic amount of order (also known as Gauss E, F, G index), H is the basis of the formula, the process of differentiol trapezoi dal el 1ipsoi d along the longitude, latitude, along the diagonal differential expression amount of one order is also on or off in the partial derivative. The determination of isometric conditions, integrability conditions and equal distance conditions also includes a series of differential and partial derivative operations.The meridional arc length from equator to latitude shows S integral:(2) in the formula, a is the long radius of the earth ellipsoid, and E is the first eccentricity.The integral formula of the sphere trapezoid area formed by the warp line and the weft line on the ellipsoid is:(3)(3) the M in the formula is the radius of curvature of the meridional circle, and N is the radius of curvature of the prime circle.In the calculation of equal projection, the spherical trapezoidal area from the equator to the latitude (in square kilometers) of longitude of 1 radians. Derivation process of X ond Y coordinale formulas for Gauss 一 gram projection,A series of complex differential and derivative and partial deri vative operations are needed. Tn our country, Professor Yang Qi he has derived the formula for the length ratio of Gausss family of projections by means of the study of the family of Gauss - and - gram projections, 13(4)The formula for the convergence angle of the meridian of Gauss projection(5)These two formulas contain, and are concerned with, the partial derivatives of (difference).4 algebraIn algebra, the equations of shape such as Fl (x, y,，z)二 F2 (x, y,. ., z) are cal led equati ons. An equation is an equation containing an unknown number.Taking the 1 ong radius a of the earth el 1 ipsoid and the short radius B, the ellipsoid equation taking the geocentric as the origin of coordinates is(6)The multi cone projection, pseudo cylindrical projection and cyl indrical project ion are al 1 symmetrical in the central meridian, and the meridian equation is expressed as a function of latitude or higher power equation 14: (7)If at least one of the f (x) and G (x) is elementaryt ran see ndental functi ons, the equati on f (x) = g (x) is called elementary transcendental equation (referred to astranscendental equation). The meridional arc length formula from equator to latitude, that is, the (2) formula of this paper, is transformed (using IUGG75 ellipsoid parameter):The trapezoidal area of the ellipsoid from equator to latitudeis 1 radiansIsometric representation function U formula for(10) (10) in the formula.The above (8), (9) and (10) formulas belong to transcendental equations in algebra. In the document 15, the inverse soiution of this transcendental equation (known or inverse latitude) is given.There are many spatial curves in the map representalion object, and the equation form of space curve is 16:(IDBoolean algebra, also called logical algebra, is the mathematical basis for explaining the principles of computer computation. On the map image systcm to the point of the basic elements of map symbols, to the basic elements of map symbol system and map layer as the basic element of the map layer system, the author proves that they all belong to the Boolean algebra system 17-19. The author also proves that the map editing process is essentially a Boolean operation through a finite mapp ing synthesi s operator 20.5 graph theoryThe classical definition of graphs is given in document 21.Since the map is a subset of the set of graphs, the restriction of the z/lancT character makes it differont from other kinds of graphs, such as circuit diagrams, plant maps, and so on, so that they have the basic characteristics of maps 22. Since a map must have at least one point as its content, it can be made into a graph. A point is defined by a graph as an ordinary graph. Taking into account the inevit abi lity of the ex is tence of line profile, so in any case, the map is to meet the definition of standard, this is the logic foundation of existing map In the document 23, a rigorous mathematical definition of the map is given on the basis of the map contents and mathematical expressions.Map symbols can be di vided into three kinds: dotted symbols, linear symbols and surface symbols. The dot and line symbol formation G, and the surface symbol is the planar embedding ofFig G,That is, the planar embedding of planar graph G divides the plane into several connected enclosed regions, each of which is called a surface of the graph G, which is called the outside of the graph G. From the point of view of graph theory, this paper also reveals the feature of planar embedding of figure G, which is composed of point map, line symbol and point line map symbol 24. Graph theory also illustrates the essential difference between the generative principle and the visual perception of a graph (con si sting of dots, Un es, symbols) and the background (consisting of surface symbols)6 set theoryThe geological entities represented by maps, such as residential areas, road networks, water systems, landscapes, etc. , are essentially collections of points of different natureMap symbols have three basic character! sties: the nature, the characteristic I, the representation feature (color) J and the shading layer t. These basic features are the basis of the map set model such as bl ack and white maps and col or maps- 25.Classification is a basic step in understanding things and dealing with information- Each classifiedtion method in the map symbol classification system divides the symbol into several set families according to a certain mark, and different marks form different classifications. For example, according to the geometric proper! ies of symbol positioning part, can be di vided into point, 1 inc and surface map symbols; according to the correlation bet ween symbols and map scale, can be divided into seale symbol, non seale symbol and semi scale symbol 26; according to whether reflect the reality of map symbols can be divided into simulation and virtual map symbol, etc. 27, 28 Set theory provides a mathematical tool for classification of map contents.Different geomorphic patterns can be regarded as the set of points at any point in a given area, 1, to determine the elevation of the geomorphic featuro point, P, to satisfy a given condition.The neighborhood set landform features for P A, for, if, according to the conditions of different D, respectively on slopes, hills, ridges, valleys, canyons, such as saddle def i nit ion of 29. Mounta ins and pla ins can be def ined as 30 by elevation and relief constraints. By applying the neighborhood concept of set theory, the topographic morphology can be established by means of the different constraints of the geomorphic feature points, and the mathematical system is strictly defined. 31.Set on the X anti reflexive, symmetric and transitive relation called partial order, said the collection is less than partial relation called 32 poset. Name scale, ordinal scale, scale interval and ratio scale and geographical scale, its essenee is the source of data ordering satisfies a condition set (X =) simplified, simplified partial order different sets (A, less than or equal to (X), is less than or equal) con tains the relationship and has the form di fferent type 33.7 probability theory and mathematical statisticsProbability theory is a discipline that studies the statistical laws of random phenomeno from the quantity side. Geomorphology i s sometimes expressed as a normal distribution in probabi 1 i ty theory, but most of which are represented by Pearson III distribution. The third curve of Pierre line was founded by British scholar Pearson, and is quite common in map making.The main object of mathematical statistics research is the correlation. In cartography, linear correlation and curvilinear correlation are often used The study of the correlation between a random variable and another non tandom variable is called regression analysis. The correlationbetween two random variables is called correlation analysis. Regression analysis and correlation analysis are applied in cartography. Residential areas, for example, are one of the most closely related elements of the road network,The choice of residential land has import a nt inf lue nee on road network An example is given in Rof 34, through the measurement of 100 samples, the number of samples in each n mesh to the number of residents and Q Road, according to the distribution of the coordinate paper given n and Q, using the linear fitting equation, regression equations were obtained by regression calculation(12)When Q二 1, there are n二0. 3, indicating that only 1 residential sites, generally do not constitute mesh, the average variance Sn= + 0.76, the deviation coefficient Cv二0.066. The author selects the road according to this formula and gets bet ter effect. J .| I | BIIIProbabi 1 ity theory and mathematical statistics are appl ied to the determination of residential areas, the selection of quantitative indicators, and the selection criteria of rivers.8 fractal geometryEuclidean geometry is quite effective in the study of regular and smooth shapes (or ordered systems). However, there are many problems in the real world that cannot be solved by Euclidean geometry. British L Richardson examines the length of coastline, found that in Spain, Portugal, Belgium, Holland and other countries publ ished encyclopedia records of some coastal 1 ength difference of 20%. The French mathematician Mendel Rob (B. Mandelbrot) by Swedish mathematician Kirk (H.Von. Koch) found that the Kirk curve as a mathematical model on coast 1ine problems, through in-depth research and the introduction of the concept of fractal dimension, 1977 will be officially fractionality graphics called Fractal (fractal), and the establishment of this kind of graphic as the object of mathematics branch 一 fractal geometry 35 Fractal is the form composed of parts,Each part is similar in some way to the whole Fractal geometry reveals that the length of an irregular curve, such as a coastl ine, is re la ted to the len gth of a measuri ng ruler, and the smal ler the rul er, the greater the measurement resul t. Similarly, taking into account the measuromont of irregular surface area, then you can put the ruler as a side of the cube. Count the number of squares (N) that intersect the curve or surface. Then the length of the curve L () and the area of the surface A () can be calculated In these two cases, the line 1 ength L () or the surface area A () is satisfied:Figure 2 Sketch of irregular curve length by fractal geometryFig2., Sketch, of, Measure, non-rule, Curve, Length, with, Geometry, Fractal仃3)For the straight line, the N () index T denotes the dimension of the measurement system is 1; for the plane, N () : index 一2 denotes the dimension of the system is 2. But for the coastl ine shown in Figure 2, it meetsN 0 (14)D can be an integer or a fraction, and it is the fractal dimension of the system. If D and the topological dimension of the system are consistent, then most (not al 1) of such systems are Euclidean or non fractal 36.On the basis of studying the fractal phenomerm of water system, He Zongyi proposed the method of determining the fractal dimension of water system elements, and used the fractal dimension rule of water system to map synthesis, and obtained better results 37.There are many irregular curves like the coastline on real space and maps. Fractal geometry provides a mathenuitical tool for the measurement of such curves.9 fuzzy mathemat icsIn 1965, the United States L.After A. Zadel put forward the fuzzy set/ thesis, it produced fuzzy mathematics. The fuzzy set and the characteristic function are defined as follows:Define X as a common set. Mappings, called fuzzy sets (Fuzzy, Set), referred to as F sets. A (x) is called X, relative to themembership degree of F