Projekt współfinansowany ze środków Unii Europejskiej w ramach Europejskiego Funduszu Społecznego MATERIAŁY DYDAKTYCZNE DO PRZEDMIOTU.

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1 MATERIAŁY DYDAKTYCZNE DO PRZEDMIOTU Cartography Wydział Inżynierii Środowiska, Geomatyki i Energetyki Opracował: prof. dr hab. inż. Jacek Szewczyk

2 1. The concept of cartography Cartography in a discipline dealing with graphic, communicative, visual-conceptual and technological presentation of spatial information based on maps and other cartographic representations (globes, panoramas, block-diagrams, cartographic animations, virtual reality). According to the British Cartographic Society, cartography is "science, art and technology of map making and studying them as scientific documents and works of art". The methods of cartographic studies include: using maps as models of the studied phenomena to the stages of studying reality using mathematic and geometric methods in the studies on deformations and geo-reference data models (data structure, making objects) rules of designing maps (semiotic, graphical) studies of visual and cognitive processes (socio-psychological methods) The branches of cartography include: cartometric mapping, including measuring the accuracy of the situation of the objects on the map, transferring geometric values on the map, the measurement of angles (horizontal and vertical) on the maps, distances, areas, elevations (relative and absolute) and the calculation of co-ordinates, geo-visualization, understood as private activity, during which so far unknown spatial information is revealed in highly interactive graphical computer environment (A.M. MacEarchen), this involves using visual representation of geo-spatial information to facilitate thinking, understanding and constructing knowledge on the aspects of human and physical environment and creating visual representation for these aspects; the key technology is GIS, knowledge of maps and their properties, mathematic cartography, referring to the theories of cartographic projections, editing and processing of maps, cartographic reproduction, history of cartography. The object of the interest of cartography, according to the definition of this discipline, is a map. The map is usually defined as graphical, mathematically defined model of reality referred to the plane, according to the accepted scale, presenting the features of the objects and the special relationships between them, using symbols. In the geometric aspects, a map is a perpendicular parallel projection on the defined horizontal reference surface, miniaturized in a certain scale. The classification of maps can be presented according to six criteria: Classification of maps by their content: 1. General geographic maps (reference maps): general distribution of phenomena; divided by scale into: Large-scale (scale larger than 1:10000); including: o Basic maps o Inventory maps

3 Medium-scale (topographic) (1: :200000) Small-scale (review) (below 1:200000) 2. Thematic maps (statistical) Natural maps (of the ecological environment, physical phenomena) Social and economic maps (demographic, communication, industrial, historical, linguistic etc.) Classification of maps by the method of mapping: Cartograms Carto-diagrams Signature maps Dot maps Isocline maps etc. Classification of maps by the way of presentation: Analogue map (a graphic form; also on the monitor) Digital map: 3D model of space written in a digital form as information on the situation and features of the object Vector map: in the form of the set of geometric objects (points, lines and surfaces) Raster map: in the form of matrixes of pixels Classification of maps by the form of presentation: Static maps Interactive maps Multimedia maps Animated maps Classification of maps by function: Didactic maps Scientific maps Town and country planning maps Documentation maps Navigation maps etc. The most important features of maps include: mathematically defined reference surface, mathematically expressed way of the projection of situation objects and relief on the plane, mathematically defined system of co-ordinates, accepted map scale, the way of presenting the map content with conventional signs and information descriptions, the degree and way of the generalization of terrain traits. 2. Reference surfaces The real surface of the Earth is the surface of the continuous irregular solid, of the shape similar to ellipsoid of revolution. To make a map, it is necessary to define the mathematically

4 definable surface to which the location of situation details and relief will be defined. The approximations of the physical surface of the Earth will be (Fig. 1): the geoid, the surface of the flattened ellipsoid of revolution, the surface of a sphere, the plane. Fig. 1. Reference surfaces. Legend: 1 ocean, 2 ellipsoid, 3 local vertical, 4 continent, 5 geoid. Geoid. The geoid on a selected territory or on the whole globe, in a very short time interval, is a surface of almost constant gravity potential. It is a closed surface, surrounding the whole globe, of a strongly folded and usually regular structure, not possible to be described in a mathematical way. Thus the application of geoid, as the reference surface is difficult. This concept is used to determine the height of the Earth surface over the sea level (determined locally). Orthogonal trajectories (perpendicular) to geoid are called gravity lines. Geometric structure of geoid in a great approximation of the structure of the surface of three-axis ellipsoid. Flattening the ellipsoid equator is, however, relatively small and can practically be negligible. Earth ellipsoid is a flattened elipsoid the surface of which is the closest to the hydrostatic surface of the Earth. Ellipsoid of revolution is defined by two constant parameters, including at least one length parameter, e.g. by two semi-axes a and b or by semi-axis a and flattening f. The necessary observations are reduced to the plane. Nowadays, because of the GNSS technology, only the distance between points is reduced. Global ellipsoid is the ellipsoid referring to the whole globe. It should be adjusted to the Earth in such a way that describes the whole surface most accurately. Such an ellipsoid should fulfil the following conditions: the mass of ellipsoid equals the mass of the Earth, the centre of ellipsoid is in the middle of the mass of the Earth, the axis of the rotation of ellipsoid overlaps with the axis of the rotation of geoid,

5 metric parameter and structural parameter of the surface of ellipsoid is selected from the minimum condition of the sum of differences of the value of the potential on the ellipsoid and geoid in corresponding points attributed by the lines of the work of the gravity. The local ellipsoid (reference ellipsoid) refers to the limited area of the Earth. The local ellipsoid corresponds in the best way to these areas, on which the measurements were made to mark it. For other areas, this might not be the best adjusted ellipsoid. The reference ellipsoid of the physical surface of the Earth is determined empirically on a so-called geodesy network of zero order by the definition of the following in the knots of this network: geodetic latitudes B, geodetic longitudes L, azimuths A, making precise levelling and the measurement of the acceleration of the gravity, using the parameters of the movement of artificial satellites in the field of the Earth s gravity. In the history of cartography there are many ellipsoids of a mostly local character. The parameters of the most important ellipsoids are presented in tab. 1. Ellipsoid s Year of Semi-major Semi-minor Flattening Name Definition axis a, m axis b, m Bessel : Clarke : Hayford : Krasowski : GRS : WGS : In Poland, before WW2 there was a local Bessel ellipsoid. After WW2, until 2010, the Krasowski ellipsoid was accepted (also local, reasonably well adjusted to the area of Poland). Since 2010, the global ellipsoid has been used on WGS 84 (practically equivalent to GRS 80). The characteristic parameters of the ellipsoid of revolution include apart from semi-axes a and b also include: o eccentricity, o second eccentricity, o third eccentricity, o flattening o second flattening o third flattening o normal main cross-sections normal cross-sections of the highest and the smallest curvature, including: meridional cross-section of the smallest curvature radius (the biggest curvature), transverse cross-section of the biggest curvature radius (the smallest curvature). A sphere as a reference surface is used, first of all, in small-scale work. The radius of the sphere is determined, accepting the following assumptions: the area of the sphere equals the area of the ellipsoid, the volume of the sphere equals the volume of the ellipsoid, as the arithmetic mean of three semi-axes of the ellipsoid, as the geometric mean of the curvature radius of main cross-sections.

6 The plane as reference surface is usually applied in the presentation of small areas. The difference between the area of the sphere and the plane (in a static situation) is expressed by the formulae: where: L distance, R the radius of the Earth, dh the difference between the heights determined on the surface of the sphere and on the plain, dl the difference between the distances determined on the surface of the sphere and on the plain. The ranges of the applicability of the reference surface: Plain: up to 80 km². Sphere: 50 km² km². Ellipsoid: above km². 3. Map projection 3.1. The concept of projection The acceptation of the reference surface does not eliminate all the problems connected with map making. This is because the map makes a plane. Among the listed reference surface, only the projection of situation details and relief on the plane allows direct map making. The range of such operation is limited to a small area. The surfaces of the ellipsoid of revolution and sphere belong to the so-called surfaces unable to be developed on the plane. This means that we are not able to present the projection of the real surface of the Earth into the mentioned surfaces in the form of a plane. The solution is the application of map projections. The projection of one surface into another is called any unambiguous and mutual point adequacy between the surface, which is presumed as the surface of the original and the surface presumed the surface of the image. This adequacy is decided with the function of relationships between the parametric variables on the surface of the image and parametric variables on the surface of the original: Functions u (U, V) and v (U, V) are called projection functions. They can be established under different conditions, receiving different projections. Cartography deals only with regular projections of surfaces. The projections are called regular, if the projection functions u (U, V) and v(u, V) are: one-to-one: every pair of the values of parameters U, V is corresponding to one and only one pair of the values of parameters u, v and vice-versa

7 continuous and at least twice subdued to differentiation mutually independent, which takes place when the function indicator, called Jacobian in function u (U,V) and v (U,V): is other than zero for all the pairs of value U, V. In a general case, the surface of the original and the image can be written as: where: x, y, z, X, Y, Z co-ordinates in the perpendicular spatial system u, v parameters In the projection of a sphere into the plane, parameters (u, v) correspond to geographic coordinates (ϕ, λ), while for the plane co-ordinate Z(u, v) = 0 (the surface of the image). The projection of a sphere or ellipsoid on a plane is carried out with (local) deformations Classification of projections There are many projections. They can be classified by different criteria. 1. Due to the character of projected deformations: conformal (preserve angles without deformations; there is a conformality), equiareal (preserve the area without deformations), equidistant (preserve the distance in one of the main directions without deformations), compromise (where the deformations of lengths, angles and areas occur). 2. Due to the shape of the normal grid of meridians and parallels: - azimuthal: the images of parallels are circles, the centre of which is in the image of the pole, images of meridians are simple, concentrating in the pole, Fig. 2. The normal azimuthal projection of the sphere (source: Banasik et alt., 2011) - cylindrical: the images of parallels are sections of straight lines mutually parallel, the images meridians are straight lines or sections perpendicular to the image of parallels,

8 Fig. 3. The normal cylindrical projection of the sphere (source: Banasik et alt., 2011) - conic: the images of parallels are arcs of the concentric circles, the images of meridians are sections or half-lines perpendicular to the image of parallels, Fig. 4. The normal conic projection of the sphere (source: Banasik et alt., 2011) - compromise: pseudo-azimuthal, pseudo-cylindrical, pseudo-conic, multi-conic, circular, derivative.

9 Fig. 5. Examples of the grids in compromise projections: a) the Mollweide projection (pseudocylindrical, equal-area) in normal situation; b) the Mollweide projection in the skew situation; c) the Peterman star-like projection; d) the equal-area Goode projection (source: Banasik et alt., 2011) 3. Due to the mutual situation of the rotation axis of the original and the image: - normal: the axis of the revolution of the plane, cylinder or cone overlaps the revolution axis of the sphere (ellipsoid), - skewed: the axis of the revolution of the plane, cylinder or cone cuts the axis of the rotation of the sphere (ellipsoid) in a certain angle, - transverse: the axis of the rotation of the plane, cylinder, conic is perpendicular to the axis of the sphere (ellipsoid) rotation. 4. Due to mutual situation of the surface of the original and the image: - tangents: plane, cylinder, cone are tangent to the sphere (ellipsoid) in the point or along a certain line, - secant: plane, cylinder and cone cut the surface of the sphere (ellipsoid), - remote: not applied in practice. 5. Due to the situation of the centre of projection towards the projection plane: - gnomonic (geocentric): projection from the centre of the Earth, - stereographic: projection from the opposite pole (main point), - orthographic: parallel projection, perpendicular to the projection plane Projection deformations Introducing projections causes the situation that every map based on them is deformed in the relation to the reality. Thus there is no an ideal map, presenting this reality without deformations. Deformations refer to the lengths, angles and surfaces; only one of these elements can be projected to the image (map) preserving the shape of the original. Thus we differentiate elementary deformations lengths, areas and angles. Elementary scale of lengths, elementary deformation of lengths. Elementary scale of lengths (or directly: scale of lengths) in the reproduction is the relationship:

10 where: ds the element of the arc on the surface of the original, the corresponding arc element on the surface of the image. This scale is the function of 3 variables: two (U,V) marking the situation of the point on the surface of the original and its corresponding point on the surface of the image, and the directional angle β of the considered element of arc ds on the surface of the original, or directional angle β of the corresponding element of arc on the surface the image: Sometimes it is the function of only 2 variables (conformal projections) or 1 variable (linear projections of plane to plane). Elementary deformation lengths (or directly: deformation lengths) are expressed with the formula: Given as fraction, in %, in. Elementary scale of areas, elementary deformation of areas An elementary scale of areas (or directly: scale of areas) is the relationship: where: dp the element of the area on the surface of the original, the element of the area on the surface the image. This scale is the function of 2 parametric variables (U, V) and depends only on the situation of the point on the surface: An elementary deformation of areas (or directly: deformation of areas) is: Given as a fraction, in %, w. Deformation angles The measure of deformations ω (extreme deformations in a given point) is the biggest positive difference between the angle in the image and the angle in the original (regarding all the possible openings and situation of its arms). where: α the angle between two random curves in the point on the surface of the original, α the angle between the images of these curves on the surface of the image. The deformation angle is the difference between α and α. The maximal deformation angle is the function of 2 variables U,V, determining the situation of the point on the surface.

11 3.4. Main curves, main directions, Tissot s theorems At the projection of one surface onto the other, the right angle between two curves, usually the projection, goes on the obtuse or acute angle. If the areas of the original and the image are regular surfaces and projection functions fulfil the conditions of regular projections, in every point of the original area one can find such two directions making a right angle, which in the projection will also make the right angle. Thus it is always possible to find such a grid of curves intersecting also on straight angles on the original surface. First Tissot s theorem: in any regular projection of one regular surface on another, there is always at least one and if the projection is not conformal there is only one, orthogonal grid on the original surface, the image on the other surface will always be an orthogonal grid. Such grids are called the grids of main curves. In the conformal projection there are no grids of main curves, because every orthogonal grid on the original surface is projected also into an orthogonal grid. Straight lines tangent to main curves in any point of the original surface, as well as straight lines tangent to main curves in a corresponding to this point - the point of the image surface, are called main tangents, and their directions main directions. Elementary scales of lengths in main directions, in any point of projections, are extreme scales in this point; the values of scales in all the remaining directions in this point are within the borders of the value of extreme scales. Using the scales in main directions in any point projections is the most convenient to calculate the scale of areas and deformation of angles occurring in this point. Second Tissot s theorem: in the regular projection of the area on the surface, with the graphic image of elementary scales of lengths in all directions getting out of a given point of the image of the original surface, is an ellipse, the semi-axes of which are equal to elementary scales of length in main directions. The ellipse should be oriented towards parametric lines in the projection calculate angles between the direction of the image of the meridian in the considered point, and the direction of the semi-axes Gauss-Krüger projection Fig. 6. The ellipse of deformations (Tissot s Indicatrix) One of the commonly applied maps of projections is Gauss-Krüger projection (tangent or secant). This is a base for map making in Poland.

12 Gauss-Krüger projection is a conformal, transverse, cylindrical projection of the revolution ellipsoid on a plane, carried out in narrow meridian zones, fulfilling the condition that the central meridian of the zone is exactly projected on the section of the straight line. The surface of the image is the surface of the elliptic cylinder, tangent in the transverse situation to a chosen meridian of the ellipsoid. Maximal deformations of length and area will occur on the verge of the projected zone (width 3 0 or 6 0 ). The shape of the map grid is presented in the following way: the central meridian is projected into the section of the straight line, the other meridians into the curves symmetric towards the central meridian, directed with its concavity to the image of this meridian, the equator is projected on the section straight of the straight line, and parallels are projected into curves symmetric towards the image of the equator, directed to this with convexity. Gauss-Krüger projection in a tangent aspect The calculation of co-ordinates xgk, ygk, based on ellipsoid co-ordinates ϕ, λ, is carried out in two stages: conformal projections of the ellipsoid on the surface of the cylinder in a normal situation projection of the surface of the cylinder into the plane preserving the conditions of conformity and exactly preserving the central meridian (as X axis). Using isometric co-ordinates and complex numbers as well as their development into the Taylor polynomial, we obtain: where: - expressed in radians, N radius of the transverse cross-section of the ellipsoid, Spol length of the meridian s arc on the ellipsoid e, e first and second eccentricity of the ellipsoid: a, b semi-axes of the ellipsoid. Length of meridian Spol from the equator to width ϕ: Reverse passage (from-ordinates xgk, ygk int co-ordinates ϕ, λ)

13 - latitude corresponding to co-ordinate xgk, calculated with iteration procedure. Fig. 7. Longitude corresponding to co-ordinate xgk (source: Banasik et alt., 2011) Between axis X of the perpendicular system and the image of meridian, there is a small angle called the convergence of the meridians: γ, between the tangent to the image of the meridian in a given point and the straight line parallel to axis X, intersecting a given point. Fig. 8. Convergence of meridians in Gauss-Krüger projection (source: Banasik et alt., 2011) The convergence of meridians is measured from the tangent to the image of the meridian. The value of the convergence: - in the function of co-ordinates ϕ, λ: - in the function of co-ordinates x, y: Gauss-Krüger projection is conformal: the scale length is the same in all the directions and equals: Replacing the radius of the transverse section N with the mean radius of curvature R:

14 and we obtain: The elementary scale of the area (square of the elementary scale of length) equals: The image of the geodetic line (the shortest distance from the surface of the ellipsoid joining two points) is a curve with its concavity directed to the axis meridian of the projected zone. Fig. 9. The shape of the geodetic line on the Gauss-Krüger plane (source: Banasik et alt., 2011) The azimuth of the geodetic line equals the azimuth of the image of this line after the projection. The geodetic line joining two points is replaced on the plane with a section and regards the reduction to direction δ a small angle between this line and the straight line. In the case of the angle, one should reduce both its arms to the arms being straight lines (two reductions of the direction): Gauss-Krüger projection in a secant aspect Changes in the distribution of elementary scales and deformations look in the following way: along the line of intersecting both surfaces, the scales will equal one, and the deformations will equal zero, in the area between the secant lines, the deformations will be smaller in their absolute values, but negative (contraction); maximal size of deformations will occur on the central meridian,

15 in the remaining area deformations will be smaller in their absolute values, but positive, the bigger distance from the intersecting line the bigger deformations. Co-ordinates in the modified projection are calculated from the formulae: The elementary scale length equals: 4. Systems of co-ordinates 4.1. Reference system and reference frame The situation of a randomly selected point on the surface of the Earth, in a relatively known relation to this surface can be defined in a different way, giving appropriate linear, angular, linear-angular or descriptive values. One should differentiate several basic terms: Reference system: the set of recommendations and decisions with the description of the models necessary to define the beginning, the scale and orientation of the axes and their variability in time. Reference frame: practical implementation of the reference system; made by the marked (from measurements) values of parameters describing the beginning of the frame, the scale and axis orientation as well as their changeability in time. Co-ordinate system): unambiguously defines the way of attributing the set of numeric values co-ordinates of the point in the relation to the reference frame. In many countries including Poland local systems of co-ordinates are usually applied. There are, however global references, especially Geodetic Reference System 1980 (GRS80), and also: ITRS International Terrestrial Reference System: global kinematic spatial system (a geocentric system (the centre of the Earth is defined with the oceans and atmosphere), metric, orientation of axis is towards , no rotation of network in the relation to horizontal tectonic movements). ITRF - International Terrestrial Reference Frame practical implementation of ITRS, defined by giving the co-ordinates of points on a defined era with the rate of changes (resulting, among other, from the movement of tectonic plates). The world network of points realizing system ITRF is presented in Fig. 10:

16 Fig. 10. The network of points forming ITRF (source: Banasik et alt., 2011) In Europe, the compulsory system is European Terrestrial Reference System (ETRS) and its realization European Terrestrial Reference System (ETRF) European reference system, overlapping with the realization of ITRS into era It is connected with the stable part of the Eurasian Plate; the changes of co-ordinates reflect only local movements. The implementation is possible due to the points of EPN. It is also compulsory in Poland.

17 Fig. 11. The network of points realizing UTRF (according to the state in 2010) (source: Banasik et alt., 2011) 4.2. Geographic co-ordinates (astronomic and geodetic) and perpendicular One of the best known frames of co-ordinates are geographic co-ordinates. The following can be differentiated: geographic (astronomic) co-ordinates, geodetic co-ordinates, azimuthal co-ordinates (spherical coordinates). The application of the mentioned above co-ordinates, depends on the assumed reference surface and the kind of projections. Geographic (astronomic) co-ordinates refer to the situation points on the surface of the sphere. The geographic latitude is an angle, the vertex of which is in the gravity centre of the Earth. Geodetic co-ordinates refer to the situation of the point on the surface of the ellipsoid of revolution. The geodetic latitude is an angle, the vertex of which does not overlap (except of polar points) with the centre of the Earth. The azimuthal co-ordinates refer to situation points on the surface of the sphere in the case of skewed projections in the relation to the the Earth axis. The differences between the kinds of co-ordinates are presented in the figures below.

18 Fig. 12. Geographic (astronomic) co-ordinates (source: Banasik et alt., 2011) Fig. 13. Geodetic co-ordinates (source: Banasik et alt., 2011) Fig. 14. Azimuthal co-ordinates (source: Banasik et alt., 2011) Perpendicular co-ordinates start from the middle of the Earth sphere or in the middle of the spherical ellipsoid; axes Xi Y lie in the plane of the equator, while axis X lies in the plane of zero meridian and axis Y is perpendicular to it. Axis Z overlaps with axis of the revolution of the Earth. Depending on the accepted reference surface, the following co-ordinates can be differentiated: on the ellipsoid of revolution and sphere.

19 The frames of co-ordinates on the ellipsoid of revolution a. Geodetic co-ordinates are defined as: Longitude: the angle between the plane of the meridian taken as zero and the plane of the meridian intersecting a given point. Latitude: the angle between the plane of the equator and the straight line perpendicular to the surface of the ellipsoid in a given point. Geocentric latitude: the angle between the plane of the equator and the radius of the ellipsoid in a given point: or: tg The biggest difference between them equals 11.8 (for = 45 0 ) tg b. Perpendicular straight-line co-ordinates The right system of perpendicular straight-line co-ordinates OXYZ, with the beginning in the centre of the ellipsoid of revolution (accepted as mathematic reference surface of the Earth). Axis Z overlaps with the axis of the revolution of the Earth, axis X lies in the plane of the initial meridian. Point P is determined by co-ordinates X,Y,Z, which fulfil the equation of the ellipsoid of revolution: These co-ordinates are used only in certain theoretical speculation, but not in calculation practice. c. Spheroid perpendicular co-ordinates Meridian NAS on the surface of the ellipsoid of revolution is taken as axis. From point P, the geodetic line PB is led perpendicularly to the axis meridian. Situation point P on the surface of the ellipsoid is determined with the axis meridian AB (spheroid abscissa) and the arc of the geodetic line BP (spheroid ordinate). These values are determined as XS and YS and expressed in m. Abcissae XS to the north from the equator are positive, ordinates YS to the east from the axis meridian are also positive. The axis meridian is the meridian intersecting the centre of the considered area or the extreme west point. Fig. 15. Perpendicular spherical co-ordinates on the ellipsoid d. The relationship between the perpendicular straight line and geodetic co-ordinates:

20 Point P has perpendicular co-ordinates X, Y, Z and geodetic co-ordinates ϕ, λ. Axis X lies in the plane of the zero meridian. The following goes on: Because: then: tg Frames of co-ordinates on the sphere a. Geographic co-ordinates are defined as: Longitude: the angle between the plane of the meridian taken as the initial and the plane of the meridian intersecting a given point. Latitude: the angle between the plane of the equator and the radius of the sphere brought to a given point. The polar distance of the point: the angle between semi-axis ON and the radius carried out to a given point. b. Azimuthal co-ordinates: G and G1 opposite points on the surface of the sphere, do not overlap with the Earth poles N and S. The plane is moved perpendicularly to axis GG1; the trace of the intersecting with the surface of the sphere will be a large disc (great horizontal disc). Almucantarats: circles of discs made as a result of intersecting planes perpendicul to axis GG1 (parallel to the great horizontally disc) with the sphere; equivalent to parallels) Verticals: semi-circles formed as a result of intersecting the surface of the sphere of planes crossing axis GG1; the equivalent of meridians. Initial vertical: vertical containing the North Pole. Point P: defined by: - azimuth angle α between the plane of the initial vertical and the plane of the vertical of point P, - height angle h, or zenith distance angle z: the angle between the plane of the big horizontal disc and the radius of the sphere led to point P; zenith distance is different from the height by c. Perpendicular straight-line co-ordinates

21 The right frame of perpendicular straight-line co-ordinates oxyz with the beginning in the centre of the sphere, axis z overlaps with the axis of the Earth, axis x lies in the plane of zero meridian. The situation of point P on the surface of the sphere is defined by co-ordinates x, y, z, while: d. perpendicular spherical co-ordinates NAS The axis meridian on the sphere The arc of the big disc, perpendicular to axis meridian is led from point P. Point P is defined by: - arc AB of axis meridian (spherical abscissa) xs, - arc BP of the big disc (spherical ordinate) ys. Fig. 16. Perpendicular spherical co-ordinates on the sphere e. The relation between azimuth co-ordinates and geographic co-ordinates N,S poles of the Earth G, G1 poles (zenith and nadir) of the azimuthal co-ordinates; co-ordinates G: ϕ0, λ0. GN the arc of initial vertical equals ϕ0 NP the arc of the meridian of point P, corresponding angle 90 0 ϕ GP the arc of the vertical of point P corresponding to zenith distance z Angle NGP azimuth α point P Angle GNP equals λ - λ0 According to the formulae with spherical trigonometry: Thus: Introducing the auxiliary angle: One obtains the formulae to calculate z and α from co-ordinates ϕ and

22 or: If pole G lies on the equator, ϕ0 = 0, sin ϕ0 = 0, cos ϕ0 = 1; then: f. The relationship between co-ordinates of the perpendicular straight line and geographic co-ordinates Because OP0 = O1P = R cos : 4.3. Frames of flat co-ordinates applied in Poland The coordinate system of Borowa Góra was introduced in The reference frame was Bessel ellipsoid of 1841; the point of applying ellipsoid was in Borowa Góra (φ = , λ = ); thus the name. Gauss-Krüger projection was applied in 5 2-step zones (with axis meridians 17 0, 19 0, 21 0, 23 0, 25 0 ), scale on meridians = 1. System 1942 was introduced in 1953 in the framework of the unification of the systems of the Warsaw Pact. The reference surface was the Krasowski ellipsoid, with the point of application in Pułkowo (φ = ,55, λ = ,0 ) (so-called Pułkowo42 system). Two versions of Gauss-Kruger projections were applied: - in 6-degree spheres of axis meridians 15 0 and 21 0, - in 3- degree spheres of axis meridians 15 0, 18 0, 21 0 and In 1983, a new version of the system (known as ) was applied. Since1968, it was only a military system. System 1965 was introduced in Its basis was reference system Pułkowo42. The following assumptions for the frame were made: co-ordinates as many numbers, as the frame 1942, linear dislocation compared to the frame 1942 minimum 100 km, co-ordinates possible to be counted directly based on co-ordinates in frame 1942, projection reductions: based on flat perpendicular co-ordinates, 5 independent zones: in 4 quasi-stereografic projections with the main point in the centre of the zone (scale of secancy: ); in the fifth zone: Gauss-Kruger projection one zone, with axis meridian in the centre of the zone (secancy scale: ). maximal deformations in the centre of the zone: to -20 cm/km, in zone V: to -1.7 cm/km.

23 Fig. 17. Zones of frame 1965 with the isolines of deformations (source: Banasik et alt., 2011) The division into 5 irregular zones allowed covering the country with a uniform map in the scale from 1: to 1: In 1986 there was a new equalization of the network and new co-ordinates; the name frame was used. Maps in scales 1: : and the basic map of Poland (1:500 1:5000) are made in this frame. System GUGiK-80 was made for maps in scale 1: This covered one zone with axis meridian19 0, in Gauss-Krüger quasi-stereographic secant projection (secancy scale equalled , the secancy circle had radius of 215 km. The main point of the frame had coordinates: φ = 52 0, λ = , Deformations reached +70 cm/km (at the ends).

24 Fig. 18. The distribution of linear deformations of GUGiK-1980 (source: znieksztalcenia.png) UTM System (Universal Transverse Mercator) is a frame initially applied by NATO, then modified and also made available for civilian users. Universal transverse secant Mercator projection was applied (co-efficient of scale on meridian was ; secant discs distant from the central meridian by 180 km), with 6-degree zones. The reference surface is now the surface of ellipsoid WGS-84. Since 1/01/2010 there have been two frames of co-ordinates in Poland: 1992 and Both were introduced in System of co-ordinates 1992 is compulsory for maps in scales below 1: It was based on geodetic co-ordinates in EUREF-89 (European Reference Frame 1989). Perpendicular coordinates x, y are defined in Gauss-Krüger projection in 10-degree-scale projection zone with the central meridian 19 0, at the scale co-efficient Secant discs are 240 km from the central meridian. Axis X is dislocated 500 km westwards, axis Y 5300 km northwards (300 km south from the southern border of Poland).

25 Fig. 19. Distribution of linear deformations in frame 1992 (source: ediaviewer/file:puwg1992_znieksztalcenia.png) System of co-ordinates 2000 is compulsory for large-scale maps (to scale 1:10 000). Perpendicular co-ordinates are defined in modified Gauss-Krüger projection, in 3-degree zones (central meridians: 15 0, 18 0, 21 0, 24 0 ), with the scale of length on central meridian Deformations reach -7,7 cm/km and -154 m 2 /km 2. The values of co-ordinates are calculated with the formulae: While introducing the frame, it was assumed that the borders of the zone will go alongside the county (powiat) borders (Fig. 20).

26 Fig. 20. The division into zones (pol.: strefa) in frame 2000 (source: ediaviewer/file:powiaty_puwg2000.png) 4.4. Map identification numbers The map identification number is the sign of the map sheet, allowing its localization on the surface of the Earth. It is usually the series of numbers and letters, which, with the name of the map sheet, make a basic unit of the map nomenclature. The map identification number is established according to the accepted section division in a given system of projection. The identification number can be: order number in the accepted system of numeration, separate for each of the scales, geographic or perpendicular co-ordinates of the selected (usually left bottom) corner of the respective sheet, determination of the belt and post, in the intersection of which the initial sheet is situated. For the systems of co-ordinates required in Poland, the following systems of the sheet determination were accepted: System 1992: The base to define formats and sheets of the topographic map is a sheet in the scale of 1: and dimension 4 x 6. Division: 1: identification number made according to the formula a-b-cc (e.g. N-M-34) o a signature of the hemisphere (N or S) often skipped

27 o b letter symbol of the belt (capital letter) o c post number (1-60) 1: sheet division 1: into 4 parts (capital letter A-D e.g. N-33-C) 1: sheet division 1: into 36 parts (Roman number I-XXXVI e.g. N-33- XXIV) 1: sheet division 1: into 144 parts (number e.g. N ) 1: sheet division 1: into 4 parts (capital letter A-D e.g. N C) 1: sheet division 1: into 4 parts (lower case a-d e.g. N C-a) 1: sheet division 1: into 4 parts (number 1-4 e.g. N C-a-4) System 2000: The base to define the formats and sheets of the basic map is a sheet in scale 1: of dimensions 5 km x 8 km. Division: 1: identification number made according to the formula a.bbb.cc (e.g ) o a number of the projection zone (the value of the axis meridian divided by three) o bbb the result of the ratio (xi 4920 km)/5 o cc the result of the ratio (yi km)/8 1:5 000 sheet division 1: into 4 parts (e.g ) 1:2 000 sheet division 1: into 25 parts (e.g ) 1:1 000 sheet division 1:2 000 into 4 parts (e.g ) 1:500 sheet division 1:1 000 into 4 parts (e.g ) 4.5. Height co-ordinates The height points are determined from the accepted reference surface, which can be the (local) sea level, geoid (or quasi-geoid), ellipsoid of revolution. Thus several terms referring to the height co-ordinate can be differentiated (Fig. 21): Ellipsoid height: distance point from the surface of the ellipsoid measured along the straight line, normal to this surface. Orthometric height: distance from geoid to the Earth surface measured along the real line of the vertical (in principle: the altitude above the sea level). Normal height: referred to the sea level, measured to the centre of the Earth. Undulation of the geoid (N): the difference between the ellipsoid height and orthometric height. Quasi-geoid: surface created by determining normal heights in the line of the vertical from a given point to the centre of the Earth. In Poland, height points are determined in the Kronstadt system. There are three frame of the same name: Kronstadt 60 the result of referring to the mareograph in Kronstadt by making height measurements during a so-called 2 nd measurement campaign of required based on the enactment of 2000 r. Kronstadt 86 the result of a so-called 3 rd measurement campaign of , when the reference of network was measured and levelled once again - required based on the enactment of Mean differences between these frames in Poland range from 2 cm to 8.5 cm. Kronstadt 2006 (provisional name) the result of the so-called 4 th measurement campaign of The frame functions in theory, it does not function in a legal area. The differences between this and previous frame range from -19 mm to +21 mm. In 2019 there will be (as before the WW2) frame Amsterdam in Poland. The difference between this and Kronstadt equals m.

28 Fig. 21. Ellipsoid height h, orthometric height Hort, normal height Hnorm, undulation of geoid N (source: Banasik et alt., 2011) Fig. 22. Quasi-geoid 2005 for the area of Polski (based on materials of IGiK) (source: Banasik et alt., 2011) 4. Main scale and graduation

29 The scales of the map express the value of linear decrease on the map compared to the real length. The denominator of the map is the number informing on multiple diminishing: it is the ration of the real length D and corresponding length on the map d. The scale is the smaller, when the smaller the describing fraction is. The scale is written in the topographic maps in the following way: - numeric (in the form of fraction 1:M, where M scale denominator), - nominated, - in the form of linear graduation. The main scale of the map is expressing the degree of linear dimensions: The main scale is only preserved in points or lines, in which length deformations do not occur. In the map the main scale is written (preserved only in definite directions) The local scale takes different values in different places of the image; the more diverts from the main scale the further from the places of the preservation of this scale. The scale co-efficient is the ratio of the local scale and the main scale. E.g. the scale coefficient for the central meridian in the projection UTM equals ; on secant discs equals 1, on the verge of the zone grows to Graduation is a graphic presentation of the scale (construction: drawing a straight line, measuring the section of the sections of the same length, corresponding integer number of the length units in a given scale, description of the lines of division). On the monitors of the computers we can freely change the scale without changing the granularity. In the digital maps, the value of the scale is indicated in the reference meaning. The scale of the map is connected with the size of the cartographic signs (which does not influence the level of the granularity of the presentation, although it determines the accuracy of the localization of objects and graphic load of the map. 5. System of conventional signs The system of conventional (graphic) signs on the maps is specific for an individual country and covers point signs, linear signs and surface signs. In Poland, the list of conventional signs, applied in large-scale maps (the basic map and inventory map) contains Instruction K-1. The graphic sign can, but doesn t have to, refer to spatial reference data, which it presents. Usually, but not always, data referred to the surface are shown with surface signs, point data - with point signs, and linear with the use of linear signs. Every graphic sign can be additionally distinguished with graphic variables. Different level of the measurement data involves the application of different solutions. Every event is presented by a sign, having certain graphic features, expressed graphic signs, to which (according to Bertin) belongs: situation, shape;

30 size; brightness; granularity; tone; orientation. The situation, as an important feature of the map, is the supreme variable referring to six variables. This value can be understood in three ways: as the size of the sign (e.g. height of column diagram, disc area; as the number of uniform signs (e.g. dots) put in segments; or distributed on the surface of a given unit of reference. Fig. 23. The size of the sign and the ways of its understanding (source: Medyńska-Gulij, 2011) Variable of brightness is the relationship of white to black, thus different values of the phenomenon can be shown by different degrees of grey (from white to black). Granularity, defined as texture, is a variable, the subsequent degrees of which preserve the same proportion of white to black. The impression of fine or coarse granularity is obtained by diminishing or increasing of the raster, by handling the size of the grain. Tone is understood as the component of the colour. In cartography, colour is defined by three attributes: brightness, tone and intensity. Tone (colour) is connected with the wave length in a visible electromagnetic spectrum (e.g. 0.4 µm - violet, 0.5 µm - blue, 0.6 µm orange). The concept of tone is understood as commonly used to define colours. Orientation is a variable, in which the direction of the sign shows different features of the studied phenomenon. The application of this variable, however, is not as the drawing of a meandering river, because it results from the situation of subsequent section of this river. The shape allows the differentiation of the presented events (disc, square, triangle, rectangular and others). A graphic way of presentation. In the process of visualization, a graphic way of the presentation of the phenomenon is important and understood as the degree of the engagement of the surface of map by the presentation of the content. It is directly connected with the way of the occurrence of the phenomenon. A natural way of the presentation of continuous phenomena is a continuous presentation, e.g. in case of temperature. Dispersed phenomena are best presented in a discrete way, e.g. in case of settlement. Regarding the purpose of the map, the transformation of the graphic way of presentation can be carried out, e.g. dispersed phenomena can be presented in a continuous way, if justifiable.

31 6. Cartographic generalization Cartographic generalization is the process of selection and generalization of spatial information on the map. The most important factors of generalization include: - map scale, - map function, - character of the presented phenomenon, - purpose of the map. In the map there should be only signs necessary in the presentation of the defined phenomenon. One should avoid the excess of the content. Excessive content means cartographic redundancy: the application of different signs or graphic features to the presentation of the same spatial information. In the process of generalization, there is a contradiction between getting the accuracy of geometric situation of objects and geographic compliance (reflecting main features and connected phenomena) (Salishchev, 2003). The solution of this contradiction means the acceptance of the following rules: - in the maps of scale bigger than 1:5000 geometric accuracy should be preserved, - in the topographic maps (1: : ) the compromise between the correctness of the situation of objects in the map and geographic correctness should be sought, - in the maps in scales smaller than 1: m the correctness of geographic relations should have a priority before the geometric accuracy. The assumptions of the digital generalization model according to R.B. McMaster and K.S. Shea (1992) Cartographic generalization Why to generalize? When to generalize? How to generalize? Theoretical bases Cartometric assessment Geometric and attribute transformations Theoretical elements; elements of specialist applications; Calculation elements Geometric conditions; Spatial and holistic maps, transformation control Geometric (spatial) transformations: simplification; aggregation; combination; selection; smoothening; combining; translation; enhancement Attribute transformations: Classification; symbolization An important factor of generalization is the definition of the recognisability of the drawing. The norm of the recognisability of the drawing of the width of the line 0.1 mm is defined by the elementary triangle of the lengths of the shorter arm and basis: for analogue maps: arm a0 = 0.5[mm], and basis [ ] mm, or arm a0 0.5[mm], and base [ ] mm, for digital maps: o for natural objects arm ε01 = 0.5[mm]*M and base [0.6 mm *M 0.7 mm *M],

32 o for anthropogenic objects arm ε02 0.5[mm]*M and base [0.4 mm *M 0.5 mm *M]. where: M map denominator. The methods of generalization include: simplification smoothing aggregation combination translation selection enhancement Simplification is most often applied in generalization in digital transformation. The range of algorithms is applied, e.g.: Lang algorithm procedure of conditional extending of local transformation; Reumann-Witkam algorithm the procedure of unconditional extended local transformation; Douglas-Peucker algorithm global procedure; Chrobak algorithm, where the defined norms of the recognisability of a drawing are applied, and the elimination of points is based on the elementary triangle. Extreme points are determined by iteration, regarding elementary triangle. As a result of the carried out generalization, a conflict can arise, i.e. improperly selected sets, localizations or shapes of cartographic symbols (objects), giving the user wrong information on geographic relations between the objects or different parts of one object (Fei, 2002). One can differentiate the following causes of conflicts (Shea, McMaster, 1989): too small surfaces, too small sections, too narrow objects, objects situated too close to each other. Their identification and removal is possible, when the criterion of the recognisability of a drawing and its modification is applied (Żukowska 2008). 7. The ways of mapping 7.1. The order of carting large scale maps In case of a large scale maps (including the basic map) for analogue maps, the following order of carting is taken: Putting the grid of squares and the frame of the map. Putting the points of the geodetic reference line. Carting the situation details. Interpolation of the layers. Drawing the map. Measurement of the surface. Digital maps are made in a way specific for the applied program Methods of thematic mapping

33 The methods of mapping (mapping techniques, methods of cartographic presentation) are the procedures of cartographic presentation of spatial phenomena and relationships between them. The map type depends on the accepted method of data mapping; the base for the typology is: the definition of the geometric base of objects, the definition of the features of the phenomena on the measurement level, the application of visual variables. The classification of attributes is carried out by the attribution of numbers or measures to the objects, i.e. the presentation of the relationship between the objects by properly selected measurement scales. One can differentiate the following scales: nominal scale: e.g. sex, official language etc. ordinal scale: hierarchy features without giving the numeric values (e.g. classes of the water purity) interval scale (absolute or relative): informs about the order and the differences between the numeric values. The following methods of mapping are applied: Signature method (having many options) applied for the presentation of the layers of the general geographic or topographic map. Signature is the name of a graphic presentation of spatial objects in the form of symbols and cartographic signs. Signatures make a system of conventional signs. The following signatures can be differentiated: image (refer to the appearance of the presented object), symbolic (refer to the associations referring to the features of the phenomenon or object, e.g. an anchor as a symbol of a port), geometric (can have shapes of simple figures not referring to the real shapes, e.g. hexagon as a symbol of hard coal), letters (inform about different features of objects, e.g. the road number). Signatures can be graded: quantitative, information is mad by the changes in a graphical form. Cartogram (choropleth map) is a statistic map presenting the mean intensity of the phenomenon within the borders of the reference areas (administrative or natural units). Numeric values are put discretely in class intervals (the recommended number: 6). To determine classes a histogram is constructed. Classes are selected according to the intervals of equal span, equal number of observations (e.g. quantiles), based on the geometric or arithmetic, or individual qualitative intervals. The ranges method presents qualitative characteristics of the area or its part in the map with the possibility of the overlapping of ranges of different phenomena. The borders of ranges are present as lines with triangles or lines to the inside; also by hatching or transparency. The chorochromatic method (surface, mosaic) shows the intensity of a given phenomenon using colours within the limits of basic fields, which can be natural units, administrative units and geometric units. In the latter case, we talk about chorochromatic network maps made e.g. based on satellite images, which give information in pixels regular square cells. The interpretation of the cells of the image, a network chorochromatic map is made (chorochromatic-qualification every pixel gets qualified to some category)

34 Fig. 24. Differentiation of basic fields in the chorochromatic map (source: Medyńska-Gulij, 2011) Fig. 25. The chorochromatic network map for the present land use with a different size of pixels (source: Medyńska-Gulij, 2011) Dasymetric map is applied in the presentation of mean intensity of the phenomenon, taken in class intervals (skipping the non-classified fields). Dasymetric map (dasymetric cartograms) are similar to carthograms; the difference is the rejection of administrative units and acceptance of units corresponding to real intensity of the phenomenon. Dot distribution map: the sign of the presentation is a small disc (dot), the size of which allows more precise localization of the phenomenon. The definite value for a given dot (weight of the dot) is presumed. Cartodiagram is a statistic map, on which quantitative data are presented with diagrams (geometric figures representing the numeric value of the phenomenon; usually discs or squares). There are simple cartodiagrams (only situation and size of objects), summary and structural cartodiagrams (the total value and the value of components), structural cartodiagrams (proportions of the phenomena). Isolines method is used for the presentation of the phenomena of a continuous character. Isolines (contour lines) i.e. lines joining the points of the same numeric value are used.