Cylindrical map projection. Map projections and distortions

The use of the results of topographic and geodetic work is significantly simplified if these results are related to the simplest - a rectangular coordinate system on a plane. In such a coordinate system, many geodetic problems on small areas of terrain and on maps are solved by applying simple formulas of analytical geometry on a plane. The law of the image of one surface on another is called projection. Cartographic projections are based on the formation of a specific display of the parallels of latitude and meridians of longitude of the ellipsoid on some leveled or unfolded surface. In geometry, as is known, the simplest developable surfaces are a plane, a cylinder and a cone. This determined three families of map projections: azimuthal, cylindrical and conical . Regardless of the chosen type of transformation, any mapping of a curved surface onto a plane entails errors and distortions. For geodetic projections, they prefer projections that ensure a slow increase in distortions of the elements of geodetic constructions with a gradual increase in the area of ​​the projected territory. Particularly important is the requirement that the projection ensure high accuracy and ease of accounting for these distortions, using the simplest formulas. Errors in projection transformations arise based on the accuracy of four characteristics:

equiangularity - the truth of the shape of any object;

equal area – equality of areas;

equidistance – the truth of distance measurement;

truth of directions.

None of the map projections can provide accurate displays on the plane for all of the listed characteristics.

By nature of distortion cartographic projections are divided into equiangular, equal-area and arbitrary (in particular cases equidistant).

Equiangular (conformal) ) projections are those in which there are no distortions in the angles and azimuths of linear elements. These projections preserve angles without distortion (for example, the angle between north and east should always be straight) and the shapes of small objects, but their lengths and areas are sharply deformed. It should be noted that maintaining corners for large areas is difficult to achieve and can only be achieved in small areas.

Equal in size (equal area) projections are projections in which the areas of the corresponding areas on the surface of the ellipsoids and on the plane are identically equal (proportional). In these projections, the angles and shapes of objects are distorted.

free projections have distortions of angles, areas and lengths, but these distortions are distributed across the map in such a way that they are minimal in the central part and increase at the periphery. A special case of arbitrary projections are equidistant (equidistant), in which there are no length distortions in one of the directions: along the meridian or along the parallel.

Equidistant are called projections that preserve length along one of the main directions. As a rule, these are projections with an orthogonal map grid. In these cases, the main directions are along the meridmans and parallels. Accordingly, equidistant projections along one of the directions are determined. The second way to construct such projections is to maintain a unit scale factor along all directions from one point or two. Distances measured from such points will exactly correspond to real ones, but for any other points this rule will not apply. When choosing this type of projection, the choice of points is very important. Typically, preference is given to points from which the largest number of measurements are taken.

a) conical
b) cylindrical
c) azimuthal

Figure 11. Classes of projections by construction method

Equal Azimuth projections most often used in navigation, i.e. when the greatest interest is in maintaining directions. Similar to equal area projection, true directions can only be preserved for one or two specific points. Straight lines drawn only from these points will correspond to the true directions.

By construction method(unfolding a surface onto a plane) there are three large classes of projections: conical (a), cylindrical (b) and azimuthal (c).

Conic projections are formed based on the projection of the earth's surface onto the lateral surface of a cone, oriented in a certain way relative to the ellipsoid. In direct conic projections, the axes of the globe and the cone coincide, and a secant or tangent cone is selected. After design, the side surface of the cone is cut along one of the generatrices and unfolded into a plane. Depending on the size of the depicted area in conical projections, one or two parallels are adopted, along which the lengths are maintained without distortion. One parallel (tangent) is adopted for a short latitude; two parallels (secant) for a large extent to reduce deviations of scales from unity. Such parallels are called standard. A special feature of conical projections is that their central lines coincide with the middle parallels. Consequently, conic projections are convenient for depicting territories located in mid-latitudes and significantly elongated in longitude. That is why many maps of the former Soviet Union are drawn up in these projections.

Cylindrical projections are formed on the basis of projecting the earth's surface onto the side surface of a cylinder, oriented in a certain way relative to the earth's ellipsoid. In straight cylindrical projections, parallels and meridians are depicted by two families of straight parallel lines perpendicular to each other. Thus, a rectangular grid of cylindrical projections is specified. Cylindrical projections can be considered as a special case of conical ones, when the vertex of the cone is at infinity ( = 0). There are different ways to form cylindrical projections. The cylinder can be tangent to or secant to the ellipsoid. In the case of using a tangent cylinder, the accuracy of length measurement is maintained along the equator. If a secant cylinder is used - along two standard parallels, symmetrical relative to the equator. Straight, oblique and transverse cylindrical projections are used, depending on the location of the imaged area. Cylindrical projections are used when compiling maps of small and large scales.

Azimuthal projections are formed by projecting the earth's surface onto a certain plane, oriented in a certain way relative to the ellipsoid. In them, parallels are depicted as concentric circles, and meridians as a bunch of straight lines emanating from the center of the circle. The angles between the meridians of the projections are equal to the corresponding differences in longitude. The spaces between the parallels are determined by the accepted nature of the image (equiangular or other). The normal projection grid is orthogonal. Azimuthal projections can be considered as a special case of conic projections, in which =1.

Direct, oblique and transverse azimuthal projections are used, which is determined by the latitude of the central point of the projection, the choice of which, in turn, depends on the location of the territory. Depending on the distortion, azimuthal projections are divided into equiangular, equal-area, and with intermediate properties.

There is a wide variety of projections: pseudocylindrical, polyconical, pseudoazimuthal and others. The possibility of optimal solution of the tasks depends on the correct choice of map projection. The choice of projections is determined by many factors, which can be roughly grouped into three groups.

The first group of factors characterizes the object of mapping from the point of view of the geographical location of the territory under study, its size, configuration, and the significance of its individual parts.

The second group includes factors characterized by the map being created. This group includes the content and purpose of the map as a whole, methods and conditions for its use in solving GIS problems, and requirements for the accuracy of their solution.

The third group includes factors that characterize the resulting map projection. This is a condition for ensuring a minimum of distortions, the permissible maximum values ​​of distortions, the nature of their distribution, the curvature of the image of meridians and parallels.

The choice of map projections is proposed to be carried out in two stages.

At the first stage, a set of projections is established taking into account the factors of the first and second groups. In this case, it is necessary that the central lines or projection points, near which the scales change little, be located in the center of the territory under study, and the central lines coincide, if possible, with the direction of the greatest distribution of these territories. At the second stage, the desired projection is determined.

Let's consider the choice of different projections depending on the location of the study area. Azimuthal projections are chosen, as a rule, to depict the territories of the polar regions. Cylindrical projections are preferable for areas located close to and symmetrically relative to the equator and elongated in longitude. Conic projections should be used for the same areas, but not symmetrical relative to the equator or located in mid-latitudes.

For all projections of the selected population, partial scales and distortions are calculated using mathematical cartography formulas. Preference should be given, naturally, to the projection that has the least distortion, a simpler form of the cartographic grid, and, under equal conditions, a simpler mathematical apparatus of projection. When considering using equal area projections, you should consider the size of the area of ​​interest and the amount and distribution of angular distortion. Small areas appear with much less angular distortion when using equal area projections, which can be useful when the area and shapes of objects are important. In the case when the problem of determining the shortest distances is solved, it is better to use projections that do not distort directions. Selecting a projection is one of the main processes in creating a GIS.

When solving mapping problems in subsoil use in Russia, the two projections described below are most often used.

Modified simple polyconic projection used as multifaceted, i.e. Each sheet is defined in its own version of the projection.

Figure 12. Nomenclature trapezoids of sheets of scale 1:200000 in polyconic projection

The features of the modified simple polyconic projection and the distribution of distortions within individual million-scale sheets are as follows:

all meridians are depicted as straight lines, there are no distortions of lengths on extreme parallels and on meridians located ±2º from the average,

the extreme parallels of each sheet (north and south) are arcs of circles, the centers of these parallels are located on the middle meridian, their length is not distorted, the middle parallels are determined by proportional division in latitude along straight meridians,

The earth's surface, taken as the surface of an ellipsoid, is divided by lines of meridians and parallels into trapezoids. Trapezes are depicted on separate sheets in the same projection (for a map at a scale of 1: 1,000,000 in a modified simple polyconic). The sheets of the International World Map, scale 1: 1,000,000, have certain trapezoid sizes - 4 degrees along the meridians, 6 degrees along the parallels; at a latitude from 60 to 76 degrees, the sheets are doubled, they have parallel dimensions of 12; above 76 degrees four sheets combine and their parallel size is 24 degrees.

The use of projection as multifaceted is inevitably associated with the introduction of nomenclature, i.e. systems for designating individual sheets. For a million-scale map, the designation of trapezoids along latitude zones is accepted, where in the direction from the equator to the poles the designation is carried out in letters of the Latin alphabet (A, B, C, etc.) and along the columns in Arabic numerals, which are counted from the meridian with longitude 180 (according to Greenwich) counterclockwise. The sheet on which the city of Yekaterinburg is located, for example, has the nomenclature O-41.

Figure 13. Nomenclature division of the territory of Russia

The advantage of a modified simple polyconic projection, applied as a polyhedral one, is the small amount of distortion. Analysis within the map sheet showed that distortions in length do not exceed 0.10%, area 0.15%, angles 5´ and are practically imperceptible. The disadvantage of this projection is the appearance of gaps when connecting sheets along meridians and parallels.

Conformal (conformal) pseudocylindrical Gauss-Kruger projection. To use such a projection, the surface of the earth's ellipsoid is divided into zones enclosed between two meridians with a longitude difference of 6 or 3 degrees. Meridians and parallels are depicted as curves, symmetrical relative to the axial meridian of the zone and the equator. The axial meridians of the six-degree zones coincide with the central meridians of the map sheets at a scale of 1: 1,000,000. The serial number is determined by the formula

where N is the column number of the map sheet at a scale of 1: 1,000,000.

D The values ​​of the axial meridians of six-degree zones are determined by the formula

L 0 = 6n – 3, where n is the zone number.

Rectangular x and y coordinates within the zone are calculated relative to the equator and central meridian, which are depicted as straight lines

Figure 14. Conformal pseudocylindrical Gauss-Kruger projection

Within the territory of the former USSR, the abscissas of the Gauss-Kruger coordinates are positive; ordinates are positive to the east, negative to the west of the axial meridian. To avoid negative ordinate values, the points of the axial meridian are conventionally given the value y = 500,000 m with the obligatory indication of the corresponding zone number in front. For example, if a point is located in zone number 11, 25,075 m east of the axial meridian, then the value of its ordinate is written as follows: y = 11,525,075 m: if the point is located west of the axial meridian of this zone at the same distance, then y = 11,474,925 m.

In a conformal projection, the angles of the triangulation triangles are not distorted, i.e. remain the same as on the surface of the earth's ellipsoid. The scale of the image of linear elements on the plane is constant at a given point and does not depend on the azimuth of these elements: linear distortions on the axial meridian are equal to zero and gradually increase with distance from it: at the edge of the six-degree zone they reach their maximum value.

In countries of the Western Hemisphere, the Universal Transverse Mercator (UTM) projection in six-degree zones is used to compile topographic maps. This projection is close in its properties and distribution of distortions to the Gauss-Kruger projection, but on the axial meridian of each zone the scale is m=0.9996, not unity. The UTM projection is obtained by double projection - an ellipsoid onto a ball, and then a ball onto a plane in the Mercator projection.

Figure 15. Coordinate conversion in geographic information systems

The presence of software in GIS that performs projection transformations makes it easy to transfer data from one projection to another. This may be necessary if the received source data exists in a projection that does not coincide with the one selected in your project, or if you need to change the projection of the project data to solve a specific problem. The transition from one projection to another is called projection transformations. It is possible to translate the coordinates of digital data originally entered in the conventional coordinates of the digitizer or raster substrate using plane transformations.

Each spatial object, in addition to the spatial reference, has some meaningful essence, and in the next chapter we will consider the possibilities of describing it.

Date: 24.10.2015

Map projection- a mathematical method of depicting the globe (ellipsoid) on a plane.

For projecting a spherical surface onto a plane use auxiliary surfaces.

By appearance auxiliary cartographic surface projections are divided into:

Cylindrical 1(the auxiliary surface is the side surface of the cylinder), conical 2(lateral surface of the cone), azimuth 3(the plane called the picture plane).

Also distinguished polyconical

pseudocylindrical conditional

and other projections.

By orientation auxiliary figure projections are divided into:

• normal(in which the axis of the cylinder or cone coincides with the axis of the Earth model, and the picture plane is perpendicular to it);
• transverse(in which the axis of the cylinder or cone is perpendicular to the axis of the Earth model, and the picture plane is or parallel to it);
• oblique, where the axis of the auxiliary figure is in an intermediate position between the pole and the equator.

Cartographic distortions- this is a violation of the geometric properties of objects on the earth's surface (lengths of lines, angles, shapes and areas) when they are depicted on a map.

The smaller the map scale, the more significant the distortion. On large-scale maps, distortion is negligible.

There are four types of distortions on maps: lengths, areas, corners And forms objects. Each projection has its own distortions.

Based on the nature of distortion, cartographic projections are divided into:

• equiangular, which store the angles and shapes of objects, but distort lengths and areas;

• equal in size, in which areas are stored, but the angles and shapes of objects are significantly changed;

• arbitrary, in which lengths, areas and angles are distorted, but they are distributed evenly on the map. Among them, alignment projections are especially distinguished, in which there are no distortions of lengths either along parallels or along meridians.

Zero Distortion Lines and Points- lines along which and points at which there are no distortions, since here, when projecting a spherical surface onto a plane, the auxiliary surface (cylinder, cone or picture plane) was tangents to the ball.

Scale indicated on the maps, preserved only on lines and at points of zero distortion. It's called the main one.

In all other parts of the map, the scale differs from the main one and is called partial. To determine it, special calculations are required.

To determine the nature and magnitude of distortions on the map, you need to compare the degree grid of the map and the globe.

On the globe all parallels are at the same distance from each other, All meridians are equal to each other and intersect with parallels at right angles. Therefore, all cells of the degree grid between adjacent parallels have the same size and shape, and the cells between the meridians expand and increase from the poles to the equator.

To determine the magnitude of distortion, distortion ellipses are also analyzed - ellipsoidal figures formed as a result of distortion in a certain projection of circles drawn on a globe of the same scale as the map.

In conformal projection Distortion ellipses have the shape of a circle, the size of which increases depending on the distance from the points and lines of zero distortion.

In equal area projection Distortion ellipses have the shape of ellipses whose areas are the same (the length of one axis increases and the second decreases).

In equidistant projection Distortion ellipses have the shape of ellipses with the same length of one of the axes.

The main signs of distortion on the map

1. If the distances between the parallels are the same, then this indicates that the distances along the meridians (equidistant along the meridians) are not distorted.
2. Distances are not distorted by parallels if the radii of the parallels on the map correspond to the radii of the parallels on the globe.
3. Areas are not distorted if the cells created by the meridians and parallels at the equator are squares and their diagonals intersect at right angles.
4. Lengths along parallels are distorted, if lengths along meridians are not distorted.
   
5. Lengths along meridians are distorted if lengths along parallels are not distorted.

The nature of distortions in the main groups of map projections

Map projections	Distortions
Conformal	They preserve angles and distort areas and lengths of lines.
Equal size	They preserve areas and distort angles and shapes.
Equidistant	In one direction they have a constant length scale, the distortions of angles and areas are in equilibrium.
free	They distort corners and areas.
Cylindrical	There are no distortions along the equator line, but they increase as you approach the poles.
Conical	There are no distortions along the parallel of contact between the cone and the globe.
Azimuthal	There are no distortions in the central part of the map.

Map projections

mapping the entire surface of the Earth's ellipsoid (See Earth's ellipsoid) or any part of it onto a plane, obtained mainly for the purpose of constructing a map.

Scale. Control stations are built on a certain scale. Mentally reducing the earth's ellipsoid into M times, for example 10,000,000 times, we get its geometric model - Globe, the life-size image of which on a plane gives a map of the surface of this ellipsoid. Value 1: M(in example 1: 10,000,000) determines the main, or general, scale of the map. Since the surfaces of an ellipsoid and a ball cannot be developed onto a plane without breaks and folds (they do not belong to the class of developable surfaces (see developable surface)), any compositing surface is inherent in distortions in the lengths of lines, angles, etc. , characteristic of any map. The main characteristic of a space system at any point is the partial scale μ. This is the reciprocal of the ratio of the infinitesimal segment ds on the earth's ellipsoid to its image dσ on the plane: μ min ≤ μ ≤ μ max, and equality here is possible only at individual points or along some lines on the map. Thus, the main scale of the map characterizes it only in general terms, in some average form. Attitude μ/M called relative scale, or increase in length, the difference M = 1.

General information. Theory of K. p. - Mathematical cartography - Its goal is to study all types of distortions in mapping the surface of the earth's ellipsoid onto a plane and to develop methods for constructing projections in which the distortions would have either the smallest (in any sense) values ​​or a predetermined distribution.

Based on the needs of cartography (See Cartography), in the theory of cartography, mappings of the surface of the earth's ellipsoid onto a plane are considered. Because the earth's ellipsoid has a low compression, and its surface slightly deviates from the sphere, and also due to the fact that elliptical elements are necessary for drawing up maps on medium and small scales ( M> 1,000,000), then they are often limited to considering mappings onto the plane of a sphere of some radius R, deviations of which from the ellipsoid can be neglected or taken into account in some way. Therefore, below we mean mappings onto the plane xOy sphere referred to geographical coordinates φ (latitude) and λ (longitude).

The equations of any QP have the form

x = f 1 (φ, λ), y = f 2 (φ, λ), (1)

Where f 1 and f 2 - functions that satisfy some general conditions. Meridian images λ = const and parallels φ = const in a given map they form a cartographic grid. K.p. can also be determined by two equations in which non-rectangular coordinates appear X,at planes, but any other. Some projections [for example, perspective projections (in particular, orthographic, rice. 2 ) perspective-cylindrical ( rice. 7 ) etc.] can be determined by geometric constructions. A map is also determined by the rule for constructing the corresponding cartographic grid or by its characteristic properties, from which equations of the form (1) that completely determine the projection can be obtained.

Brief historical information. The development of the theory of cartography, as well as all cartography, is closely related to the development of geodesy, astronomy, geography, and mathematics. The scientific foundations of cartography were laid in Ancient Greece (6th-1st centuries BC). The gnomonic projection, used by Thales of Miletus to construct maps of the starry sky, is considered to be the oldest CG. After its establishment in the 3rd century. BC e. spherical shape of the Earth. C. began to be invented and used in the compilation of geographical maps (Hipparchus, Ptolemy, etc.). The significant rise in cartography in the 16th century, caused by the Great Geographical Discoveries, led to the creation of a number of new projections; one of them, proposed by G. Mercator, It is still used today (see Mercator projection). In the 17th and 18th centuries, when the broad organization of topographic surveys began to supply reliable material for compiling maps over a large territory, maps were developed as the basis for topographic maps (French cartographer R. Bonn, J. D. Cassini), and also studies were carried out on individual most important groups of quantum fields (I. Lambert, L. Euler, J. Lagrange etc.). The development of military cartography and the further increase in the volume of topographic work in the 19th century. required the provision of a mathematical basis for large-scale maps and the introduction of a system of rectangular coordinates on a basis more suitable for geodetic calculations. This led K. Gauss to the development of a fundamental geodetic projection (See Geodetic projections). Finally, in the middle of the 19th century. A. Tissot (France) gave a general theory of distortions of the CP. The development of the theory of CP in Russia was closely related to the needs of practice and gave many original results (L. Euler, F. I. Schubert, P. L. Chebyshev, D. A. Grave, etc.). In the works of Soviet cartographers V. V. Kavraisky (See Kavraisky), N. A. Urmaev, and others, new groups of maps, their individual variants (up to the stage of practical use), and important questions of the general theory of maps were developed. , their classifications, etc.

Distortion theory. Distortions in an infinitesimal region around any projection point obey certain general laws. At any point on the map in a projection that is not conformal (see below), there are two such mutually perpendicular directions, which also correspond to mutually perpendicular directions on the displayed surface, these are the so-called main display directions. The scales in these directions (main scales) have extreme values: μ max = a And μ min = b. If in any projection the meridians and parallels on the map intersect at right angles, then their directions are the main ones for this projection. The length distortion at a given projection point visually represents an ellipse of distortion, similar and similarly located to the image of an infinitesimal circle circumscribed around the corresponding point of the displayed surface. The semi-diameters of this ellipse are numerically equal to the partial scales at a given point in the corresponding directions, the semi-axes of the ellipse are equal to the extreme scales, and their directions are the principal ones.

The connection between the elements of the distortion ellipse, the distortions of the QP, and the partial derivatives of functions (1) is established by the basic formulas of the theory of distortions.

Classification of map projections according to the position of the pole of the spherical coordinates used. The poles of the sphere are special points of geographic coordination, although the sphere at these points does not have any features. This means that when mapping areas containing geographic poles, it is sometimes desirable to use not geographic coordinates, but others in which the poles turn out to be ordinary coordination points. Therefore, spherical coordinates are used on the sphere, the coordinate lines of which, the so-called verticals (conditional longitude on them a = const) and almucantarates (where polar distances z = const), similar to geographic meridians and parallels, but their pole Z 0 does not coincide with the geographic pole P0 (rice. 1 ). Transition from geographic coordinates φ , λ any point on the sphere to its spherical coordinates z, a at a given pole position Z 0 (φ 0 , λ 0) carried out using the formulas of spherical trigonometry. Any QP given by equations (1) is called normal, or direct ( φ 0 = π/2). If the same projection of a sphere is calculated using the same formulas (1), in which instead of φ , λ appear z, a, then this projection is called transverse when φ 0 = 0, λ 0 and oblique if 0 . The use of oblique and transverse projections leads to a reduction in distortion. On rice. 2 shows normal (a), transverse (b) and oblique (c) orthographic projections (See Orthographic projection) of a sphere (surface of a ball).

Classification of map projections by the nature of distortions. In equiangular (conformal) points, the scale depends only on the position of the point and does not depend on the direction. Distortion ellipses degenerate into circles. Examples - Mercator projection, Stereographic projection.

In equal-sized (equivalent) spaces, the areas are preserved; more precisely, the areas of figures on maps compiled in such projections are proportional to the areas of the corresponding figures in nature, and the coefficient of proportionality is the reciprocal of the square of the main scale of the map. Distortion ellipses always have the same area, differing in shape and orientation.

Arbitrary composites are neither equiangular nor equal in area. Of these, equidistant ones are distinguished, in which one of the main scales is equal to unity, and orthodromic, in which the great circles of the ball (orthodromes) are depicted as straight.

When depicting a sphere on a plane, the properties of equiangularity, equilaterality, equidistance and orthodromicity are incompatible. To show distortions in different places of the imaged area, use: a) distortion ellipses constructed in different places of the grid or map sketch ( rice. 3 ); b) isocolas, i.e. lines of equal distortion value (on rice. 8v see isocols of the greatest distortion of angles с and isocols of the area scale r); c) images in some places of the map of some spherical lines, usually orthodromes (O) and loxodromes (L), see. rice. 3a ,3b etc.

Classification of normal map projections by the type of images of meridians and parallels, which is the result of the historical development of the theory of CP, embraces most of the known projections. It retains the names associated with the geometric method of obtaining projections, but the groups under consideration are now defined analytically.

Cylindrical projections ( rice. 3 ) - projections in which the meridians are depicted as equidistant parallel lines, and the parallels are depicted as straight lines perpendicular to the images of the meridians. Beneficial for depicting territories stretched along the equator or any parallels. Navigation uses the Mercator projection - a conformal cylindrical projection. The Gauss-Kruger projection is a conformal transverse cylindrical projection - used in the compilation of topographic maps and processing of triangulations.

Azimuthal projections ( rice. 5 ) - projections in which the parallels are concentric circles, the meridians are their radii, and the angles between the latter are equal to the corresponding differences in longitude. A special case of azimuthal projections are perspective projections.

Pseudoconic projections ( rice. 6 ) - projections in which parallels are depicted as concentric circles, the middle meridian as a straight line, and the remaining meridians as curves. Bonn's equal area pseudoconic projection is often used; Since 1847, it compiled a three-verst (1: 126,000) map of the European part of Russia.

Pseudocylindrical projections ( rice. 8 ) - projections in which parallels are depicted as parallel straight lines, the middle meridian as a straight line perpendicular to these straight lines and being the axis of symmetry of the projections, the remaining meridians as curves.

Polyconic projections ( rice. 9 ) - projections in which parallels are depicted as circles with centers located on the same straight line representing the middle meridian. When constructing specific polyconic projections, additional conditions are imposed. One of the polyconic projections is recommended for the international (1:1,000,000) map.

There are many projections that do not belong to these types. Cylindrical, conic and azimuthal projections, called the simplest, are often classified as circular projections in the broad sense, distinguishing from them circular projections in the narrow sense - projections in which all meridians and parallels are depicted as circles, for example Lagrange conformal projections, Grinten projection, etc.

Using and Selecting Map Projections depend mainly on the purpose of the map and its scale, which often determine the nature of the permissible distortions in the selected metric. Large- and medium-scale maps intended for solving metric problems are usually drawn up in conformal projections, and small-scale maps used for general surveys and determining the ratio of the areas of any territories - in equal areas. In this case, some violation of the defining conditions of these projections is possible ( ω ≡ 0 or p ≡ 1), which does not lead to noticeable errors, i.e., we allow the choice of arbitrary projections, of which projections equidistant along the meridians are more often used. The latter is also used when the purpose of the map does not provide for the preservation of angles or areas at all. When choosing projections, they start with the simplest ones, then move on to more complex projections, even possibly modifying them. If none of the known CPs meets the requirements for the map being compiled in terms of its purpose, then a new, most suitable CP is sought, trying (as far as possible) to reduce distortions in it. The problem of constructing the most advantageous CPs, in which distortions are in any sense reduced to a minimum, has not yet been completely solved.

C. points are also used in navigation, astronomy, crystallography, etc.; they are sought for the purposes of mapping the Moon, planets and other celestial bodies.

Transformation of projections. Considering two QPs defined by the corresponding systems of equations: x = f 1 (φ, λ), y = f 2 (φ, λ) And X = g 1 (φ, λ), Y = g 2 (φ, λ), it is possible, excluding φ and λ from these equations, to establish the transition from one of them to the other:

X = F 1 (x, y), Y = F 2 (x, y).

These formulas when specifying the type of functions F 1 ,F 2, firstly, give a general method for obtaining so-called derivative projections; secondly, they form the theoretical basis for all possible methods of technical methods for drawing up maps (see Geographic maps). For example, affine and fractional linear transformations are carried out using cartographic transformers (See Cartographic transformer). However, more general transformations require the use of new, in particular electronic, technology. The task of creating perfect CP transformers is an urgent problem of modern cartography.

Lit.: Vitkovsky V., Cartography. (Theory of map projections), St. Petersburg. 1907; Kavraisky V.V., Mathematical cartography, M. - L., 1934; his, Izbr. works, vol. 2, century. 1-3, [M.], 1958-60; Urmaev N. A., Mathematical cartography, M., 1941; him, Methods for finding new cartographic projections, M., 1947; Graur A.V., Mathematical cartography, 2nd ed., Leningrad, 1956; Ginzburg G. A., Cartographic projections, M., 1951; Meshcheryakov G. A., Theoretical foundations of mathematical cartography, M., 1968.

G. A. Meshcheryakov.

2. The ball and its orthographic projections.

3a. Cylindrical projections. Mercator equiangular.

3b. Cylindrical projections. Equidistant (rectangular).

3c. Cylindrical projections. Equal area (isocylindrical).

4a. Conical projections. Equiangular.

4b. Conical projections. Equidistant.

4c. Conical projections. Equal size.

Rice. 5a. Azimuthal projections. Conformal (stereographic) on the left - transverse, on the right - oblique.

Rice. 5b. Azimuthal projections. Equally intermediate (on the left - transverse, on the right - oblique).

Rice. 5th century Azimuthal projections. Equal-sized (on the left - transverse, on the right - oblique).

Rice. 8a. Pseudocylindrical projections. Mollweide equal area projection.

Rice. 8b. Pseudocylindrical projections. Equal-area sinusoidal projection of V. V. Kavraisky.

Rice. 8th century Pseudocylindrical projections. Arbitrary projection of TsNIIGAiK.

Rice. 8g. Pseudocylindrical projections. BSAM projection.

Rice. 9a. Polyconic projections. Simple.

Rice. 9b. Polyconic projections. Arbitrary projection of G. A. Ginzburg.

Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

See what “Map projections” are in other dictionaries:

Mathematical methods for depicting the surface of the earth's ellipsoid or sphere on a plane. Map projections determine the relationship between the coordinates of points on the surface of the earth's ellipsoid and on the plane. Due to the inability to expand... ... Big Encyclopedic Dictionary

MAP PROJECTIONS, systematic methods of drawing meridians and parallels of the Earth on a flat surface. Only on a globe can territories and forms be reliably represented. On flat maps of large areas, distortion is inevitable. Projections are... Scientific and technical encyclopedic dictionary

World and screen coordinates

Projections

When using any graphics devices, projections are usually used. Projection specifies the way objects are displayed on a graphics device. We will consider only projections onto the plane.

Projection is the mapping of points specified in a coordinate system with dimension N to points in a system of lower dimension.

Projectors (projecting rays) are straight segments running from the center of the projection through each point of the object until they intersect with the projection plane (picture plane).

When displaying spatial objects on a screen or on a sheet of paper using a printer, you need to know the coordinates of the objects. We will consider two coordinate systems. First - world coordinates, which describe the true position of objects in space with a given accuracy. The second is the coordinate system of the display device, in which images of objects are displayed in a given projection. Let's call the coordinate system of the graphics device screen coordinates(although this device does not have to be like a computer monitor).

Let the world coordinates be 3D rectangular coordinates. Where the center of coordinates should be located, and what the units of measurement along each axis will be, is not very important for us now. The important thing is that for display we will know any numerical values ​​of the coordinates of the displayed objects.

To obtain an image in a specific projection, it is necessary to calculate the projection coordinates. To synthesize an image on a screen plane or paper, we use a two-dimensional coordinate system. The main task is to specify transformations of coordinates from world coordinates to screen coordinates.

The image of objects on a plane (display screen) is associated with the geometric operation of design. There are several types of design used in computer graphics, but there are two main types: parallel and central.

The projecting beam of rays is directed through the object to the picture plane, onto which the coordinates of the intersection of the rays (or straight lines) with this plane are subsequently found.

Rice. 2.14. Main types of projections

With central design all lines start from one point.

With parallel- it is considered that the center of the rays (straight lines) is infinitely distant, and the straight lines are parallel.

Each of these main classes is divided into several subclasses depending on the relative position of the picture plane and coordinate axes.

Single point projection

Rice. 2.15. Classification of plane projections

For parallel projections, the center of the projection is located at infinity from the projection plane:

• orthographic (orthogonal),
• axonometric (rectangular axonometric) - projectors are perpendicular to the projection plane located at an angle to the main axis,
• oblique (oblique axonometric) - the projection plane is perpendicular to the main axis, the projectors are located at an angle to the projection plane.

For central projections, the center of the projection is at a finite distance from the projection plane. There are so-called perspective distortions.

Orthogonal projections (main views)

Rice. 2.16. Orthogonal projections

1. Front view, main view, frontal projection, (on the back face of V),
2. Top view, plan, horizontal projection, (on the lower edge of H),
3. Left view, profile projection, (on the right side of W),
4. View from the right (on the left side),
5. Bottom view (top edge),
6. Rear view (front side).

The orthogonal projection matrix onto the YZ plane along the X axis has the form:

If the plane is parallel, then this matrix must be multiplied by the shift matrix, then:

where p is the shift along the X axis;

For the ZX plane along the Y axis

where q is the shift along the Y axis;

For the XY plane along the Z axis:

where R is the shift along the Z axis.

In axonometric projection, the projecting lines are perpendicular to the picture plane.

Isometric- all three angles between the picture normal and the coordinate axes are equal.

Dimetria - two angles between the picture normal and the coordinate axes are equal.

Trimetry - the normal vector of the picture plane forms different angles with the coordinate axes.

Each of the three types of these projections is obtained by a combination of rotations followed by a parallel projection.

When rotating by an angle β relative to the Y axis (ordinates), by an angle α around the X axis (abscissa) and subsequent projection of the Z axis (applicate), a matrix appears

Isometric projection

Rice. 2.17. Isometric projections

Dimetric projection

Rice. 2.18. Dimetric projections

Oblique projections

A classic example of a parallel oblique projection is cabinet projection(Fig. 2. 26). This projection is often used in mathematical literature for drawing solid shapes. Axis at depicted tilted at an angle of 45 degrees. Along the axis at scale 0.5, along other axes - scale 1. Let's write down the formulas for calculating the coordinates of the projection plane

Here, as before, the axis Υ pr directed downwards.

For oblique parallel projections, the projection rays are not perpendicular to the projection plane.

Rice. 2.19. Oblique projections

Now regarding the central projection. Since the projection rays for it are not parallel, we will assume normal such central projection, the main axis of which is perpendicular to the plane projection. For central oblique projection the main axis is not perpendicular to the projection plane.

Let's consider an example of a central oblique projection, which shows all the vertical lines of the depicted objects as parallel lines. Let's position the projection plane vertically, set the display angle with angles a, β and the position of the vanishing point (Fig. 2.21).

Fig.2.20. Cabinet projection

Rice. 2.21. Vertical central oblique projection: a – location of the projection plane, b – view from the left end of the projection plane

We will assume that the axis Ζ view coordinates is located perpendicular to the projection plane. The center of the view coordinates is at the point ( xc, us, zc). Let's write the corresponding aspect transformation:

As for the normal central projection, the vanishing point of the projection rays is located on the Z axis at a distance Ζk from the center of the view coordinates. It is necessary to take into account the inclination of the main axis of the oblique projection. To do this, it is enough to subtract from Υ pr the length of the segment is 0-0" (Fig. 2.21). This length is equal to ( Ζ k - Ζ pl) ctgβ. Now let’s write down the result - formulas for calculating the coordinates of an oblique vertical projection

Where Phew And Pu are the projection functions for normal projection.

It should be noted that for such a projection it is impossible to make a top view (β = 0), since here сtgP = ∞.

The property of the considered vertical oblique projection, which consists in maintaining the parallelism of vertical lines, is sometimes useful, for example, when depicting houses in architectural computer systems. Compare fig. 2. 22 (top) and fig. 2.22 (bottom). In the lower picture, verticals are depicted as verticals - houses do not “fall apart”.

Rice. 2.21. Comparison of projections

Cabinet projection (axonometric oblique frontal dimetric projection)

Rice. 2.23.Cabinet projection

Free projection (axonometric oblique horizontal isometric projection)

Rice. 2.24.Free projection

Central projection

Central projections of parallel lines not parallel to the projection plane converge at vanishing point.

Depending on the number of coordinate axes that the projection plane intersects, one, two and three-point central projections are distinguished.

Rice. 2.25. Central projection

Let's consider an example of a perspective (central) projection for a vertical camera position, when α = β = 0. Such a projection can be imagined as an image on glass through which an observer located above at point ( x, y, z) = (0, 0, z k). Here the projection plane is parallel to the plane (x 0 y), as shown in fig. 2.26.

For an arbitrary point in space (P), based on the similarity of triangles, we write the following proportions:

X pr /(z k – z pl) = x/(z k – z)

Y pr /(z k – z pl) = y/(z k – z)

Let's find the coordinates of the projection, also taking into account the coordinate Ζpr:

Let us write such coordinate transformations in functional form

Where Π - function of perspective transformation of coordinates.

Rice. 2.26 Perspective projection

In matrix form, the coordinate transformation can be written as follows:

Please note that here the matrix coefficients depend on the z coordinate (in the denominator of the fraction). This means that the coordinate transformation is nonlinear (more precisely, fractional-linear), it belongs to the class projective transformations.

We have obtained formulas for calculating projection coordinates for the case when the vanishing point of the rays is on the axis z. Now let's consider the general case. Let's introduce a view coordinate system (X, Υ,Ζ), arbitrarily located in three-dimensional space (x, y, z). Let the vanishing point be on the axis Ζ view coordinate system, and the viewing direction is along the axis Ζ opposite to its direction. We will assume that the transformation to view coordinates is described by a three-dimensional affine transformation

After calculating the coordinates ( X, Y, Z) you can calculate the coordinates in the projection plane in accordance with the formulas we have already discussed earlier. Since the vanishing point is on the Z-axis of the view coordinates, then

The sequence of coordinate transformation can be described as follows:

This coordinate transformation allows you to simulate camera locations at any point in space and display any viewing objects in the center of the projection plane.

Rice. 2.27. Central projection of point P 0 into the plane Z = d

Chapter 3. Raster graphics. Basic raster algorithms

By nature of distortion projections are divided into conformal, equal-area and arbitrary.

Conformal(or conformal) projections preserve the size of angles and shapes of infinitesimal figures. The length scale at each point is constant in all directions (which is ensured by a natural increase in the distances between adjacent parallels along the meridian) and depends only on the position of the point. Distortion ellipses are expressed as circles of different radii.

For each point in conformal projections the following dependencies are valid:

/ L i= a = b = m = n; a>= 0°; 0 = 90°; k = 1 And a 0 =0°(or ±90°).

Such projections especially useful for determining directions and laying routes along a given azimuth (for example, when solving navigation problems).

Equal size(or equivalent) projections do not distort the area. In these projections the areas of the distortion ellipses are equal. An increase in the length scale along one axis of the distortion ellipse is compensated by a decrease in the length scale along the other axis, which causes a natural decrease in the distances between adjacent parallels along the meridian and, as a consequence, a strong distortion of shapes.

Such projections are convenient for measuring areas objects (which, for example, is essential for some economic or morphometric maps).

In the theory of mathematical cartography it is proven that no, and there cannot be a projection that would be both equiangular and equal in area. In general, the greater the distortion of corners, the less distortion of areas and vice versa

free projections distort both angles and areas. When constructing them, they strive to find the most beneficial distribution of distortions for each specific case, reaching, as it were, some compromise. This group of projections used in cases where excessive distortion of corners and areas is equally undesirable. According to their properties, arbitrary projections lie between equiangular and equal-area. Among them we can highlight equidistant(or equidistant) projections, at all points of which the scale along one of the main directions is constant and equal to the main one.

Classification of map projections by type of auxiliary geometric surface .

Based on the type of auxiliary geometric surface, projections are distinguished: cylindrical, azimuthal and conical.

Cylindrical are called projections in which a network of meridians and parallels from the surface of the ellipsoid is transferred to the lateral surface of a tangent (or secant) cylinder, and then the cylinder is cut along the generatrix and unfolded into a plane (Fig. 6).

Fig.6. Normal cylindrical projection

Distortion is absent on the tangency line and is minimal near it. If the cylinder is secant, then there are two lines of tangency, which means 2 LNI. Distortion between LNIs is minimal.

Depending on the orientation of the cylinder relative to the axis of the earth's ellipsoid, projections are distinguished:

– normal, when the axis of the cylinder coincides with the minor axis of the earth’s ellipsoid; meridians in this case are equidistant parallel lines, and parallels are straight lines perpendicular to them;

– transverse, when the cylinder axis lies in the plane of the equator; grid type: the middle meridian and equator are mutually perpendicular straight lines, the remaining meridians and parallels are curved lines (Fig. c).

– oblique, when the cylinder axis makes an acute angle with the ellipsoid axis; in oblique cylindrical projections, meridians and parallels are curved lines.

Azimuthal are called projections in which the network of meridians and parallels is transferred from the surface of the ellipsoid to the tangent (or secant) plane (Fig. 7).

Rice. 7. Normal azimuthal projection

The image near the point of tangency (or section line) of the plane of the earth's ellipsoid is almost not distorted at all. The tangent point is the point of zero distortion.

Depending on the position of the point of tangency of the plane on the surface of the earth's ellipsoid, azimuthal projections are distinguished:

– normal, or polar, when the plane touches the Earth at one of the poles; type of grid: meridians - straight lines diverging radially from the pole, parallels - concentric circles with centers at the pole (Fig. 7);

– transverse, or equatorial, when the plane touches the ellipsoid at one of the points of the equator; grid type: the middle meridian and equator are mutually perpendicular straight lines, the remaining meridians and parallels are curved lines (in some cases, parallels are depicted as straight lines;

– oblique, or horizontal, when the plane touches the ellipsoid at some point lying between the pole and the equator. In oblique projections, only the middle meridian on which the tangent point is located is a straight line, the remaining meridians and parallels are curved lines.

Conical are called projections in which the network of meridians and parallels from the surface of the ellipsoid is transferred to the lateral surface of the tangent (or secant) cone (Fig. 8).

Rice. 8. Normal conic projection

Distortions are little noticeable along the line of tangency or two cross-section lines of the cone of the earth's ellipsoid, which are the line(s) of zero distortion of the LNI. Like cylindrical conical projections, they are divided into:

– normal, when the axis of the cone coincides with the minor axis of the earth’s ellipsoid; The meridians in these projections are represented by straight lines diverging from the apex of the cone, and the parallels are represented by arcs of concentric circles.

– transverse, when the axis of the cone lies in the plane of the equator; grid type: the middle meridian and parallel of tangency are mutually perpendicular straight lines, the remaining meridians and parallels are curved lines;

– oblique, when the axis of the cone makes an acute angle with the axis of the ellipsoid; in oblique conical projections, meridians and parallels are curved lines.

In normal cylindrical, azimuthal and conic projections, the map grid is orthogonal - meridians and parallels intersect at right angles, which is one of the important diagnostic features of these projections.

If, when obtaining cylindrical, azimuthal and conical projections, a geometric method is used (linear projection of an auxiliary surface onto a plane), then such projections are called perspective-cylindrical, perspective-azimuthal (ordinary perspective) and perspective-conical, respectively.

Polyconical are called projections in which a network of meridians and parallels from the surface of an ellipsoid is transferred to the lateral surfaces of several cones, each of which is cut along a generatrix and unfolded into a plane. In polyconic projections, parallels are depicted as arcs of eccentric circles, the central meridian is a straight line, all other meridians are curved lines symmetrical with respect to the central one.

Conditional are called projections, the construction of which does not resort to the use of auxiliary geometric surfaces. A network of meridians and parallels is built according to some predetermined condition. Among the conditional projections we can distinguish pseudocylindrical, pseudo-azimuth And pseudoconical projections that retain the appearance of parallels from the original cylindrical, azimuthal and conical projections. In these projections the middle meridian is a straight line, the other meridians are curved lines.

To conditional projections also include polyhedral projections , which are obtained by projecting onto the surface a polyhedron touching or cutting the earth's ellipsoid. Each face is an equilateral trapezoid (less commonly, hexagons, squares, rhombuses). A variety of polyhedral projections are multi-lane projections , and the strips can be cut along both meridians and parallels. Such projections are advantageous in that the distortion within each face or stripe is very small, so they are always used for multi-sheet maps. The main disadvantage of polyhedral projections is the impossibility of combining a block of map sheets into common frames without breaks.

In practice, the division by territorial coverage is valuable. By territorial coverage map projections are allocated for maps of the world, hemispheres, continents and oceans, maps of individual states and their parts. According to this principle Tables-determinants of cartographic projections were built. Besides, last time Attempts are being made to develop genetic classifications of map projections based on the form of differential equations describing them. These classifications cover the entire possible set of projections, but are extremely unclear, because are not related to the type of grid of meridians and parallels.