Map projections. Types of map projections and their essence For which maps is a cylindrical projection used?

Map projection is a mathematically defined method of displaying the surface of the earth's ellipsoid on a plane. It establishes a functional relationship between the geographic coordinates of points on the surface of the earth's ellipsoid and the rectangular coordinates of these points on the plane, i.e.

X= ƒ 1 (B, L) And Y= ƒ 2 (IN,L).

Cartographic projections are classified by the nature of distortions, by the type of auxiliary surface, by the type of normal grid (meridians and parallels), by the orientation of the auxiliary surface relative to the polar axis, etc.

By nature of distortion The following projections are distinguished:

1. equiangular, which convey the magnitude of angles without distortion and, therefore, do not distort the shapes of infinitesimal figures, and the length scale at any point remains the same in all directions. In such projections, distortion ellipses are depicted as circles of different radii (Fig. 2 A).

2. equal in size, in which there are no area distortions, i.e. The ratios of areas of areas on the map and the ellipsoid are preserved, but the shapes of infinitesimal figures and length scales in different directions are greatly distorted. Infinitesimal circles at different points of such projections are depicted as equal-area ellipses having different elongations (Fig. 2 b).

3. arbitrary, in which there are distortions in different proportions of both angles and areas. Among them, equidistant ones stand out, in which the length scale along one of the main directions (meridians or parallels) remains constant, i.e. the length of one of the axes of the ellipse is preserved (Fig. 2 V).

By type of auxiliary surface for design The following projections are distinguished:

1. Azimuthal, in which the surface of the earth's ellipsoid is transferred to a tangent or secant plane.

2. Cylindrical, in which the auxiliary surface is the lateral surface of the cylinder, tangent to the ellipsoid or cutting it.

3. Conical, in which the surface of the ellipsoid is transferred to the lateral surface of the cone, tangent to the ellipsoid or cutting it.

Based on the orientation of the auxiliary surface relative to the polar axis, projections are divided into:

A) normal, in which the axis of the auxiliary figure coincides with the axis of the earth's ellipsoid; in azimuthal projections, the plane is perpendicular to the normal, coinciding with the polar axis;

b) transverse, in which the axis of the auxiliary surface lies in the plane of the earth's equator; in azimuthal projections, the normal of the auxiliary plane lies in the equatorial plane;

V) oblique, in which the axis of the auxiliary surface of the figure coincides with the normal located between the earth’s axis and the equatorial plane; in azimuthal projections the plane is perpendicular to this normal.

Figure 3 shows various positions of the plane tangent to the surface of the earth's ellipsoid.

Classification of projections by type of normal grid (meridians and parallels) is one of the main ones. Based on this feature, eight classes of projections are distinguished.

a B C

Rice. 3. Types of projections by orientation

auxiliary surface relative to the polar axis.

A-normal; b-transverse; V- oblique.

1. Azimuthal. In normal azimuthal projections, meridians are depicted as straight lines converging at one point (pole) at angles equal to the difference in their longitudes, and parallels are depicted as concentric circles drawn from a common center (pole). In oblique and most transverse azimuthal projections, meridians, excluding the middle one, and parallels are curved lines. The equator in transverse projections is a straight line.

2. Conical. In normal conical projections, meridians are depicted as straight lines converging at one point at angles proportional to the corresponding differences in longitude, and parallels are depicted as arcs of concentric circles with the center at the point of convergence of the meridians. In oblique and transverse ones there are parallels and meridians, excluding the middle one, there are curved lines.

3. Cylindrical. In normal cylindrical projections, meridians are depicted as equidistant parallel lines, and parallels are depicted as lines perpendicular to them, which in general are not equidistant. In oblique and transverse projections, parallels and meridians, excluding the middle one, have the form of curved lines.

4. Polyconical. When constructing these projections, the network of meridians and parallels is transferred to several cones, each of which unfolds into a plane. Parallels, excluding the equator, are depicted by arcs of eccentric circles, the centers of which lie on the continuation of the middle meridian, which looks like a straight line. The remaining meridians are curves, symmetrical to the middle meridian.

5. Pseudo-azimuth, the parallels of which are concentric circles, and the meridians are curves that converge at the pole point and are symmetrical about one or two straight meridians.

6. Pseudoconic, in which parallels are arcs of concentric circles, and meridians are curved lines symmetrical with respect to the average rectilinear meridian, which may not be depicted.

7. Pseudocylindrical, in which parallels are depicted as parallel straight lines, and meridians as curves, symmetrical with respect to the average rectilinear meridian, which may not be depicted.

8. Circular, whose meridians, excluding the middle one, and parallels, excluding the equator, are depicted by arcs of eccentric circles. The middle meridian and equator are straight lines.

Conformal transverse cylindrical Gauss–Kruger projection. Projection zones. Counting order of zones and columns. Kilometer grid. Determining the zone of a topographic map sheet by digitizing a kilometer grid

The territory of our country is very large. This leads to significant distortions when it is transferred to a plane. For this reason, when constructing topographic maps in Russia, not the entire territory is transferred to the plane, but its individual zones, the length of which in longitude is 6°. To transfer zones, the transverse cylindrical Gauss–Kruger projection is used (used in Russia since 1928). The essence of the projection is that the entire earth's surface is depicted by meridional zones. Such a zone is obtained as a result of dividing the globe by meridians every 6°.

In Fig. Figure 2.23 shows a cylinder tangent to an ellipsoid, the axis of which is perpendicular to the minor axis of the ellipsoid.

When constructing a zone on a separate tangent cylinder, the ellipsoid and the cylinder have a common line of tangency, which runs along the middle meridian of the zone. When moving to a plane, it is not distorted and retains its length. This meridian, passing through the middle of the zone, is called axial meridian.

When the zone is projected onto the surface of the cylinder, it is cut along its generatrices and unfolded into a plane. When unfolded, the axial meridian is depicted without distortion of the straight line RR′ and it is taken as an axis X. Equator HER' also depicted by a straight line perpendicular to the axial meridian. It is taken as an axis Y. The origin of coordinates in each zone is the intersection of the axial meridian and the equator (Fig. 2.24).

As a result, each zone is a coordinate system in which the position of any point is determined by flat rectangular coordinates X And Y.

The surface of the earth's ellipsoid is divided into 60 six-degree longitude zones. The zones are counted from the Greenwich meridian. The first six-degree zone will have a value of 0°–6°, the second zone 6°–12°, etc.

The 6° wide zone adopted in Russia coincides with the column of sheets of the State Map at a scale of 1:1,000,000, but the zone number does not coincide with the number of the column of sheets of this map.

Check zones is underway from Greenwich meridian, A check columns – from meridian 180°.

As we have already said, the origin of coordinates of each zone is the point of intersection of the equator with the middle (axial) meridian of the zone, which is depicted in the projection by a straight line and is the abscissa axis. Abscissas are considered positive north of the equator and negative south. The ordinate axis is the equator. The ordinates are considered positive to the east and negative to the west of the axial meridian (Fig. 2.25).

Since the abscissas are measured from the equator to the poles, for the territory of Russia, located in the northern hemisphere, they will always be positive. The ordinates in each zone can be either positive or negative, depending on where the point is located relative to the axial meridian (in the west or east).

To make calculations convenient, it is necessary to get rid of negative ordinate values ​​within each zone. In addition, the distance from the axial meridian of the zone to the extreme meridian at the widest point of the zone is approximately 330 km (Fig. 2.25). To make calculations, it is more convenient to take a distance equal to a round number of kilometers. For this purpose, the axis X conditionally assigned to the west 500 km. Thus, the point with coordinates is taken as the origin of coordinates in the zone x = 0, y = 500 km. Therefore, the ordinates of points lying west of the axial meridian of the zone will have values ​​less than 500 km, and those of points lying east of the axial meridian will have values ​​of more than 500 km.

Since the coordinates of the points are repeated in each of the 60 zones, the ordinates are ahead Y indicate the zone number.

To plot points by coordinates and determine the coordinates of points on topographic maps, there is a rectangular grid. Parallel to the axes X And Y draw lines through 1 or 2 km (taken at map scale), and therefore they are called kilometer lines, and the grid of rectangular coordinates is kilometer grid.

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 and etc.). The development of military cartography and the further increase in the volume of topographic work in the 19th century. demanded 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 geometric 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 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 and 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 chosen 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 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. 5 B. 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

MAP PROJECTION AND ITS TYPES

Justification for choosing the topic of the paragraph

For our work we chose the topic “Map Projections”. Currently, this topic is practically not discussed in geography textbooks; information about various map projections can only be seen in the 6th grade atlas. We believe that students will be interested in knowing the principles by which various projections of geographic maps are selected and constructed. Questions about map projections are often raised in Olympiad assignments. They also appear on the Unified State Exam. In addition, atlas maps, as a rule, are built in different projections, which raises questions among students. Cartographic projection is the basis for constructing maps. Thus, knowledge of the basic principles of constructing map projections will be useful to students when choosing the professions of a pilot, sailor, and geologist. In this regard, we consider it appropriate to include this material in a geography textbook. Since at the 6th grade level the mathematical preparation of students is not yet so strong, in our opinion, it makes sense to study this topic at the beginning of the 7th grade in the section “General features of the nature of the Earth” when considering material about sources of geographic information.

Map projections

It is impossible to imagine a geographical map without a system of parallels and meridians that form it degree network. It is they that allow us to accurately determine the location of objects; it is from them that the sides of the horizon on the map are determined. Even distances on a map can be calculated using a degree network. If you look at the maps in the atlas, you will notice that the degree network looks different on different maps. On some maps, parallels and meridians intersect at right angles and form a grid of parallel and perpendicular lines. On other maps, meridians fan out from one melancholy, and parallels are represented as arcs. On a map of Antarctica, the meridians look like snowflakes, and the parallels extend from the center in concentric circles.

CREATING MAPS

The creation of cartographic works is carried out by the cartography section of cartography. Cartography is a branch of science, production and technology, covering the history of cartography and the study, creation and use of cartographic works. Maps are created using map projections - a method of transition from a real, geometrically complex earth's surface to the map plane. To do this, they first move on to a mathematically correct figure of an ellipsoid or bullet, and then project the image onto a plane using mathematical dependencies.

Types of projections

What is a map projection?

Map projection - a mathematically defined way of displaying a surface ellipsoid on surface. The system of depicting the network of meridians and parallels adopted for this map projection is called cartographic grid.

According to the method of constructing a cartographic normal mesh all projections are divided into conical, cylindrical, conditional, azimuthal, etc.

On conic projections when transferring the coordinate lines of the Earth to a plane, a cone is used. After obtaining an image on its surface, the cone is cut and unfolded onto the plane. To obtain a conical grid, the axis of the cone must exactly coincide with the axis of the Earth. On the resulting map, parallels are depicted as circular arcs, meridians - as straight lines emanating from one point. In such a projection, you can depict the northern or southern hemisphere of our planet, North America or Eurasia. In the process of studying geography, conic projections will most often be found in your atlases when constructing a map of Russia.

Map projections

On cylindrical projections obtaining a normal mesh is carried out by projecting it onto the walls of a cylinder, the axis of which coincides with the Earth's axis. Then it is unfolded onto a plane. The grid is obtained from mutually perpendicular straight lines of parallels and meridians.

On azimuthal projections a normal mesh is obtained immediately on the projection plane. To do this, the center of the plane is aligned with the Earth's pole. As a result, the parallels look like concentric circles, the radius of which increases with distance from the center, and the meridians look like straight lines intersecting in the center.

Conditional projections are built according to some predetermined conditions. This category cannot be classified as other types of projection. Their number is unlimited.

Of course, it is absolutely impossible to transfer an image from the surface of a ball to a plane. If we try this, we will inevitably end up with a tear in the image. However, we do not see these gaps on the map, and even when transferring the image to the surface of a cylinder, cone or plane, the image turns out to be uniform. What's the matter?

By projecting points from the surface of the globe onto the surface of a future map, we obtain distorted images. If we imagine projecting the Earth's surface onto a plane in the form of a shadow, which is obtained when highlighting an object from the center of the Earth, then the further the object is from the place of direct contact of the map surface with the ball, the more its image will change.

Based on the nature of distortion, all projections are divided into equiangular, equal-area and arbitrary.

On conformal projections Angles on the ground between any directions are equal to angles on the map between the same directions, that is, they (angles) do not have distortions. The scale depends only on the position of the point and does not depend on the direction. An angle on the ground is always equal to an angle on the map, a line that is straight on the ground is a straight line on the map. Infinitesimal figures on the map, due to the property of equiangularity, will be similar to the same figures on Earth. But the linear dimensions on the maps of this projection will have distortions. Imagine a perfectly round lake. No matter where it is located on the resulting map, its shape will remain round, but the dimensions can change significantly. The river bed will bend in the same way as it bends on the ground, but the distance between its bends will not correspond to the real one.

Equal area projection

On equal area projections The areas are not distorted, their proportionality is maintained. But the angles and shapes are greatly distorted. When its outline is transferred to the map at the point of contact between the ball and the surface of the future map, its image will be just as round. At the same time, the further it is located from the line of contact, the more its outlines will stretch out, although the area of ​​the lake will remain unchanged.

On arbitrary projections Both angles and areas are distorted, the similarity of the figures is not preserved, but they have some special properties that are not inherent in other projections, which is why they are the most used.

Maps are created either directly as a result of topographic surveys of the area, or on the basis of other maps, that is, ultimately, again as a result of surveying. Currently, the vast majority of topographic maps are created using the aerial photography method, which allows you to quickly obtain a topographic map of a vast territory. Many photographs (aerial photographs) of the area are taken from a flying airplane using special photographic devices. Then these aerial photographs are processed using special devices. Before becoming a map, a series of aerial photographs goes through a long and complex process in production.

Ellipsoid

All small-scale general geographic and special maps (including electronic GPS maps) are created on the basis of other maps, only on a larger scale.

Terms

Degree network- a system of meridians and parallels on geographic maps and globes, which serves to count the geographic coordinates of points on the earth’s surface - longitudes and latitudes.

Ellipsoid- closed surface. An ellipsoid can be obtained from the surface of a ball if the ball is compressed (stretched) in arbitrary ratios in three mutually perpendicular directions.

Normal mesh- a cartographic grid for each class of projections, the image of meridians and parallels of which has the simplest form.

Concentric circles- circles that have a common center and lie in the same plane.

Questions

1. What is a map projection? 2. What types of map projections do you know? 3. Which branch of cartography deals with the creation of projections? 4. What determines the nature of distortions on the map?

Work at home

1. Fill out a table in your notebook showing the characteristics of various map projections.

2. Determine in what projections the atlas maps are built. Which type of projection was used most often? Why?

A task for the curious

Using additional sources of information, find in which projection the map of the hemispheres is constructed.

Information resources for in-depth study of this topic

Literature on the topic

A.M. Berlyant "Map - the second language of geography: (essays on cartography)". 192 p. MOSCOW. EDUCATION. 1985

When transitioning from the physical surface of the Earth to its display on a plane (on a map), two operations are performed: projecting the earth's surface with its complex relief onto the surface of the earth's ellipsoid, the dimensions of which are established through geodetic and astronomical measurements, and depicting the surface of the ellipsoid on a plane using one of the cartographic projections.
A map projection is a specific way of displaying the surface of an ellipsoid on a plane.
Displaying the earth's surface on a plane is done in various ways. The simplest one is perspective . Its essence is to project an image from the surface of a model of the Earth (globe, ellipsoid) onto the surface of a cylinder or cone, followed by a turn into a plane (cylindrical, conical) or directly projecting a spherical image onto a plane (azimuthal).
One simple way to understand how map projections change spatial properties is to visualize the projection of light through the Earth onto a surface called a projection surface.
Imagine that the surface of the Earth is transparent, and a map grid is applied to it. Wrap a piece of paper around the Earth. A light source at the center of the Earth will cast shadows from the coordinate grid onto a piece of paper. You can now unfold the paper and lay it flat. The shape of the coordinate grid on the flat surface of paper is very different from its shape on the surface of the Earth (Fig. 5.1).

Rice. 5.1. Map grid of a geographic coordinate system projected onto a cylindrical surface

The map projection distorted the map grid; objects located near the pole are elongated.
Constructing in a prospective manner does not require the use of mathematical laws. Please note that in modern cartography, map grids are built analytical (mathematically) way. Its essence lies in calculating the position of nodal points (points of intersection of meridians and parallels) of the cartographic grid. The calculation is performed based on solving a system of equations that relate the geographic latitude and geographic longitude of nodal points ( φ, λ ) with their rectangular coordinates ( x, y) on surface. This dependence can be expressed by two equations of the form:

called map projection equations. They allow you to calculate rectangular coordinates x, y depicted point by geographic coordinates φ And λ . The number of possible functional dependencies and, therefore, projections is unlimited. It is only necessary that each point φ , λ the ellipsoid was represented on the plane by a uniquely corresponding point x, y and that the image is continuous.

5.2. DISTORTIONS

It is no easier to flatten a spheroid than to flatten a piece of watermelon peel. When moving to a plane, as a rule, angles, areas, shapes and lengths of lines are distorted, so for specific purposes it is possible to create projections that significantly reduce any one type of distortion, for example, areas. Cartographic distortion is a violation of the geometric properties of areas of the earth's surface and the objects located on them when they are depicted on a plane. .
Distortions of all types are closely related to each other. They are in such a relationship that a decrease in one type of distortion immediately entails an increase in the other. As area distortion decreases, angular distortion increases, etc. Rice. Figure 5.2 shows how three-dimensional objects are compressed so that they can be placed on a flat surface.

Rice. 5.2. Projecting a spherical surface onto a projection surface

On different maps, distortions can be of different sizes: on large-scale ones they are almost imperceptible, but on small-scale ones they can be very large.
In the mid-19th century, the French scientist Nicolas Auguste Tissot gave a general theory of distortion. In his work, he proposed using special distortion ellipses, which are infinitesimal ellipses at any point on the map, which are a reflection of infinitesimal circles at the corresponding point on the surface of the earth's ellipsoid or globe. The ellipse becomes a circle at the point of zero distortion. Changing the shape of the ellipse reflects the degree of distortion of angles and distances, and the size - the degree of distortion of areas.

Rice. 5.3. Ellipse on the map ( A) and the corresponding circle on the globe ( b)

The distortion ellipse on the map can occupy different positions relative to the meridian passing through its center. The orientation of the distortion ellipse on the map is usually determined azimuth of its semimajor axis . The angle between the north direction of the meridian passing through the center of the distortion ellipse and its nearest semimajor axis is called the orientation angle of the distortion ellipse. In Fig. 5.3, A this angle is indicated by the letter A 0 , and the corresponding angle on the globe α 0 (Fig. 5.3, b).
Azimuths of any direction on the map and on the globe are always measured from the northern direction of the meridian in a clockwise direction and can have values ​​from 0 to 360°.
Any arbitrary direction ( OK) on a map or globe ( ABOUT 0 TO 0 ) can be determined either by the azimuth of a given direction ( A- on the map, α - on the globe) or the angle between the semimajor axis closest to the northern direction of the meridian and this direction ( v- on the map, u- on the globe).

5.2.1. Length Distortions

Length distortion is a basic distortion. The remaining distortions follow logically from it. Length distortion means the inconstancy of the scale of a flat image, which manifests itself in a change in scale from point to point, and even at the same point, depending on the direction.
This means that there are 2 types of scale on the map:

• main scale (M);
• private scale .

Main scale maps call the degree of general reduction of the globe to certain dimensions of the globe, from which the earth's surface is transferred to a plane. It allows us to judge the decrease in the lengths of segments when transferring them from the globe to the globe. The main scale is written under the southern frame of the map, but this does not mean that the segment measured anywhere on the map will correspond to the distance on the earth's surface.
The scale at a given point on the map in a given direction is called private . It is defined as the ratio of an infinitesimal segment on a map dl TO to the corresponding segment on the surface of the ellipsoid dl Z . The ratio of the private scale to the main one, denoted by μ , characterizes the distortion of lengths

(5.3)

To assess the deviation of a particular scale from the main one, the concept is used zooming in (WITH), defined by the ratio

(5.4)

From formula (5.4) it follows that:

• at WITH= 1 private scale is equal to the main scale ( µ = M), i.e. there are no length distortions at a given point on the map in a given direction;
• at WITH> 1 private scale larger than the main one ( µ > M);
• at WITH < 1 частный масштаб мельче главного (µ < М ).

For example, if the main map scale is 1: 1,000,000, the zoom WITH is equal to 1.2, then µ = 1.2/1,000,000 = 1/833,333, i.e. one centimeter on the map corresponds to approximately 8.3 km on the ground. The partial scale is larger than the main one (the size of the fraction is larger).
When depicting the surface of a globe on a plane, the partial scales will be numerically larger or smaller than the main scale. If we take the main scale equal to unity ( M= 1), then the partial scales will be numerically greater or less than unity. In this case by a particular scale, numerically equal to the increase in scale, one should understand the ratio of an infinitesimal segment at a given point on the map in a given direction to the corresponding infinitesimal segment on the globe:

(5.5)

Deviation of private scale (µ )from one determines length distortion at a given point on the map in a given direction ( V):

V = µ - 1 (5.6)

Length distortion is often expressed as a percentage of unity, i.e., of the main scale, and is called relative length distortion :

q = 100(µ - 1) = V×100(5.7)

For example, when µ = 1.2 length distortion V= +0.2 or relative length distortion V= +20%. This means that a segment of length 1 cm, taken on the globe, will be depicted on the map as a segment of length 1.2 cm.
It is convenient to judge the presence of length distortion on a map by comparing the size of the meridian segments between adjacent parallels. If they are equal everywhere, then there is no distortion of lengths along the meridians, if there is no such equality (Fig. 5.5 segments AB And CD), then there is a distortion of line lengths.

Rice. 5.4. Part of a map of the eastern hemisphere showing cartographic distortions

If a map displays such a large area that it shows both the equator 0º and the parallel of 60° latitude, then it is not difficult to determine from it whether there is a distortion of lengths along the parallels. To do this, it is enough to compare the length of the segments of the equator and the parallel with a latitude of 60° between neighboring meridians. It is known that the parallel of 60° latitude is half as long as the equator. If the ratio of the indicated segments on the map is the same, then there is no distortion of the lengths along the parallels; otherwise it is available.
The greatest indicator of length distortion at a given point (the semimajor axis of the distortion ellipse) is denoted by a Latin letter A, and the smallest (semi-minor axis of the distortion ellipse) - b. Mutually perpendicular directions along which the largest and smallest length distortion rates apply, called the main directions .
To assess various distortions on maps, of all the private scales, the most important are the private scales in two directions: along the meridians and along the parallels. Private scale along the meridian usually denoted by a letter m , and the private scale along the parallel - letter n.
Within small-scale maps of relatively small territories (for example, Ukraine), deviations of length scales from the scale indicated on the map are small. Errors in measuring lengths in this case do not exceed 2 - 2.5% of the measured length, and they can be neglected when working with school maps. Some maps include a measuring scale and explanatory text for approximate measurements.
On nautical charts , constructed in the Mercator projection and on which the loxodrome is depicted as a straight line, no special linear scale is given. Its role is played by the eastern and western frames of the map, which are meridians divided into divisions every 1′ in latitude.
In maritime navigation, distances are usually measured in nautical miles. Nautical mile - this is the average length of a meridian arc of 1′ in latitude. It contains 1852 m. Thus, the nautical chart frames are actually divided into segments equal to one nautical mile. By determining the straight line distance between two points on the map in meridian minutes, we obtain the actual distance in nautical miles along the loxodrome.

Figure 5.5. Measuring distances using a sea map.

5.2.2. Angle distortion

Distortions of angles logically follow from distortions of lengths. The difference in angles between the directions on the map and the corresponding directions on the surface of the ellipsoid is taken as a characteristic of the distortion of angles on the map.
For the corner distortion indicator between the lines of the cartographic grid, the value of their deviation from 90° is taken and denoted by a Greek letter ε (epsilon).
ε = Ө - 90°, (5.8)
where in Ө (theta) - the angle measured on the map between the meridian and the parallel.

Figure 5.4 indicates that the angle Ө is equal to 115°, therefore ε = 25°.
At the point where the angle of intersection of the meridian and the parallel remains straight on the map, the angles between other directions can be changed on the map, since at any given point the amount of distortion of the angles can change with a change in direction.
The general indicator of angle distortion ω (omega) is taken to be the greatest angle distortion at a given point, equal to the difference between its value on the map and on the surface of the earth’s ellipsoid (sphere). When known x indicators A And b size ω determined by the formula:

(5.9)

5.2.3. Area distortions

Area distortions logically follow from length distortions. The deviation of the area of ​​the distortion ellipse from the original area on the ellipsoid is taken as a characteristic of area distortion.
A simple way to identify distortion of this type is to compare the areas of the cells of the cartographic grid, limited by parallels of the same name: if the areas of the cells are equal, there is no distortion. This occurs, in particular, on the map of the hemisphere (Fig. 4.4), on which the shaded cells differ in shape, but have the same area.
Area distortion indicator (R) is calculated as the product of the largest and smallest length distortion indicators at a given location on the map
p = a×b (5.10)
The main directions at a given point on the map may coincide with the lines of the cartographic grid, but may not coincide with them. Then the indicators A And b according to known m And n calculated using the formulas:

(5.11)
(5.12)

The distortion factor included in the equations R in this case they will recognize by the work:

Where ε (epsilon) - the deviation value of the intersection angle of the cartographic grid from 9 0°.

5.2.4. Distortions of forms

Distortion of forms consists in the fact that the shape of a site or territory occupied by an object on the map is different from its shape on the level surface of the Earth. The presence of this type of distortion on the map can be established by comparing the shape of the cells of the cartographic grid located at the same latitude: if they are the same, then there is no distortion. In Figure 5.4, two shaded cells with a difference in shape indicate the presence of a distortion of this type. You can also identify the distortion of the shape of a certain object (continent, island, sea) by the ratio of its width and length on the analyzed map and on the globe.
Shape distortion index (k) depends on the difference of the largest ( A) and the smallest ( b) indicators of length distortion at a given location on the map and is expressed by the formula:

(5.14)

When researching and choosing a map projection, use isokols - lines of equal distortion. They can be plotted on the map as dotted lines to show the magnitude of the distortion.

Rice. 5.6. Isocols of the greatest angle distortions

5.3. CLASSIFICATION OF PROJECTIONS BY NATURE OF DISTORTION

For different purposes, projections with different types of distortion are created. The nature of projection distortions is determined by the absence of certain distortions in it (angles, lengths, areas). Depending on this, all cartographic projections are divided into four groups according to the nature of distortions:
— equiangular (conformal);
- equidistant (equidistant);
— equal in size (equivalent);
- arbitrary.

5.3.1. Conformal projections

Equiangular These are called projections in which directions and angles are depicted without distortion. Angles measured on conformal projection maps are equal to the corresponding angles on the earth's surface. An infinitesimal circle in these projections always remains a circle.
In equiangular projections, the length scales at any point in all directions are the same, so they do not have distortion of the shape of infinitesimal figures and no distortion of angles (Fig. 5.7, B). This general property of conformal projections is expressed by the formula ω = 0°. But the shapes of real (finite) geographical objects that occupy entire areas on the map are distorted (Fig. 5.8, a). Conformal projections exhibit particularly large area distortions (as clearly demonstrated by distortion ellipses).

Rice. 5.7. View of distortion ellipses in equal-area projections —- A, equiangular - B, arbitrary - IN, including equidistant along the meridian - G and equidistant along the parallel - D. The diagrams show 45° angle distortion.

These projections are used to determine directions and lay out routes along a given azimuth, so they are always used on topographic and navigation maps. The disadvantage of conformal projections is that their areas are greatly distorted (Fig. 5.7, a).

Rice. 5.8. Distortions in cylindrical projection:
a - equiangular; b - equidistant; c - equal in size

5.6.2. Equidistant projections

Equidistant projections are projections in which the length scale of one of the main directions is preserved (remains unchanged) (Fig. 5.7, D. Fig. 5.7, E). They are used mainly for creating small-scale reference maps and star maps.

5.6.3. Equal Area Projections

Equal in size are called projections in which there are no area distortions, i.e. the area of ​​a figure measured on a map is equal to the area of ​​the same figure on the surface of the Earth. In equal area map projections, the area scale is the same size everywhere. This property of equal area projections can be expressed by the formula:

An inevitable consequence of the equal size of these projections is a strong distortion of their angles and shapes, which is well explained by the distortion ellipses (Fig. 5.7, A).

5.6.4. Arbitrary projections

To arbitrary These include projections in which there are distortions of lengths, angles and areas. The need to use arbitrary projections is explained by the fact that when solving some problems there is a need to measure angles, lengths and areas on one map. But no projection can be both equiangular, equidistant, and equal in area at the same time. It was said earlier that as the imaged area of ​​the Earth’s surface on the plane decreases, image distortion also decreases. When depicting small areas of the earth's surface in an arbitrary projection, the magnitude of distortions of angles, lengths and areas are insignificant, and when solving many problems they can be ignored.

5.4. CLASSIFICATION OF PROJECTIONS ACCORDING TO THE TYPE OF NORMAL CARTOGRAPHIC GRID

In cartographic practice, a common classification of projections is based on the type of auxiliary geometric surface that can be used in their construction. From this point of view, projections are distinguished: cylindrical when the lateral surface of the cylinder serves as the auxiliary surface; conical, when the auxiliary plane is the lateral surface of the cone; azimuthal, when the auxiliary surface is a plane (picture plane).
The surfaces on which the globe is projected can be tangent to it or secant to it. They can be oriented differently.
Projections, during the construction of which the axes of the cylinder and cone were aligned with the polar axis of the globe, and the picture plane onto which the image was projected was placed tangentially at the pole point, are called normal.
The geometric construction of these projections is very clear.

5.4.1. Cylindrical projections

For simplicity of reasoning, we will use a ball instead of an ellipsoid. Let us enclose the ball in a cylinder tangent to the equator (Fig. 5.9, a).

Rice. 5.9. Construction of a map grid in an equal-area cylindrical projection

Let us continue the planes of the meridians PA, PB, PV, ... and take the intersections of these planes with the side surface of the cylinder as the image of the meridians on it. If we cut the side surface of the cylinder along generatrix aAa 1 and unfold it onto a plane, then the meridians will be depicted as parallel, equally spaced straight lines aAa 1 , bBBb 1 , vVv 1 ..., perpendicular to the equator ABC.
The image of parallels can be obtained in various ways. One of them is the continuation of the planes of parallels until they intersect with the surface of the cylinder, which will give in the development a second family of parallel straight lines perpendicular to the meridians.
The resulting cylindrical projection (Fig. 5.9, b) will be equal in size, since the lateral surface of the spherical belt AGED, equal to 2πRh (where h is the distance between the planes AG and ED), corresponds to the image area of ​​this belt in the scan. The main scale is maintained along the equator; partial scales along the parallel increase, and along the meridians they decrease with distance from the equator.
Another way to determine the position of parallels is based on preserving the lengths of the meridians, i.e., preserving the main scale along all meridians. In this case, the cylindrical projection will be equidistant along the meridians(Fig. 5.8, b).
For equiangular A cylindrical projection requires constancy of scale in all directions at any point, which requires an increase in scale along the meridians as one moves away from the equator in accordance with an increase in scale along parallels at the corresponding latitudes (see Fig. 5.8, a).
Often, instead of a tangent cylinder, a cylinder is used that cuts the sphere along two parallels (Fig. 5.10), along which the main scale is preserved during development. In this case, the partial scales along all parallels between the parallels of the section will be smaller, and on the remaining parallels they will be larger than the main scale.

Rice. 5.10. A cylinder cutting a ball along two parallels

5.4.2. Conic projections

To construct a conical projection, we enclose the ball in a cone tangent to the ball along the parallel ABCD (Fig. 5.11, a).

Rice. 5.11. Construction of a map grid in an equidistant conic projection

Similar to the previous construction, we will continue the planes of the meridians PA, PB, PV, ... and take their intersections with the lateral surface of the cone as the image of the meridians on it. After unfolding the lateral surface of the cone on a plane (Fig. 5.11, b), the meridians will be depicted as radial straight lines TA, TB, TV,..., emanating from point T. Please note that the angles between them (convergence of the meridians) will be proportional (but are not equal) to differences in longitude. Along the parallel of tangency ABC (circular arc of radius TA), the main scale is maintained.
The position of other parallels, depicted by arcs of concentric circles, can be determined from certain conditions, one of which - maintaining the main scale along the meridians (AE = Ae) - leads to a conical equidistant projection.

5.4.3. Azimuthal projections

To construct an azimuthal projection, we will use a plane tangent to the ball at the pole point P (Fig. 5.12). The intersections of the meridian planes with the tangent plane give an image of the meridians Pa, Pe, Pv,... in the form of straight lines, the angles between which are equal to the differences in longitude. Parallels, which are concentric circles, can be defined in various ways, for example, by drawing radii equal to the straightened arcs of the meridians from the pole to the corresponding parallel PA = Pa. This projection will be equidistant By meridians and preserves the main scale along them.

Rice. 5.12. Construction of a map grid in azimuthal projection

A special case of azimuthal projections are promising projections constructed according to the laws of geometric perspective. In these projections, each point on the surface of the globe is transferred to the picture plane along rays emanating from one point WITH, called a point of view. Depending on the position of the point of view relative to the center of the globe, projections are divided into:

• central - the point of view coincides with the center of the globe;
• stereographic - the point of view is located on the surface of the globe at a point diametrically opposite to the point of contact of the picture plane with the surface of the globe;
• external - the point of view is taken outside the globe;
• orthographic - the point of view is taken to infinity, i.e. the design is carried out by parallel rays.

Rice. 5.13. Types of perspective projections: a - central;
b - stereographic; c - external; g - orthographic.

5.4.4. Conditional projections

Conditional projections are projections for which simple geometric analogues cannot be found. They are built based on any given conditions, for example, the desired type of geographic grid, a particular distribution of distortions on the map, a given type of grid, etc. In particular, pseudo-cylindrical, pseudo-conical, pseudo-azimuthal and other projections obtained by transforming one or several initial projections.
U pseudocylindrical projections, the equator and parallels are straight lines parallel to each other (which makes them similar to cylindrical projections), and the meridians are curves that are symmetrical with respect to the average rectilinear meridian (Fig. 5.14)

Rice. 5.14. View of the map grid in pseudocylindrical projection.

U pseudoconical projections of parallels are arcs of concentric circles, and meridians are curves symmetrical with respect to the average rectilinear meridian (Fig. 5.15);

Rice. 5.15. Map grid in one of the pseudoconic projections

Building a mesh in polyconical projection can be represented by projecting sections of the globe's degree grid onto the surface several tangent cones and subsequent development into the plane of the stripes formed on the surface of the cones. The general principle of such a design is shown in Figure 5.16.

Rice. 5.16. The principle of constructing a polyconic projection:
a - position of the cones; b - stripes; c - scan

Letters S The vertices of the cones are indicated in the figure. For each cone, a latitudinal section of the globe surface is projected adjacent to the parallel of tangency of the corresponding cone.
It is typical for the external appearance of cartographic grids in a polyconic projection that the meridians have the form of curved lines (except for the middle one - straight), and the parallels are arcs of eccentric circles.
In polyconic projections used to construct world maps, the equatorial section is projected onto a tangent cylinder, so on the resulting grid the equator has the shape of a straight line perpendicular to the middle meridian.
After scanning the cones, an image of these areas is obtained in the form of stripes on a plane; the stripes touch along the middle meridian of the map. The final appearance of the mesh is obtained after eliminating the gaps between the strips by stretching (Fig. 5.17).

Rice. 5.17. Map grid in one of the polyconic

Polyhedral projections - projections obtained by projecting onto the surface of a polyhedron (Fig. 5.18), tangent or secant to a ball (ellipsoid). Most often, each face is an equilateral trapezoid, although other options are possible (for example, hexagons, squares, rhombuses). A variety of polyhedral ones are multi-lane projections, Moreover, the stripes 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. Topographical and survey-topographical ones are created exclusively in a multifaceted projection, and the frame of each sheet is a trapezoid composed of lines of meridians and parallels. You have to “pay for this” - a block of map sheets cannot be combined into common frames without breaks.

Rice. 5.18. Scheme of a polyhedral projection and arrangement of map sheets

It should be noted that nowadays auxiliary surfaces are not used to obtain map projections. No one puts a ball in a cylinder and puts a cone on it. These are just geometric analogies that allow us to understand the geometric essence of projection. The search for projections is carried out analytically. Computer modeling allows you to quickly calculate any projection with given parameters, and automatic plotters easily draw the appropriate grid of meridians and parallels, and, if necessary, an isocol map.
There are special projection atlases that allow you to select the right projection for any territory. Recently, electronic projection atlases have been created, with the help of which it is easy to find a suitable mesh, immediately evaluate its properties, and, if necessary, carry out certain modifications or transformations interactively.

5.5. CLASSIFICATION OF PROJECTIONS DEPENDING ON THE ORIENTATION OF THE AUXILIARY CARTOGRAPHIC SURFACE

Normal projections - the projection plane touches the globe at the pole point or the axis of the cylinder (cone) coincides with the axis of rotation of the Earth (Fig. 5.19).

Rice. 5.19. Normal (direct) projections

Transverse projections - the design plane touches the equator at any point or the axis of the cylinder (cone) coincides with the equatorial plane (Fig. 5.20).

Rice. 5.20. Transverse projections

Oblique projections - the design plane touches the globe at any given point (Fig. 5.21).

Rice. 5.21. Oblique projections

Of the oblique and transverse projections, oblique and transverse cylindrical, azimuthal (perspective) and pseudo-azimuthal projections are most often used. Transverse azimuthal ones are used for maps of hemispheres, oblique ones - for territories that have a rounded shape. Maps of continents are often drawn up in transverse and oblique azimuthal projections. The transverse cylindrical Gauss-Kruger projection is used for state topographic maps.

5.6. SELECTION OF PROJECTIONS

The choice of projections is influenced by many factors, which can be grouped as follows:

• geographical features of the mapped territory, its position on the globe, size and configuration;
• purpose, scale and subject of the map, expected range of consumers;
• conditions and methods of using the map, tasks that will be solved using the map, requirements for the accuracy of measurement results;
• features of the projection itself - the magnitude of distortions of lengths, areas, angles and their distribution over the territory, the shape of meridians and parallels, their symmetry, the image of the poles, the curvature of the lines of the shortest distance.

The first three groups of factors are set initially, the fourth depends on them. If a map is being compiled for navigation purposes, the equiangular cylindrical Mercator projection must be used. If Antarctica is being mapped, then the normal (polar) azimuthal projection, etc. will almost certainly be adopted.
The significance of these factors can be different: in one case, visibility is put in first place (for example, for a wall school map), in another - the features of using the map (navigation), in the third - the position of the territory on the globe (polar region). Any combinations are possible, and therefore different projection options are possible. Moreover, the choice is very large. But it is still possible to indicate some preferred and most traditional projections.
World maps usually drawn up in cylindrical, pseudocylindrical and polyconical projections. To reduce distortion, secant cylinders are often used, and pseudo-cylindrical projections are sometimes produced with discontinuities on the oceans.
Hemisphere maps always constructed in azimuthal projections. For the western and eastern hemispheres it is natural to take transverse (equatorial), for the northern and southern hemispheres - normal (polar), and in other cases (for example, for the continental and oceanic hemispheres) - oblique azimuthal projections.
Continent maps Europe, Asia, North America, South America, Australia and Oceania are most often built in equal-area oblique azimuthal projections, for Africa they take transverse ones, and for Antarctica - normal azimuthal ones.
Maps of individual countries , administrative regions, provinces, states are performed in oblique equiangular and equal-area conical or azimuthal projections, but much depends on the configuration of the territory and its position on the globe. For small areas, the problem of choosing a projection loses its relevance; you can use different conformal projections, keeping in mind that area distortions in small areas are almost imperceptible.
Topographic maps Ukraine is created in the Gaussian transverse cylindrical projection, and the USA and many other Western countries are created in the universal transverse cylindrical Mercator projection (abbreviated UTM). Both projections are similar in their properties; Essentially, both are multi-cavity.
Nautical and aeronautical charts are always given exclusively in the cylindrical Mercator projection, and thematic maps of the seas and oceans are created in a wide variety of, sometimes quite complex, projections. For example, to show the Atlantic and Arctic oceans together, special projections with oval isocoles are used, and to depict the entire World Ocean, equal-area projections with breaks on the continents are used.
In any case, when choosing a projection, especially for thematic maps, it should be borne in mind that usually distortions on the map are minimal in the center and quickly increase towards the edges. In addition, the smaller the scale of the map and the more extensive the spatial coverage, the more attention must be paid to the “mathematical” factors in choosing a projection, and vice versa - for small areas and large scales, “geographical” factors become more significant.

5.7. PROJECTION RECOGNITION

To recognize the projection in which a map is drawn means to establish its name, to determine whether it belongs to a particular type or class. This is necessary in order to have an idea about the properties of the projection, the nature, distribution and magnitude of distortions - in a word, in order to know how to use the map and what can be expected from it.
Some normal projections at once recognized by the appearance of meridians and parallels. For example, normal cylindrical, pseudocylindrical, conical, and azimuthal projections are easily recognizable. But even an experienced cartographer does not immediately recognize many arbitrary projections; special measurements on the map will be required to identify their equiangularity, equilaterality or equidistance in one of the directions. There are special techniques for this: first, they establish the shape of the frame (rectangle, circle, ellipse), determine how the poles are depicted, then measure the distances between adjacent parallels along the meridian, the areas of adjacent grid cells, the angles of intersection of the meridians and parallels, the nature of their curvature, etc. .P.
There are special projection definition tables for maps of the world, hemispheres, continents and oceans. Having carried out the necessary measurements on the grid, you can find the name of the projection in such a table. This will give an idea of ​​its properties, will allow you to evaluate the possibilities of quantitative determinations on this map, and select the appropriate map with isocols for making corrections.

Video
Types of projections according to the nature of distortions

Questions for self-control:

1. What elements make up the mathematical basis of a map?
2. What is the scale of a geographic map?
3. What is the main map scale?
4. What is a private map scale?
5. What causes the deviation of a particular scale from the main one on a geographical map?
6. How to measure the distance between points on a sea map?
7. What is a distortion ellipse and what is it used for?
8. How can you determine the largest and smallest scales from the distortion ellipse?
9. What methods exist for transferring the surface of the earth's ellipsoid onto a plane, what is their essence?
10. What is a map projection called?
11. How are projections classified according to the nature of their distortions?
12. What projections are called conformal, how to depict an ellipse of distortion on these projections?
13. What projections are called equidistant, how to depict a distortion ellipse on these projections?
14. What projections are called equal area, how to depict an ellipse of distortion on these projections?
15. What projections are called arbitrary?

People have been using geographic maps since ancient times. The first attempts to depict it were made in Ancient Greece by scientists such as Eratosthenes and Hipparchus. Naturally, cartography as a science has come a long way since then. Modern maps are created using satellite imagery and computer technology, which, of course, helps to increase their accuracy. And yet, on every geographical map there are some distortions regarding the natural shapes, angles or distances on the earth's surface. The nature of these distortions, and therefore the accuracy of the map, depends on the types of map projections used to create a particular map.

Concept of map projection

Let us examine in more detail what a cartographic projection is and what types of them are used in modern cartography.

A map projection is an image on a plane. A more profound definition from a scientific point of view sounds like this: a cartographic projection is a method of displaying points on the Earth’s surface on a certain plane, in which some analytical relationship is established between the coordinates of the corresponding points of the displayed and displayed surfaces.

How is a map projection constructed?

The construction of any type of map projections occurs in two stages.

1. First, the geometrically irregular surface of the Earth is mapped onto some mathematically regular surface, which is called the surface of relevance. For the most accurate approximation, the geoid is most often used in this capacity - a geometric body limited by the water surface of all seas and oceans that are interconnected (sea level) and have a single water mass. At each point on the surface of the geoid, the force of gravity is applied normally. However, the geoid, like the physical surface of the planet, also cannot be expressed by a single mathematical law. Therefore, instead of the geoid, an ellipsoid of revolution is taken as the surface of reference, giving it maximum similarity to the geoid using the degree of compression and orientation in the body of the Earth. This body is called the earth's ellipsoid or reference ellipsoid, and different countries take different parameters for them.
2. Secondly, the accepted surface of relevance (reference ellipsoid) is transferred to the plane using one or another analytical dependence. As a result, we get a flat map projection

Projection distortion

Have you ever wondered why the outlines of continents are slightly different on different maps? Some map projections make some parts of the world appear larger or smaller relative to some landmarks than others. It's all about the distortion with which the projections of the Earth are transferred to a flat surface.

But why do map projections appear distorted? The answer is quite simple. It is not possible to unfold a spherical surface on a plane without folds or tears. Therefore, the image from it cannot be displayed without distortion.

Methods for obtaining projections

When studying map projections, their types and properties, it is necessary to mention the methods of their construction. So, map projections are obtained using two main methods:

• geometric;
• analytical.

At the core geometric method are the laws of linear perspective. Our planet is conventionally assumed to be a sphere of some radius and projected onto a cylindrical or conical surface, which can either touch or cut through it.

Projections obtained in this way are called perspective. Depending on the position of the observation point relative to the Earth’s surface, perspective projections are divided into types:

• gnomonic or central (when the point of view is combined with the center of the earthly sphere);
• stereographic (in this case, the observation point is located on the surface of reference);
• orthographic (when the surface is observed from any point outside the Earth’s sphere; the projection is constructed by transferring points of the sphere using parallel lines perpendicular to the mapping surface).

Analytical method construction of map projections is based on mathematical expressions connecting points on the sphere of relevance and the display plane. This method is more universal and flexible, allowing you to create arbitrary projections according to a predetermined nature of the distortion.

Types of map projections in geography

Many types of Earth projections are used to create geographic maps. They are classified according to various criteria. In Russia, the Kavraisky classification is used, which uses four criteria that determine the main types of map projections. The following are used as characteristic classification parameters:

• nature of distortion;
• form of displaying coordinate lines of a normal grid;
• location of the pole point in the normal coordinate system;
• mode of application.

So, what types of map projections exist according to this classification?

Classification of projections

By nature of distortion

As mentioned above, distortion is essentially an inherent property of any Earth projection. Any surface characteristic can be distorted: length, area or angle. By type of distortion there are:

• Conformal or conformal projections, in which azimuths and angles are transferred without distortion. The coordinate grid in conformal projections is orthogonal. Maps obtained in this way are recommended to be used to determine distances in any direction.
• Equal area or equivalent projections, where the scale of the areas is preserved, which is taken equal to one, i.e. the areas are displayed without distortion. Such maps are used to compare areas.
• Equidistant or equidistant projections, during the construction of which the scale is preserved along one of the main directions, which is assumed to be unit.
• Arbitrary projections, which may contain all types of distortions.

According to the form of displaying the coordinate lines of the normal grid

This classification is as clear as possible and, therefore, easiest to understand. Note, however, that this criterion applies only to projections oriented normal to the observation point. So, based on this characteristic feature, the following types of map projections are distinguished:

Circular, where parallels and meridians are represented by circles, and the equator and middle meridian of the grid are represented by straight lines. Similar projections are used to depict the surface of the Earth as a whole. Examples of circular projections are the Lagrange conformal projection, as well as the arbitrary Grinten projection.

Azimuthal. In this case, parallels are represented in the form of concentric circles, and meridians in the form of a bundle of straight lines diverging radially from the center of the parallels. This type of projection is used in a direct position to display the Earth’s poles with adjacent territories, and in a transverse position as a map of the western and eastern hemispheres, familiar to everyone from geography lessons.

Cylindrical, where meridians and parallels are represented by straight lines intersecting normally. With minimal distortion, territories adjacent to the equator or stretched along a certain standard latitude are displayed here.

Conical, representing a development of the lateral surface of the cone, where the lines of parallels are arcs of circles with a center at the apex of the cone, and the meridians are guides diverging from the apex of the cone. Such projections most accurately depict territories located in mid-latitudes.

Pseudoconic projections are similar to conical ones, only the meridians in this case are depicted by curved lines, symmetrical with respect to the rectilinear axial meridian of the grid.

Pseudocylindrical projections resemble cylindrical ones, only, just like in pseudoconical ones, the meridians are depicted by curved lines symmetrical to the axial rectilinear meridian. Used to depict the entire Earth (for example, Mollweide's elliptical projection, Sanson's equal-area sinusoidal, etc.).

Polyconical, where parallels are depicted in the form of circles, the centers of which are located on the middle meridian of the grid or its extension, meridians in the form of curves located symmetrically to a rectilinear

By the position of the pole point in the normal coordinate system

• Polar or normal- the pole of the coordinate system coincides with the geographic pole.
• Transverse or transversion- the pole of the normal system is aligned with the equator.
• Oblique or inclined- the pole of a normal coordinate grid can be located at any point between the equator and the geographic pole.

By method of application

According to the method of use, the following types of map projections are distinguished:

• Solid- projection of the entire territory onto a plane is carried out according to a single law.
• Multiband- the mapped area is conditionally divided into several latitudinal zones, which are projected onto the display plane according to a single law, but with changing parameters for each zone. An example of such a projection is the trapezoidal Müfling projection, which was used in the USSR for large-scale maps until 1928.
• Multifaceted- the territory is conditionally divided into a certain number of zones according to longitude, projection onto a plane is carried out according to a single law, but with different parameters for each zone (for example, the Gauss-Kruger projection).
• Composite, when some part of the territory is displayed on a plane using one pattern, and the rest of the territory using another.

The advantage of both multi-lane and multi-faceted projections is the high accuracy of display within each zone. However, a significant drawback is the impossibility of obtaining a continuous image.

Of course, each map projection can be classified using each of the above criteria. Thus, the famous Mercator projection of the Earth is conformal (equiangular) and transverse (transversion); Gauss-Kruger projection - conformal transverse cylindrical, etc.