Types of projection geography. Classifications of map projections

Date of: 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 projection is a way of transitioning from the real, geometrically complex earth's surface.

A spherical surface cannot be rotated onto a plane without deformation - compression or stretching. This means that every map has some distortion or another. Distortions of lengths of areas, angles and shapes are distinguished. On large-scale maps (see), distortions can be almost imperceptible, but on small-scale maps they can be very large. Map projections have different properties depending on the nature and size of the distortions. Among them are:

Conformal projections. They preserve the angles and shapes of small objects without distortion, but the lengths and areas of objects are sharply deformed in them. Using maps compiled in such a projection, it is convenient to plot ship routes, but it is impossible to measure areas;

Equal area projections. They do not distort areas, but the angles and shapes in them are greatly distorted. Maps in equal area projections are convenient for determining the size of a state;
Equidistant. They have a constant length scale in one direction. Distortions of angles and areas are balanced in them;

Arbitrary projections. They have distortions of both angles and areas in any ratio.
Projections differ not only in the nature and size of distortions, but also in the type of surface that is used when moving from the geoid to the map plane. Among them are:

Cylindrical, when projection from the geoid goes to the surface of the cylinder. Cylindrical projections are most often used in. They have the least distortion at the equator and mid-latitudes. This projection is most often used to create world maps;

Conical. These projections were most often chosen to create maps of the former USSR. The least amount of distortion is with 47° conical projections. This is very convenient, since the main economic zones of this state were located between the indicated parallels and the maximum load of maps was concentrated here. But in conical projections, areas located in high latitudes and water areas are greatly distorted;

Azimuthal projection. This is a type of map projection when the design is carried out on a plane. This type of projection is used when creating maps of any other region of the Earth.

As a result of map projections, each point on the globe that has certain coordinates corresponds to one and only one point on the map.

In addition to cylindrical, conical and cartographic projections, there is a large class of conditional projections, in the construction of which they use not geometric analogues, but only mathematical equations of the required type.

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

Conformal(or conformal) projections preserve the magnitude 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 equatorial plane; 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 axis of the cylinder makes an acute angle with the axis of the ellipsoid; 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; meridians in these projections are represented by straight lines diverging from the apex of the cone, and parallels 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 conical 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 remaining 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 constructed. 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.

Map projection– a method of constructing an image of the Earth’s surface and, above all, a grid of meridians and parallels (coordinate grid) on a plane. In each projection, the coordinate grid is depicted differently, the nature of the distortions is also different, i.e. projections have certain differences, which makes it necessary to classify them. All map projections are usually classified according to two criteria:

By the nature of the distortions;

By the appearance of a normal grid of meridians and parallels.

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

1. Equiangular (comfortable) ) - projections in which infinitesimal figures on maps are similar to the corresponding figures on the earth's surface. These projections are widely used in air navigation, as they make it possible to most easily determine directions and angles. In addition, the configuration of small area landmarks is transmitted without distortion, which is essential for visual orientation.

2. Equal size (equivalent)– projections in which the ratio of areas on maps and on the earth’s surface is preserved. These projections have found application in small-scale overview geographic maps.

3. Equidistant– projections in which meridian distances and parallels are depicted without distortion. These projections are used to create reference maps.

4. free– projections that do not have any of the properties listed above. These projections are widely used in air navigation, as they have practically small distortions of angles, lengths and areas, which allows them to be ignored.

Based on the type of normal coordinate grid of meridians and parallels, projections are divided into: conical, polyconical, cylindrical and azimuthal.

The construction of a cartographic grid can be presented as the result of projecting the Earth's surface onto an auxiliary geometric figure: a cone, cylinder or plane (Fig. 2.2).

Rice. 2.2. Location of the auxiliary geometric figure

Depending on the location of the auxiliary geometric figure relative to the Earth’s axis of rotation, there are three types of projections (Fig. 2.2):

1. Normal– projections in which the axis of the auxiliary figure coincides with the axis of rotation of the Earth.

2. Transverse– projections in which the axis of the auxiliary figure is perpendicular to the axis of rotation of the Earth, i.e. coincides with the plane of the equator.

3. Oblique– projections in which the axis of the auxiliary figure makes an oblique angle with the axis of rotation of the Earth.

Conical projections. To solve air navigation problems, the normal equiangular conic projection, built on a tangent or secant cone, is used from all conic projections.

Normal conformal conic projection on a tangent cone. On maps compiled in this projection, the meridians look like straight lines converging towards the pole (Fig. 2.3). Parallels are arcs of concentric circles, the distance between which increases as they move away from the tangent parallel. In this projection, maps of scale 1: 2,000,000, 1: 2,500,000, 1: 4,000,000 and 1: 5,000,000 are published for aviation.

Rice. 2.3. Normal conformal conic projection on a tangent cone

Normal conformal conic projection on a secant cone. On maps compiled in this projection, meridians are depicted as straight converging lines, and parallels as circular arcs (Fig. 2.4). In this projection, maps of scale 1: 2,000,000 and 1: 2,500,000 are published for aviation.

Rice. 2.4. Normal conformal conic projection on

secant cone

Polyconic projections. Polyconic projections have no practical application in aviation, but it forms the basis of the international projection in which most aviation maps are published.

Modified polyconic (international) projection. In 1909 in London, an international committee developed a modified polyconic projection for maps at a scale of 1: 1,000,000, which was called international. Meridians in this projection look like straight lines converging to the pole, and parallels look like arcs of concentric circles (Fig. 2.5).

Rice. 2.5. Modified polyconic projection

The map sheet occupies 4° in latitude and 6° in longitude. Currently, this projection is the most common and most aviation maps are published in it at scales 1: 1,000,000, 1: 2,000,000 and 1: 4,000,000.

Cylindrical projections. Cylindrical projections have found application in air navigation normal, transverse And oblique projection.

Normal conformal cylindrical projection. This projection was proposed in 1569 by the Dutch cartographer Mercator. On maps compiled in this projection, the meridians look like straight lines, parallel to each other and spaced from each other at distances proportional to the difference in longitude (Fig. 2.6). Parallels are straight lines, perpendicular to the meridians. The distance between parallels increases with increasing latitude. Marine navigation charts are published in a normal conformal cylindrical projection.

Rice. 2.6. Normal conformal cylindrical projection

Conformal transverse cylindrical projection. This projection was proposed by the German mathematician Gauss. The projection is built according to mathematical laws. To reduce length distortion, the Earth's surface is cut into 60 zones. Each such zone occupies a longitude of 6°. From Fig. 2.7 it can be seen that the middle meridian in each zone and the equator are depicted by straight mutually perpendicular lines. All other meridians and parallels are depicted by curves of minor curvature. Maps of scales 1: 500,000, 1: 200,000 and 1: 100,000 and larger are compiled in a conformal transverse cylindrical projection.

Rice. 2.7. Conformal transverse cylindrical projection

Oblique conformal cylindrical projection. In this projection, the inclination of the cylinder to the axis of rotation of the Earth is selected so that its lateral surface touches the axis of the route (Fig. 2.8). Meridians and parallels in the projection under consideration have the form of curved lines. On maps in this projection, in a strip of 500–600 km from the center line of the route, the distortion of lengths does not exceed 0.5%. Maps of scales 1: 1,000,000, 1: 2,000,000 and 1: 4,000,000 are published in an oblique equiangular cylindrical projection to support flights on individual long routes.

Rice. 2.8. Oblique conformal cylindrical projection

Azimuthal projections. Of all the azimuthal projections, central and stereographic polar projections are mainly used for air navigation purposes.

Central polar projection. On maps compiled in this projection, the meridians look like straight lines diverging from the pole at an angle equal to the difference in longitude (Fig. 2.9). Parallels are concentric circles, the distances between which increase as they move away from the pole. Maps of the Arctic and Antarctic at scales 1:2,000,000 and 1:5,000,000 were previously published in this projection.

Rice. 2.10. Stereographic polar projection

Maps of the Arctic and Antarctic at scales 1: 2,000,000 and 1: 4,000,000 are published in stereographic polar projection.

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

Laying out a spheroid on a plane is no easier than flattening 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?