On the world map of the hemispheres, the greatest distortion. Map projections and distortions. The westernmost point of Asia is the cape

When moving 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 an earth ellipsoid, the dimensions of which are established by means of geodetic and astronomical measurements, and the image of the ellipsoid surface 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.
The display of the earth's surface on a plane is made different ways. The simplest one is perspective . Its essence lies in projecting an image from the surface of the Earth model (globe, ellipsoid) onto the surface of a cylinder or cone, followed by a turn into a plane (cylindrical, conical) or direct projection of a spherical image onto a plane (azimuth).
One of simple ways understanding how map projections change spatial properties is to visualize the projection of light through the Earth onto a surface called the projection surface.
Imagine that the surface of the Earth is transparent and has a map grid on it. Wrap a piece of paper around the earth. A light source at the center of the earth will cast shadows from the grid onto the piece of paper. You can now unfold the paper and lay it flat. The shape of the coordinate grid on a flat surface of paper is very different from its shape on the surface of the Earth (Fig. 5.1).

Rice. 5.1. Geographic coordinate system grid projected onto a cylindrical surface

The map projection distorted the cartographic grid; objects near the pole are elongated.
Building in a perspective way does not require the use of the laws of mathematics. Please note that in modern cartography, cartographic grids are built analytical (mathematical) way. Its essence lies in the calculation of the position of nodal points (points of intersection of meridians and parallels) of the cartographic grid. The calculation is performed on the basis of 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 displayed 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 depicted on the plane by a uniquely corresponding point x, y and that the image is continuous.

5.2. DISTORTION

Decomposing a spheroid onto a plane is no easier than flattening a piece of watermelon peel. When going 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 will significantly reduce any one type of distortion, for example, areas. Cartographic distortion is a violation of the geometric properties of sections of the earth's surface and objects located on them when they are depicted on a plane. .
Distortions of all kinds are closely related. They are in such a relationship that a decrease in one type of distortion immediately leads to an increase in another. As area distortion decreases, angle distortion increases, and so on. Rice. Figure 5.2 shows how 3D objects are compressed to fit 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 maps they are almost imperceptible, but on small-scale maps they can be very large.
In the middle of the 19th century, the French scientist Nicolas August Tissot gave a general theory of distortions. In his work, he proposed to use special distortion ellipses, which are infinitesimal ellipses at any point on the map, representing infinitesimal circles at the corresponding point on the surface of the earth's ellipsoid or globe. The ellipse becomes a circle at the zero distortion point. 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 take a different position relative to the meridian passing through its center. The orientation of the distortion ellipse on the map is usually determined by azimuth of its semi-major axis . The angle between the north direction of the meridian passing through the center of the distortion ellipse and its nearest semi-major axis is called the orientation angle of the distortion ellipse. On fig. 5.3, a this corner is marked with the letter AND 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 north direction of the meridian in a clockwise direction and can have values ​​from 0 to 360°.
Any arbitrary direction ( OK) on a map or on a globe ( O 0 To 0 ) can be determined either by the azimuth of a given direction ( AND- on the map, α - on the globe) or the angle between the semi-major axis closest to the northern direction of the meridian and the given direction ( v- on the map, u- on the globe).

5.2.1. Length distortion

Length distortion - basic distortion. The rest of the distortions follow logically from it. Length distortion means the inconsistency 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 a certain size of the globe, from which the earth's surface is transferred to the plane. It allows you to judge the decrease in the length of the segments when they are transferred 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 W . 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, use the concept zoom in (With) defined by the relation

(5.4)

From formula (5.4) it follows that:

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

For example, if the main scale of the map is 1: 1,000,000, zoom in With equals 1.2, then µ \u003d 1.2 / 1,000,000 \u003d 1/833,333, i.e. one centimeter on the map corresponds to approximately 8.3 km on the ground. The private scale is larger than the main one (the value 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 one ( M= 1), then the partial scales will be numerically greater or less than unity. In this case under the private scale, numerically equal to the scale increase, 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)

Partial Scale Deviation (µ )from unity determines the length distortion at a given point on the map in a given direction ( V):

V = µ - 1 (5.6)

Often the length distortion is expressed as a percentage of unity, i.e., to 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 displayed on the map as a segment of length 1.2 cm.
It is convenient to judge the presence of length distortion on the map by comparing the size of the meridian segments between adjacent parallels. If they are everywhere equal, then there is no distortion of the lengths along the meridians, if there is no such equality (Fig. 5.5 segments AB and CD), then there is a distortion of the line lengths.

Rice. 5.4. Part of a map of the Eastern Hemisphere showing cartographic distortions

If a map depicts such a large area that it shows both the equator 0º and the parallel 60° of 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 parallels with a latitude of 60 ° between adjacent meridians. It is known that the parallel of 60° latitude is two times shorter than 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 exists.
The largest indicator of length distortion at a given point (the major semi-axis of the distortion ellipse) is denoted by the Latin letter a, and the smallest one (semi-minor axis of the distortion ellipse) - b. Mutually perpendicular directions in which the largest and smallest indicators of length distortion act, called the main directions .
To assess various distortions on maps, of all partial scales, partial scales in two directions are of greatest importance: along meridians and along parallels. private scale along the meridian usually denoted by the letter m , and the private scale parallel - letter n.
Within small scale maps relatively small territories (for example, Ukraine), the deviations of the 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 for approximate measurements are accompanied by a measuring scale, accompanied by explanatory text.
On nautical charts , built in the Mercator projection and on which the loxodrome is depicted by a straight line, no special linear scale. Its role is played by the eastern and western frames of the map, which are meridians divided into divisions through 1′ in latitude.
In maritime navigation, distances are measured in nautical miles. Nautical mile is the average length of the meridian arc of 1′ in latitude. It contains 1852 m. Thus, the frames of the sea chart are actually divided into segments equal to one nautical mile. By determining in a straight line the distance between two points on the map in minutes of the meridian, the actual distance in nautical miles along the loxodrome is obtained.

Figure 5.5. Distance measurement by sea ​​map.

5.2.2. Corner distortion

Angular distortions follow logically from length distortions. The angle difference 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 the angles on the map.
For angle distortion between the lines of the cartographic grid, they take the value of their deviation from 90 ° and designate it with 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 a point where the angle of intersection of the meridian and the parallel remains right on the chart, the angles between other directions can be changed on the chart, since at any given point the amount of angle distortion can change with direction.
For the general indicator of the distortion of angles ω (omega) take greatest distortion angle at a given point, equal to the difference between its value on the map and on the surface of the earth's ellipsoid (ball). When known x indicators a and b value ω determined by the formula:

(5.9)

5.2.3. Area distortion

Area distortions follow logically 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 the area distortion.
A simple way to identify the 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 takes place, 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 Index (R) is calculated as the product of the largest and smallest indicators of length distortion in this place cards
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 famous m and n calculated according to the formulas:

(5.11)
(5.12)

The distortion factor included in the equations R recognize in this case by the product:

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

5.2.4. Form distortion

Shape distortion consists in the fact that the shape of the site or the territory occupied by the object on the map is different from their 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 cartographic grid cells 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. It is also possible to 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 least ( b) indicators of length distortion in a given location of the map and is expressed by the formula:

(5.14)

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

Rice. 5.6. Isocoles of the greatest distortion of angles

5.3. CLASSIFICATION OF PROJECTIONS BY THE NATURE OF DISTORTIONS

For various purposes, projections of various types of distortion are created. The nature of the projection distortion 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 (equivalent);
- arbitrary.

5.3.1. Equangular projections

Equangular such projections are called in which directions and angles are depicted without distortion. The angles measured on the conformal projection maps are equal to the corresponding angles on the earth's surface. An infinitely small circle in these projections always remains a circle.
In conformal projections, the scales of lengths at any point in all directions are the same, therefore they have no 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 forms of real (final) geographical objects occupying entire sections on the map are distorted (Fig. 5.8, a). Conformal projections have especially large area distortions (which is clearly demonstrated by distortion ellipses).

Rice. 5.7. View of distortion ellipses in equal-area projections — AND, equiangular - B, arbitrary - AT, 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 plot routes along a given azimuth, so they are always used on topographic and navigational maps. The disadvantage of conformal projections is that areas are greatly distorted in them (Fig. 5.7, a).

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

5.6.2. Equidistant projections

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

5.6.3. Equal Area Projections

Equal-sized projections are called in which there are no area distortions, that is, the area of ​​\u200b\u200bthe figure measured on the map is equal to the area of ​​\u200b\u200bthe same figure on the surface of the Earth. In equal area map projections, the scale of the area has the same value everywhere. This property of equal-area projections can be expressed by the formula:

An inevitable consequence of the equal area 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 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, it becomes necessary to measure angles, lengths and areas on one map. But no projection can be at the same time conformal, equidistant, and equal area. It has already been said earlier that with a decrease in the imaged area of ​​the Earth's surface on a plane, image distortions also decrease. When depicting small areas of the earth's surface in an arbitrary projection, the distortions of angles, lengths and areas are insignificant, and in solving many problems they can be ignored.

5.4. CLASSIFICATION OF PROJECTIONS BY THE TYPE OF NORMAL GRID

In cartographic practice, the classification of projections according to the type of auxiliary geometric surface, which can be used in their construction, is common. From this point of view, projections are distinguished: cylindrical when the side 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).
Surfaces to be designed Earth, can be tangent to it or secant to it. They can also be oriented differently.
Projections, in the construction of which the axes of the cylinder and the cone were aligned with the polar axis of the globe, and the picture plane on 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, instead of an ellipsoid, we use a ball. We enclose the ball in a cylinder tangent to the equator (Fig. 5.9, a).

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

We continue the planes of the meridians PA, PB, PV, ... and take the intersection 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 the generatrix aAa 1 and deploy it on a plane, then the meridians will be depicted as parallel equally spaced straight lines aAa 1 , bBB 1 , vVv 1 ... perpendicular to the equator ABV.
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 a second family of parallel straight lines in the development, perpendicular to the meridians.
The resulting cylindrical projection (Fig. 5.9, b) will be equal, 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 area of ​​the image of this belt in the scan. The main scale is maintained along the equator; private scales increase along the parallel, and decrease along the meridians as they move away from the equator.
Another way to determine the position of the parallels is based on the preservation of the lengths of the meridians, i.e., on the preservation of the main scale along all meridians. In this case, the cylindrical projection will be equidistant along the meridians(Fig. 5.8, b).
For equiangular The cylindrical projection requires at any point the constancy of the scale in all directions, which requires an increase in scale along the meridians as you move away from the equator in accordance with the increase in scale along the 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 sweeping. In this case, partial scales along all parallels between the parallels of the section will be smaller, and on the remaining parallels - larger than the main scale.

Rice. 5.10. Cylinder that cuts the ball along two parallels

5.4.2. Conic projections

To construct a conic 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 cartographic grid in an equidistant conic projection

Similarly to the previous construction, we 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 unrolling the lateral surface of the cone on a plane (Fig. 5.11, b), the meridians will be depicted by radial straight lines TA, TB, TV, ..., emanating from the point T. Please note that the angles between them (the convergence of the meridians) will be proportional (but are not equal) to differences in longitudes. Along the tangent parallel ABV (arc of a circle with radius TA) the main scale is preserved.
The position of other parallels, depicted by arcs of concentric circles, can be determined from certain conditions, one of which - the preservation of the main scale along the meridians (AE = Ae) - leads to a conic equidistant projection.

5.4.3. Azimuthal projections

To construct an azimuthal projection, we will use a plane tangent to the ball at the point of the pole P (Fig. 5.12). Intersections of meridian planes with a 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, drawn with radii equal to straightened arcs of meridians from the pole to the corresponding parallel PA = Pa. Such a projection would equidistant on meridians and preserves the main scale along them.

Rice. 5.12. Construction of a cartographic grid in the azimuthal projection

A special case of azimuthal projections are promising projections built 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 the rays emerging from one point With called point of view. Depending on the position of the point of view relative to the center of the globe, the projections are divided into:

• central - 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 out of the globe;
• orthographic - the point of view is taken out to infinity, i.e. the projection is carried out by parallel rays.

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

5.4.4. Conditional projections

Conditional projections are projections for which it is impossible to find simple geometric analogues. They are built on the basis of some given conditions, for example, the desired type of geographic grid, one or another 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 converting one or several original projections.
At pseudocylindrical equator and parallel projections are straight lines parallel to each other (which makes them similar to cylindrical projections), and meridians are curves symmetrical about the average rectilinear meridian (Fig. 5.14)

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

At pseudoconical parallel projections are arcs of concentric circles, and meridians are curves symmetrical about the average rectilinear meridian (Fig. 5.15);

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

Building a grid in polyconic projection can be represented by projecting segments of the globe's graticule onto the surface several tangent cones and subsequent development into the plane of the stripes formed on the surface of the cones. General principle such a design is shown in Figure 5.16.

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

in letters S the tops 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 the touch of the corresponding cone.
For the external appearance of cartographic grids in a polyconic projection, it is characteristic that the meridians are in the form of curved lines (except for the middle one - straight), and the parallels are arcs of eccentric circles.
In polyconic projections used to build world maps, the equatorial section is projected onto a tangent cylinder, therefore, on the resulting grid, the equator has the form of a straight line perpendicular to the middle meridian.
After scanning the cones, these sections are imaged as stripes on a plane; the stripes touch along the middle meridian of the map. The mesh receives its final form after the elimination of gaps between the strips by stretching (Fig. 5.17).

Rice. 5.17. A cartographic grid in one of the polycones

Polyhedral projections - projections obtained by projecting onto the surface of a polyhedron (Fig. 5.18), tangent or secant to the ball (ellipsoid). Most often, each face is an isosceles trapezoid, although other options are possible (for example, hexagons, squares, rhombuses). A variety of polyhedral are multi-lane projections, moreover, the strips can be "cut" both along the meridians and along the parallels. Such projections are advantageous in that the distortion within each facet or band is very small, so they are always used for multi-sheet maps. Topographic and survey-topographic are created exclusively in a multifaceted projection, and the frame of each sheet is a trapezoid composed by lines of meridians and parallels. You have to "pay" for this - a block of map sheets cannot be combined along a common frame without gaps.

Rice. 5.18. Polyhedral projection scheme and arrangement of map sheets

It should be noted that today 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 the projection. The search for projections is performed analytically. Computer modeling allows you to quickly calculate any projection with the given parameters, and automatic graph plotters easily draw the appropriate grid of meridians and parallels, and, if necessary, an isocol map.
There are special atlases of projections that allow you to choose the right projection for any territory. AT recent times electronic atlases of projections have been created, with the help of which it is easy to find a suitable grid, immediately evaluate its properties, and, if necessary, carry out certain modifications or transformations in an interactive mode.

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 projection plane touches the equator at some point or the axis of the cylinder (cone) coincides with the plane of the equator (Fig. 5.20).

Rice. 5.20. Transverse projections

oblique projections - the projection 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, azimuth (perspective) and pseudo-azimuth projections are most often used. Transverse azimuths are used for maps of the hemispheres, oblique - for territories that have a rounded shape. Maps of the continents are often made in transverse and oblique azimuth projections. The Gauss-Kruger transverse cylindrical 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;
• the purpose, scale and subject of the map, the intended 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 the 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 drawn up for navigation, the Mercator conformal cylindrical projection must be used. If Antarctica is being mapped, the normal (polar) azimuthal projection will almost certainly be adopted, and so on.
The significance of these factors may be different: in one case, visibility is put in the first place (for example, for a wall school card), in the other - the features of using the map (navigation), in the third - the position of the territory on the globe (the polar region). Any combination is possible, and therefore different variants projections. Moreover, the choice is very large. But still, some preferred and most traditional projections can be indicated.
World Maps usually compose in cylindrical, pseudocylindrical and polyconical projections. To reduce distortion, secant cylinders are often used, and pseudocylindrical projections are sometimes given with discontinuities on the oceans.
Hemispheric maps always built in azimuthal projections. For the western and eastern hemispheres, it is natural to take transverse (equatorial) projections, 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 with Oceania are most often built in equal-area oblique azimuth projections, for Africa they take transverse, and for Antarctica - normal azimuth.
Maps of selected countries , administrative regions, provinces, states are performed in oblique conformal and equal-area conic or azimuth 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; different conformal projections can be used, bearing in mind that area distortions in small areas are almost imperceptible.
Topographic maps Ukraine is created in the transverse cylindrical projection of Gauss, and the United States and many other Western countries - in the universal transverse cylindrical projection of Mercator (abbreviated as UTM). Both projections are close in their properties; in fact, both are multi-cavity.
Maritime and aeronautical charts are always given exclusively in the cylindrical Mercator projection, and thematic maps of the seas and oceans are created in the most diverse, sometimes quite complex projections. For example, for the joint display of the Atlantic and Arctic oceans, special projections with oval isocols are used, and for the image of the entire World Ocean, equal projections with discontinuities on the continents are used.
In any case, when choosing a projection, especially for thematic maps, it should be borne in mind that map distortion is usually minimal in the center and increases rapidly towards the edges. Besides than smaller scale maps and more extensive spatial coverage, the more attention has to be paid to the "mathematical" factors of projection selection, and vice versa - for small areas and large scales, "geographical" factors become more significant.

5.7. PROJECTION RECOGNITION

To recognize the projection in which the map is drawn means to establish its name, to determine whether it belongs to one or another species, class. This is necessary in order to have an idea about the properties of the projection, the nature, distribution and magnitude of distortion - in a word, in order to know how to use the map, 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, azimuth 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 reveal their equiangularity, equivalence, or equidistance in one of the directions. For this, there are special techniques: first, the shape of the frame is set (rectangle, circle, ellipse), determine how the poles are depicted, then measure the distance between adjacent parallels along the meridian, the area of ​​\u200b\u200bneighboring cells of the grid, the angles of intersection of the meridians and parallels, the nature of their curvature, etc. .P.
There are special projection tables for maps of the world, hemispheres, continents and oceans. After carrying 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 isocoles for making corrections.

Video
Types of projections by the nature of distortions

Questions for self-control:

1. What elements make up the mathematical basis of the map?
2. What is the scale of a geographic map?
3. What is the main scale of a map?
4. What is the private scale of a map?
5. What is the reason for the deviation of the private scale from the main one on geographical map?
6. How to measure the distance between points on a sea chart?
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 are the methods of transferring the surface of the earth's ellipsoid to a plane, what is their essence?
10. What is a map projection?
11. How are projections classified according to the nature of distortion?
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 an ellipse of distortions on these projections?
14. What projections are called equal areas, how to depict an ellipse of distortions on these projections?
15. What projections are called arbitrary?

1. Explain why the globe is called a three-dimensional model of the Earth.

The globe almost completely repeats the shape of the earth, the position of objects and its surface.

How does the shape of a globe differ from the actual shape of the Earth?

The globe is a sphere, while the earth is flattened at the poles.

2. Establish in which two hemispheres the boy depicted in this photo is standing at the same time.

Western and Eastern

3. Determine which type of territory coverage the presented maps belong to. Using the atlas, give examples of maps of each type.

1 - Maps of countries (physical map of Russia).

2 - World maps (political map of the world, physical map of the world)

4. Arrange the parallels from longest to shortest.

45° S 25°N, 0°N, 70°S, 30°S 60°N 20°N

0

20 N

25 N

30 N

45 S

60 N

70 S

5. In the figure, the ships of the Russian Antarctic expedition "Vostok" and "Mirny" are depicted at noon off the coast of Peter I Island (68 ° S). Determine in which direction the ships are moving.

In the southern hemisphere at noon, the sun tends to go north, as the ship sails towards the sun, it sails north.

6. Give examples of maps from your atlas, made in such ways as shown in the figures.

7. Determine in which parts of these maps the image of the Earth is most distorted. Explain why.

On the world map. The length of the latitudes is less towards the equator. The smaller the scale, the greater the distortion.

8. Determine which of the figures shows:

a) only parallels;

b) only meridians;

c) degree grid.

All-Russian Olympiad for schoolchildren in geography

I municipal stage, 2014

Class.

Total time - 165 min

The maximum possible score is 106

Test round (time to complete 45 min.)

It is forbidden to use atlases, cellular communications and the Internet! Good luck!

I. From the proposed answers, choose one correct

At what scale can the map be drawn? natural areas of the world" in the atlas for grade 7?

a) 1:25000; b) 1:500000; c) 1:1000000; d) 1:120,000,000?

2. On the world map of the hemispheres, the least distortion is:

a) Fiery Island Earth; b) the Hawaiian Islands; c) the peninsula of Indochina; d) Kola Peninsula

3. In one degree of the circumference of the equator, in comparison with other parallels, contains:

a) the largest number of kilometers, b) the smallest number of kilometers, c) the same as on the other parallels

On the territory of which bay is the point of reference for latitude and longitude on the map?

a) Guinea, b) Biscay, c) California, d) Genoa.

5. Kazan has coordinates:

a) 45 about 13 / s.sh. 45 o 12 / E, b) 50 o 45 / N 37 about 37 / o.d.,

c) 55 about 47 / s.sh. 49 o 07 / east, d) 60 o 13 / n. 45 about 12 / o.d.,

On the ground, tourists move based on

a) magnetic azimuth, b) geographic azimuth, c) true azimuth, d) rhumb.

What azimuth corresponds to the direction to the SE?

a) 135º; b) 292.5º; c) 112.5º; d) 202.5º.

What azimuth should you move in if the path lies from a point with coordinates

55 0 N 49 0 east to the point with coordinates 56 0 n.l. 54 0 o.d.?

a) 270 0 ; b) 180 0 ; c) 45 0 ; d) 135 0 .

Which meridian can be used to navigate when surveying by eye?

a) geographical, b) axial, c) magnetic, d) zero, e) all together

10. What is the time of the year on the Spitsbergen Islands when the earth's axis is facing the Sun with its northern end? a) autumn b) winter c) summer c) spring

11. At the time when the Earth is the most distant from the Sun, in Kazan:

a) the day is longer than the night, b) the night is longer than the day, c) the day is equal to the night.

In which hemisphere does the polar day last longer?

a) in the South, b) in the North, c) in the West, d) in the East

13. In what month do the tropical latitudes of the southern hemisphere receive the most solar heat? a) January, b) March, c) June, d) September.

In what weather is the daily amplitude of air temperature the largest?

a) cloudy, b) cloudless, c) cloudiness does not affect the average daily temperature amplitude.

15. At what latitudes are the highest absolute air temperatures recorded?

a) equatorial, b) tropical, c) temperate, d) arctic.

16. Determine the relative humidity of air at a temperature of 21 ° C, if its 4 cubic meters contain 40 g of water vapor, and the density of saturated water vapor at 21 ° C corresponds to 18.3 g / m 3.

a) 54.6%, b) 0.55%, c) 218.5%, d) 2.18%.

17. At the airport in Sochi, the air temperature is +24 °C. The plane took off and took the direction to Kazan. Determine the altitude at which the aircraft flies if the air temperature overboard is -12 °C.

a) 6 km, b) 12 km, c) 24 km, d) 36 km.

What will be the atmospheric pressure on the thalweg of the ravine if the atmospheric pressure equal to 760 mm Hg was recorded in the upper part of the slope, and the depth of the incision of the ravine is 31.5 m.

a) 3 mm Hg, b) 757 mm Hg, c) 760 mm Hg, d) 763 mm Hg

a) St. Lawrence, b) Fundy, c) Gulf of Ob, d) Penzhinskaya Bay.

20. Name the continent, which is both part of the world and a continent, and is located in four hemispheres:

a) America, b) Africa, c) Australia, d) Antarctica, e) Europe, f) Asia, g) Eurasia, h) South America, i) N. America

The westernmost point of Asia is the cape

a) Piai, b) Chelyuskin, c) Baba, d) Dezhneva.

The continental shelf is practically absent

a) off the western coast of South America, b) off the northern coast of Eurasia,

c) off the western coast of S. America, d) off the northern coast of Africa.

The earth's crust is younger in the area

a) lowlands, b) mid-ocean ridges, c) low mountains, d) oceanic basins.

The source of the Volga River is located

a) on the Central Russian elevation, b) in the Kuibyshev reservoir, c) on the Valdai elevation, d) in the Caspian Sea.

25. Air circulation in Antarctica is characterized by:

a) trade winds, b) monsoons, c) katabatic winds, d) breezes.

26. Specify the analogue of the Gulf Stream in the Pacific Ocean:

a) Canary, b) Kuril, c) Kuroshio, d) North Pacific

27. Glacier ice is formed from

a) fresh water, b) sea water, c) atmospheric solid precipitation, d) atmospheric liquid precipitation.

Which traveler was the first to reach South Pole?

a) R. Scott, b) F. Bellingshausen, c) R. Amundsen, d) J. Cook.

29. Arrange the objects as far as they are from the audience where you are:

a) West Siberian Plain, b) Amazon lowland, c) Cordillera, d) Sahara desert.

30. Find a match:

Continent - plant - animal - bird

Analytical round (Time to complete 120 min)

Topic 6. Symbols on a topographic map

TASK 9. On sheets of drawing paper (A4 format) draw conventional signs topographic maps (a model for the implementation of conventional signs is topographic map scale 1: 10,000 (SNOV)).

The surface of the Earth cannot be depicted on a plane without distortion. Cartographic distortion is a violation of the geometric properties of areas of the earth's surface and objects located on them.

There are four types of distortion: length distortion, angle distortion, area distortion, shape distortion.

Line length distortion It is expressed in the fact that distances that are the same on the surface of the Earth are depicted on the map as segments of different lengths. The map scale is therefore a variable value. But on any map there are points or lines of zero distortion, and the image scale on them is called main. AT other places the scales are different, they are called private.

It is convenient to judge the presence of length distortion on the map by comparing the size of the segments between the parallels (Figure 11). Segments AB and CD (Figure 11) should be equal, but they are different in length, therefore, there is a distortion of the meridian lengths (τ) on this map. The segments between two adjacent meridians along one of the parallels must also be equal and correspond to a certain length. The segment EF is not equal to the segment GH (Figure 11), therefore, there is a distortion in the lengths of the parallels ( P). The largest distortion indicator is denoted by the letter a, and the smallest - the letter b.

Figure 11– Examples of distortions of lengths, angles, areas, shapes

Corner distortion very easy to install on the map. If the angle of intersection of the parallel and the meridian deviates from the angle of 90°, then the angles are distorted (Figure 11). The angle distortion indicator is denoted by the letter ε (epsilon):

ε = θ + 90º,

where θ is the angle measured on the map between the meridian and the parallel.

Area distortion it is easy to determine by comparing the areas of cells of the cartographic grid, limited by parallels of the same name. In Fig. 1, the area of ​​the shaded cells is different, but should be the same, therefore, there is a distortion of the areas ( R). Area distortion index ( R) is calculated by the formula:

p = n m cos ε.

Shape distortion is that the shape of the area on the map is different from the shape on the surface of the Earth. The presence of distortion can be established by comparing the shape of the cartographic grid cells located at the same latitude. In Figure 11, the shape of the two shaded cells is different, which indicates the presence of this type of distortion. Shape Distortion Index ( To)depends on the difference of the largest ( a) and least ( b) indicators of distortion of lengths and is expressed by the formula:

K=a:b

TASK 10. But physical map hemispheres, scale 1: 90,000,000 (atlas "Elementary Geography Course" for grades 6 (6–7) of secondary school) to determine private scales, the degree of length distortion along the meridian ( t), parallel ( n), angle distortion ( ε ), area distortion ( R) for two points indicated in one of the options (Table 11). Record the data of measurements and calculations in the table according to the form (table 10).

Table 10– Determining the amount of distortion

Before filling in the table, indicate the name of the map, its main scale, the name and output data of the atlas.

1). Find partial length scales along parallels and meridians.

For determining n necessary:

1 measure on the map the length of the arc of the parallel on which the given point lies with an accuracy of 0.5 mm l 1 ;

2 find the actual length of the corresponding arc of the parallel on the surface of the earth's ellipsoid according to table 12 "The length of the arcs of parallels and meridians on the Krasovsky ellipsoid" L1;

3 calculate private scale n = l 1 /L 1, while presenting the fraction in the form 1: xxxxxxx.

For determining t:

1 measure on the map the length of the arc of the meridian on which the given point lies l 2 .

2 find the actual length of the corresponding meridian arc on the surface of the earth's ellipsoid according to table 12 L2;

3 calculate private scale: m \u003d l 2 /L 2, while presenting the fraction in the form: 1: ххххххх.

4 express the private scale in fractions of the principal. To do this, divide the denominator of the main scale by the denominator of the quotient.

2). Measure the angle between the meridian and the parallel and calculate its deviation from the straight line ε, the measurement accuracy is up to 0.5º.

To do this, draw tangents to the meridian and parallels at a given point. The angle θ between the tangents is measured with a protractor.

3). Calculate the area distortion using the formula above.

Table 11– Task options 10

Table 12– Length of arcs of parallels and meridians on the Krasovsky ellipsoid

Goals and objectives of studying the topic:

To give an idea of ​​the distortions on the maps and the types of distortions:

To form an idea of ​​distortions in lengths;

- form an idea of ​​distortions in areas;

- to form an idea of ​​distortions in the corners;

- form an idea of ​​distortions in forms;

The result of mastering the topic:

The surface of an ellipsoid (or sphere) cannot be turned into a plane while maintaining the similarity of all outlines. If the surface of the globe (model of the earth's ellipsoid), cut into strips along the meridians (or parallels), is turned into a plane, in cartographic image there will be gaps or overlaps, and with distance from the equator (or from the middle meridian) they will increase. As a result, it is necessary to stretch or compress the strips in order to fill the gaps along the meridians or parallels.

As a result of stretching or compression in the cartographic image, distortions occur in lengthsm (mu) , areas p, cornersw and forms k. In this regard, the scale of the map, which characterizes the degree of reduction of objects in the transition from nature to the image, does not remain constant: it changes from point to point and even at one point in different directions. Therefore, one should distinguish main scale ds , equal to the given scale in which the earth ellipsoid decreases.

The main scale shows the overall reduction rate adopted for this map. The main scale is always signed on maps.

In all other places map scales will differ from the main one, they will be larger or smaller than the main one, these scales are called private and denoted by the letter ds 1.

The scale in cartography is understood as the ratio of an infinitely small segment taken on a map to the corresponding segment on the earth's ellipsoid (globe). It all depends on what is taken as the basis for constructing the projection - the globe or the ellipsoid.

The smaller the change in scale within a given area, the more perfect the map projection will be.

To perform cartographic work, you need to know distribution on a map of partial scales so that corrections can be made to the measurement results.

Private scales are calculated using special formulas. Analysis calculation of particular scales shows that among them there is one direction with largest scale , and the other with least.

largest the scale, expressed in fractions of the main scale, is denoted by the letter " a", a least - letter « in" .

The directions of the largest and smallest scales are called main directions . The main directions only coincide with the meridians and parallels when the meridians and parallels intersect under right angles.

In such cases scale by meridians denoted by the letter « m" , and by parallels - letter « n" .

The ratio of the private scale to the main one characterizes the distortion of lengths m (mu).

In other words, the value m (mu) is the ratio of the length of an infinitesimal segment on the map to the length of the corresponding infinitesimal segment on the surface of an ellipsoid or ball.

m(mu) = ds 1

Area distortion.

Area distortion p defined as the ratio of infinitesimal areas on a map to infinitesimal areas on an ellipsoid or ball:

p= dp 1

Projections in which there are no area distortions are called equal.

While creating physical and geographical and socio-economic cards, it may be necessary to save correct area ratio. In such cases, it is advantageous to use equal-area and arbitrary (equidistant) projections.

In equidistant projections, the area distortion is 2-3 times less than in conformal projections.

For political maps world, it is desirable to maintain the correct ratio of the areas of individual states without distorting the external contour of the state. In this case, it is advantageous to use an equidistant projection.

The Mercator projection is not suitable for such maps, since areas are greatly distorted in it.

Corner distortion. Let's take the angle u on the surface of the globe (Fig. 5), which on the map is represented by the angle u ′ .

Each side of the angle on the globe forms an angle α with the meridian, which is called the azimuth. On the map, this azimuth will be represented by the angle α ′.

In cartography, two types of angular distortions are accepted: direction distortions and angle distortions.

A A ′

α α ′

0 u 0 ′ u ′

B B ′

Fig.5. Corner distortion

The difference between the azimuth of the side of the corner on the map α ′ and the azimuth of the side of the angle on the globe is called direction distortion , i.e.

ω = α′ - α

The difference between the angle u ′ on the map and the value u on the globe is called angle distortion, those.

2ω = u - u

The distortion of the angle is expressed by the value 2ω because the angle consists of two directions, each of which has a distortion ω .

Projections in which there are no angle distortions are called equiangular.

The distortion of shapes is directly related to the distortion of angles (specific values w match certain values k ) and characterizes the deformation of the figures on the map in relation to the corresponding figures on the ground.

Form distortion will be the greater, the more the scales differ in the main directions.

As shape distortion measures accept coefficient k .

k = a / b

where a and in are the largest and smallest scales at a given point.

Distortions on geographical maps are the greater, the larger the depicted territory, and within the same map, distortions increase with distance from the center to the edges of the map, and the slew rate changes in different directions.

In order to visualize the nature of distortions in different parts of the map, they often use the so-called ellipse of distortion.

If we take an infinitely small circle on the globe, then when moving to the map, due to stretching or contraction, this circle will be distorted like the outlines of geographical objects and will take the form of an ellipse. This ellipse is called ellipse distortion or Tissot's indicatrix.

The dimensions and degree of elongation of this ellipse compared to the circle reflect all kinds of distortions inherent in the map in this place. Type and dimensions ellipse are not the same in different projections and even at different points of the same projection.

The largest scale in the distortion ellipse coincides with the direction of the major axis of the ellipse, and the smallest scale coincides with the direction of the minor axis. These directions are called main directions .

The distortion ellipse is not displayed on the maps. It is used in mathematical cartography to determine the magnitude and nature of distortions at some projection point.

The directions of the axes of the ellipse may coincide with the meridians and parallels, and in some cases the axes of the ellipse may occupy an arbitrary position relative to the meridians and parallels.

Determination of distortions for a number of map points and subsequent drawing on them isocol - lines connecting points with the same distortion values ​​gives a clear picture of the distribution of distortions and allows you to take into account distortions when using the map. To determine the distortions within the map, you can use special tables or diagrams isokol. Isocols can be for angles, areas, lengths, or shapes.

No matter how you deploy earth's surface on the plane, gaps and overlaps will necessarily occur, which in turn leads to tensions and compressions.

But on the map, at the same time, there will be places where there will be no compressions and tensions.

Lines or points on a geographical map that are not distorted and the main scale of the map is preserved, called lines or zero-distortion points (LNI and TNI) .

As you move away from them, the distortion increases.

Questions for repetition and consolidation of the material

1. What causes cartographic distortions?

2. What types of distortions occur during the transition from the surface
ellipsoid to plane?

3. Explain what is the point and line of zero distortion?

4. On which maps does the scale remain constant?

5. How to determine the presence and magnitude of distortion in certain areas of the map?

6. What is Tissot's indicatrix?

7. What is the purpose of the distortion ellipse?

8. What are isocoles and what is their purpose?