On the world map of the hemispheres, it has the greatest distortion. In what parts of the world map is the distortion the greatest? Educational and analytical information

Colombia is a country located in South America bordering Panama, Peru, Ecuador, Venezuela and Brazil. It is washed by the waters of the Pacific Ocean and the Caribbean Sea.

interactive maps

Convenient interactive map of Colombia, which can be moved and zoomed in on the right place to get the necessary information. It can also be switched to the mode of satellite, relief and weather information display.

Also, you can use another interactive map Colombia, adapted for Russian travelers.

Geographic map

Geographic map of Colombia, which shows the relief and natural features of the country, the main cities and roads, as well as borders with neighboring countries.

Educational and analytical information

Applying signs of distortion on the maps, students establish:

1. The map has a distortion of the line lengths, since the 20-degree segments of the meridians increase from the center of the map and along the middle meridian and away from it; length distortions are also observed on the parallels (the 20-degree segment of the parallel 60 ° N near the middle meridian is not two times smaller than the 20-degree segment of the equator); there is no distortion of lengths along the equator, its segments are equal. Conclusion: both meridians and parallels, being distorted, stretch with the distance from the central point of the map. The equator is not distorted.
2. The map has shape distortions, since the shapes of the cartographic grid cells at the same latitude (for example, along the equator) are different.
3. The map has a distortion of the corners, which is clearly seen in many of its sections by the deviation of the angles of intersection of the meridians and parallels from 90°.
4. The map has area distortion. This can be seen by eye from the increase in the area of ​​the cells of the cartographic grid to the edge of the map. For example, along the equator, the bases of the cells remain unchanged, and their heights are greater, the closer the cell is to the edge of the map. It follows from this that the cell areas grow in the same direction.

Distortions on maps of the hemispheres, the mainland, and the USSR can be analyzed in the same way. At the same time, a regularity is revealed that with a decrease in the coverage of the territory depicted on the map, as a rule, the amount of distortion also decreases. This conclusion can also be suggested by the teacher.

The general concept and definition of cartographic projection are given in the textbook. Here, with sufficient completeness, the three main types of projection are characterized, distinguished by their inherent distortions (equiangular, equal-sized and arbitrary), and a variety of arbitrary ones - equidistant.

A practically important task is to develop in students the ability, based on the analysis of the distortion of the map, to determine which of the named groups the projection in which this map is built belongs to. This conclusion should be the end of the analysis of distortions on the maps. The teacher needs to know belonging to one or another group of map projections by their distortions. In arbitrary projections built: all world maps in the atlas for class VI, map North America us. 4 in the atlas for class VII; an arbitrary equidistant projection is represented by a world map in the same atlas.

Neither the program nor the textbook obliges seventh graders to study the distortion indicators on the maps. But in the atlas for class VII, these indicators are shown in the form of so-called distortion ellipses (in a graphic table called "Geometric representation of distortions"). This table shows how the shape, radius lengths and area change under the influence of distortion. geometric figure a circle away from the midpoint on the map where it is not distorted. From the top of the three figures, it can be seen that in conformal projections the shape of the circle will change, but its area increases; in the middle figure it is proved that with the distance from the undistorted image of the circle, its shape turns into an ellipse with an area equal to the area of ​​the circle. The bottom drawing highlights how the shape and area of ​​the initial circle increases. The information provided may be useful to the teacher if the students are interested in this drawing.

Differences (classification) of cartographic projections of educational maps are shown in the atlas. Us. 4 of the atlas for class VII there are drawings explaining how cylindrical, conical and azimuth projections can be obtained, using the surfaces of a cylinder, cone or plane as auxiliary surfaces, respectively.

To explain to students how to build map projections using an auxiliary geometric surface, it is useful in a lesson on this topic to use geographic globe, a sheet of plywood or cardboard to depict a plane, and a sheet of drawing paper that can be folded into a cylinder or cone. For example, when explaining how to obtain a conic projection, in which many maps of the USSR are compiled, the teacher puts a sheet of paper folded into a cone on the globe so that the side surface of the cone is in contact with the globe along one of the parallels, and the top of the cone would be above the pole, on the continuation of the axis rotation of the earth. Holding the cone in this position, the teacher outlines with a soft pencil on the outer side of the cone the parallel of contact, two or three other parallels and several meridians. At the same time, he says that when designing (transferring) the lines of the degree grid onto the surface of the cone, the parallels take the form of circles, and the meridians take the form of straight lines directed towards the top of the cone.

Having finished drawing the lines of the degree grid on the paper cone, the teacher unfolds it in a plane and fixes it on the board so that the students see the characteristic shape of the cartographic grid in a conic projection. Of course, the grid lines with this method of drawing cannot be even. You can draw them in advance reverse side paper and, attaching the sheet to the board, turn it to the side on which the grid was previously drawn. N. V. Malakhov recommends linking the study of the projection of maps with the projections of objects that students use in the drawing course. He writes: “Students, starting from the 7th grade, may mistakenly associate map projections with parallel (orthogonal) projections known to them from the drawing course, which, as you know, are obtained as a result of projecting objects onto a plane with parallel rays. The projections of the maps used in the school have different design principles than in drawing.

In order for students to correctly understand map projections, it is useful to compare the image of one of the hemispheres, for example, the eastern one, on the map with the image of the same hemisphere, but obtained according to the principle of orthogonal projection. A similar representation of the eastern hemisphere is used to show the Earth as a planet and, in particular, in the atlas for teachers.

Of course, the concepts of cartographic projections are formed especially effectively by building sleds in different projections. For lack of time in geography lessons, such work can be offered to participants in a school geographical circle or as an individual independent assignment. How to build a cartographic grid in different projections can be found in the manual for teachers "Production of geographical maps at school."

Without such a consolidation of the acquired knowledge, the names of projection groups alone and the information given about their obtaining by geometric projection onto an auxiliary surface of one form or another do not sufficiently reveal these concepts. In order for this information to be fixed, it is necessary to record and remember the features of the distribution of distortion in each group:

• in cylindrical projections there is usually no distortion along the equatorial line, which is therefore the line of zero distortion. Distortions increase with distance from the equator to the north and south;
• in azimuthal projections, there is no distortion at the central point of the map. In all directions from this point of zero distortion, they grow.

1. Distortions are less, the smaller part of the surface of the globe is shown on the map. Topographic maps covering very small areas earth's surface, in which the bulge of the Earth is not noticeable, give the most accurate images.

2. In different parts of the same map, the scale is different. The scale in points or lines of zero distortion is called the main scale. Usually it is indicated on the maps. As you move away from points or lines of zero distortion, the scale of the map differs more and more from the main one. Only on topographic maps the scale indicated on them is valid for all their parts.

3. The least distortion on the cards is in their middle parts, with the distance to the edges (frame) of the card, the distortion increases.

Distortions on maps of the hemisphere. To find out what distortions have turned out on the map of the hemispheres, it is necessary to compare the degree grid of the globe and the cartographic grid of the map. On the globe, all meridians have the same length, which is true. On the map of the hemispheres, the length of the meridians is different. The middle meridian is shown as a straight line, the others are curved. The farther the meridians are located from the middle one, the more they are curved, and the extreme ones form semicircles and are almost one and a half times longer than the middle meridian. Parallels on the globe are depicted as circles parallel to each other. On the map of the hemispheres, the equator is a straight line, and the parallels are arcs, and the distances between adjacent parallels are not the same and increase towards the edges of the map.

Let's see what this arrangement of meridians and parallels leads to on the map of the hemispheres and how it affects the depicted objects. On the globe, a section of the earth's surface (ocean or land) near the equator, having a length of 10 ° in latitude, everywhere has a figure similar to a square. On the map of the hemispheres, these areas at different longitudes have different shapes. In the center they have a shape close to a square, as on a globe, and towards the edge of the map their shape changes greatly. At the same time, the segments of the meridians are lengthened, and the segments of the equator are shortened.

From all this it follows that distances that are the same on the globe (Earth), in different places on the map are depicted by segments of different lengths, i.e., the scale of the map is not the same in its different parts. This results in a different scale of the cartographic image.

The scale indicated on the maps is not accurate for the entire map, but only for certain parts of it. Therefore, it cannot be used when measuring distances and areas across the entire map. On the map of the hemispheres, the scale corresponds to that indicated only at the central point, namely at the intersection of the equator and the middle meridian. This is the point of zero distortion. In all other parts of the map, the scale is greater or less than indicated on it. On other maps, there may not be points, but lines of zero distortion.

Distortions on world maps. On world maps, the distortions are the greatest, since they depict the surface of the entire ball at once. For example, on a globe, 1° longitude at 60° N. sh. and yu. sh. is 55.8 km, i.e., two times less than at the equator. On the world map, this distance is only 1.5 times. 1° longitude at 80° N sh. and yu. sh. less than at the equator, already 6.5 times, and on the world map only 2 times. The scale indicated on these world maps is maintained along the parallels of 45 ° N. sh. and yu. sh. According to the parallels lying from them towards the equator, it is less, and towards the poles - more. Moreover, it increases rapidly towards the poles. Therefore, in the northern and southern parts of our world maps, the geographical maps are noticeably stretched from west to east. According to the meridians, the scale indicated on world maps is preserved only in the center - at the intersection of the middle meridian and the equator. With the removal in all directions, the scale of lengths along the meridians increases. Therefore, the length of the meridian segments between the parallels also increases.

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 - Earth or 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 one deploys the earth's surface onto a plane, gaps and overlaps will inevitably 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 dots on geographical map, in which there are no distortions 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?

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 °С. 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 most western point Asia - 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