On the world map of the hemispheres, it has the greatest distortion. Which parts of the world map have the greatest distortions? In which hemisphere does the polar day last longer?

1. The smaller the smaller part of the surface of the globe is depicted on the map, the smaller the distortion. Topographic maps covering very small areas earth's surface, in which the Earth's bulge is not noticeable, provide the most accurate images.

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

3. The least distortion on cards is in their middle parts; as you move towards the edges (frame) of the card, the distortion increases.

Distortions on maps of the hemisphere. To find out what distortions occurred on the map of the hemispheres, you need 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 lengths of the meridians are different. The middle meridian is depicted as a straight line, the rest are curved. The further 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 average meridian. Parallels on the globe are depicted in the form of circles parallel to each other. On a map of the hemispheres, the equator is a straight line, and the parallels are arcs, and the distances between adjacent parallels are unequal and increase towards the edges of the map.

Let's see what this arrangement of meridians and parallels on the map of the hemispheres leads to and how this is reflected in the depicted objects. On the globe, a section of the earth's surface (ocean or land) near the equator, having an extension 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, like on a globe, but towards the edge of the map their shape changes greatly. In this case, the meridian segments lengthen, and the equator segments shorten

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

The scale indicated on the maps turns out to be accurate not 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 a 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 larger or smaller than that 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 greatest, since they depict the surface of the entire ball at once. For example, on a globe there is 1° longitude at 60° N. w. and Yu. w. 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° north. w. and Yu. w. less than on 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 45° N parallels. w. and Yu. w. According to the parallels lying from them towards the equator, it is smaller, and towards the poles it is larger. Moreover, it increases rapidly towards the poles. Therefore, in the northern and southern parts of our world maps, the geographic maps are noticeably stretched from west to east. Along 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 distance in all directions, the scale of lengths along the meridians increases. Therefore, the length of the meridian segments between parallels also increases.

All-Russian Olympiad for schoolchildren in geography

Municipal stage, 2014

Class.

Total time – 165 min

The maximum possible score is 106

Test round (time to complete 45 min.)

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

I. From the proposed answer options, choose one correct

On what scale can the map “Natural areas of the world” be made 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 has:

A) Ognennaya Island Earth; b) Hawaiian Islands; c) Indochina Peninsula; d) Kola Peninsula

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

a) the greatest number of kilometers, b) the least number of kilometers, c) the same as on other parallels

In which bay is the latitude and longitude reference point on the map located?

a) Guinean, b) Biscay, c) Californian, d) Genoese.

5. Kazan has coordinates:

a) 45 o 13/N. 45 o 12 / east, b) 50 o 45 / north. 37 o 37/E,

c) 55 o 47 / N. 49 o 07 / east, d) 60 o 13 / north. 45 o 12/E,

Tourists move around the area based on

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

What azimuth corresponds to the direction to the SE?

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

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

55 0 N 49 0 E. to a point with coordinates 56 0 N. 54 0 east?

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

Which meridian can you use to navigate when shooting by eye?

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

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

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

a) day is longer than night, b) night is longer than day, c) day is equal to 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 which month do the tropical latitudes of the southern hemisphere receive the most solar heat? a) January, b) March, c) June, d) September.

Under what weather conditions is the daily range of air temperature large?

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 having a temperature of 21 o C, if its 4 cubic meters contain 40 g of water vapor, and the density of saturated water vapor at 21 o C corresponds to 18.3 g/m 3.

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

17. At the Sochi airport the air temperature is +24 °C. The plane took off and headed for Kazan. Determine the altitude at which the plane flies if the air temperature outside 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 an atmospheric pressure of 760 mm Hg was recorded in the upper part of the slope, and the depth of the ravine incision 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) Ob Bay, d) Penzhina Bay.

20. Name a continent that 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) North America

The westernmost point of Asia - the cape

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

There is virtually no continental shelf

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

c) off the western coast of North 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) ocean 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. Indicate an 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 objects as they move away from the audience where you are:

a) West Siberian Plain, b) Amazonian lowland, c) Cordillera city, d) Sahara Desert.

30. Find a match:

Continent – ​​plant – animal – bird

Analytical round (Completion time 120 min)

Goals and objectives of studying the topic:

Give an idea of ​​distortions on maps and types of distortions:

Form an idea of ​​distortions in lengths;

- form an idea of ​​​​distortions in areas;

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

- form an idea of ​​distortions in forms;

Result of mastering the topic:

The surface of an ellipsoid (or ball) cannot be turned into a plane while maintaining the similarity of all outlines. If the surface of a globe (a model of the earth's ellipsoid), cut into strips along meridians (or parallels), is turned into a plane, gaps or overlaps will occur in the cartographic image, 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 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) , squares p, cornersw And forms k. In this regard, the scale of the map, which characterizes the degree of reduction of objects during the transition from life to image, does not remain constant: it changes from point to point and even at one point in different directions. Therefore it is necessary to distinguish main scale ds , equal to a given scale at which the Earth's ellipsoid decreases.

The main scale shows the overall degree of reduction adopted for a given map. The main scale is always indicated on maps.

In all other places maps, the 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.

In cartography, scale is understood as the ratio of an infinitesimal segment taken on a map to its corresponding segment on the earth’s ellipsoid (globe). It all depends on what is taken as a basis when constructing a projection - a globe or an 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 scale quantities so that corrections can be made to the measurement results.

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

Largest a scale expressed in fractions of the main scale is denoted by the letter “ A", A least – letter « V" .

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

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

The ratio of the particular 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

Distortion of areas.

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

p= dp 1

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

While creating physiographic 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, area distortion is 2-3 times less than in equiangular 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

Distortion of corners. Let us take an angle u on the surface of the globe (Fig. 5), which on the map will be represented by an angle u ′ .

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

There are two types of angular distortions used in cartography: directional distortions and angular distortions.

A A ′

α α ′

0 u 0 ′ u ′

B B ′

Fig.5. Angle 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 directional 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 angle distortion is expressed by the quantity 2ω because the angle consists of two directions, each of which has a distortion ω .

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

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

Distortions of forms will be greater, the more the scales differ in the main directions.

As measures of shape distortion take the coefficient k .

k = a/b

Where A And V – the largest and smallest scales at a given point.

The larger the territory depicted, the greater the distortion on geographic maps, and within one map the distortion increases with distance from the center to the edges of the map, and the rate of increase varies in different directions.

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

If you take a circle of infinitesimal size on a globe, then when you go to the map due to stretching or compression, this circle will be distorted like the outlines geographical objects and will take the shape of an ellipse. This ellipse is called distortion ellipse or indicatrix Tissot.

The dimensions and degree of elongation of this ellipse compared to the circle reflect all types of distortions inherent in the map in this place. Type and dimensions ellipses 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 with the direction of the minor axis. These directions are called main directions .

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

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

Determining distortions for a number of map points and then following them isokol - lines connecting points with the same distortion values ​​gives a clear picture of the distribution of distortions and allows you to take distortions into account when using the map. To determine 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 unfold the earth's surface onto a plane, gaps and overlaps will certainly arise, which in turn leads to stretching and compression.

But at the same time there will be places on the map in which there will be no compression or expansion.

Lines or points on geographical map, in which there are no distortions and the main scale of the map is preserved, called lines or points of zero distortion (LNI and TNI) .

As you move away from them, the distortion increases.

Questions for repetition and consolidation of material

1. What causes cartographic distortions?

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

3. Explain what the zero distortion point and line are?

4. On which maps does the scale remain constant?

5. How to determine the presence and magnitude of distortions in certain places on the map?

6. What is Tissot's indicatrix?

7. What is the purpose of the distortion ellipse?

8. What are isokols and what is their purpose?

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 minor.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The main signs of distortion on the map

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

The nature of distortions in the main groups of map projections

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

Topic 6. Symbols on a topographic map

TASK 9. Draw on sheets of drawing paper (A4 format) conventional signs topographic maps (an example for the implementation of conventional signs is topographic map scale 1: 10,000 (SNOW)).

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 the objects located on them.

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

Distortion of line lengths 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 scale of the map is therefore variable. But on any map there are points or lines of zero distortion, and the image scale on them is called the main thing. IN in other places the scales are different, they are called private.

It is convenient to judge the presence of length distortion on a map by comparing the size of the segments between parallels (Figure 11). Segments AB and CD (Figure 11) should be equal, but they are different in length, therefore, on this map there is a distortion of the lengths of the meridians (τ). 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 rate is indicated by the letter A, and the smallest one is a letter b.

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

Distortion of corners very easy to install using 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 designated by the letter ε (epsilon):

ε = θ + 90º,

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

Area distortion easy to determine by comparing the areas of the 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 indicator ( R) is calculated using the formula:

p = n · m · cos ε.

Distortion of forms is that the shape of the area on the map differs from the shape on the surface of the Earth. The presence of distortion can be determined by comparing the shape of map 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 indicator ( TO)depends on the difference of the largest ( A) and the smallest ( b) length distortion indicators 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 6th (6–7) grades of secondary school) determine partial scales, the degree of distortion of length along the meridian ( T), along the parallel ( n), angle distortion ( ε ), area distortion ( R) for two points specified in one of the options (Table 11). Enter the measurement and calculation data into a table according to the form (Table 10).

Table 10– Determination of the magnitude of distortion

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

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

For determining n necessary:

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

2 find the actual length of the corresponding parallel arc on the surface of the earth’s ellipsoid according to table 12 “Length of parallel arcs and meridians on Krasovsky’s ellipsoid” L 1;

3 calculate private scale n = l 1 /L 1, and present the fraction in the form 1: xxxxxxxx.

For determining T:

1 measure on a map the length of the arc of the meridian on which a 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 L 2;

3 calculate the private scale: m = l 2 /L 2, and present the fraction in the form: 1: xxxxxxxx.

4 express the partial scale in fractions of the main one. To do this, divide the denominator of the main scale by the denominator of the particular scale.

2). Measure the angle between the meridian and the parallel and calculate its deviation from the straight line ε, measurement accuracy 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 previously given formula.

Table 11– Task options 10

Option	Geographic coordinates of point 1	Geographic coordinates of point 2
latitude	longitude,	latitude	longitude
	0º	90º in. d.	60º	150ºv. d.
	10ºs. w.	90º in. d.	70º C. w.	150ºv. d.
	10ºs. w.	80º W d.	70º C. w.	30º W d.
	0º	60º in. d.	20º C. w.	0º
	10º S w.	100º in. d.	30º S w.	150ºv. d.
	0º	120º W. d.	50ºS. w.	120ºv. d.
	30ºs. w.	140º in. d.	40º C. w.	160º W d.
	20º S w.	100ºw d.	0º	0º
	60º w.	140 v. d.	40º C. w.	80ºc. d
	50ºs. w.	160º in. d.	20º C. w.	60º in. d.

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