Which parts of the world are the most distorted. Map projections and distortions. The westernmost point of Asia is the cape

The date: 24.10.2015

map projection- a mathematical way of depicting the globe (ellipsoid) on a plane.

For projecting a spherical surface onto a plane use auxiliary surfaces.

By type auxiliary cartographic projection surface is divided into:

Cylindrical 1(auxiliary surface is the side surface of the cylinder), conical 2(lateral surface of the cone), azimuth 3(the plane, which is called the picture plane).

Also allocate polyconical

pseudocylindrical conditional

and other projections.

Orientation auxiliary figures of the projection 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 distortion- 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 displayed on a map.

The smaller the scale of the map, the more significant the distortion. On large scale maps, distortion is negligible.

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

According to the nature of distortions, map projections are divided into:

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

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

• arbitrary, in which the distortions of lengths, areas and angles, but they are evenly distributed on the map. Among them, 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 there are also points where 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 cards, persists only on lines and at zero-distortion points. 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 distortion 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 and intersect with parallels at right angles. Therefore, all the 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 amount 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.

Conformal projection distortion ellipses are shaped like a circle, the size of which increases depending on the distance from the points and lines of zero distortion.

In an equal area projection distortion ellipses have the shape of ellipses, the areas of which are the same (the length of one axis increases, and the second decreases).

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 are not distorted (equidistant along the meridians).
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. The lengths along the parallels are distorted, if the lengths along the meridians are not distorted.
   
5. The lengths are distorted along the meridians, if the lengths along the parallels are not distorted.

The nature of distortions in the main groups of cartographic projections

Map projections	distortion
Equangular	Preserve angles, distort areas and lengths of lines.
isometric	They preserve areas, distort angles and shapes.
Equidistant	In one direction they have a constant length scale, the distortions of angles and areas are in equilibrium.
Arbitrary	Distort corners and squares.
Cylindrical	There are no distortions along the line of the equator, but they increase with the degree of approach to 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.

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

On what scale can the map "Natural zones 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 is:

a) the island of Tierra del Fuego; 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 the 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 of topographic maps (a topographic map of scale 1: 10,000 (SNOV) serves as a model for the implementation of conventional symbols).

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) must 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 the physical map of the hemispheres, scale 1: 90,000,000 (atlas "Elementary Geography Course" for 6 (6-7) grades of secondary school) 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

Option	Geographic coordinates of point 1	Geographic coordinates of point 2
latitude	longitude,	latitude	longitude
	0º	90º in. d.	60º	150º in. d.
	10º s. sh.	90º in. d.	70º s. sh.	150º in. d.
	10º s. sh.	80º W d.	70º s. sh.	30º W d.
	0º	60º in. d.	20º s. sh.	0º
	10º S sh.	100º in. d.	30º S sh.	150º in. d.
	0º	120º W d.	50º sh.	120º in. d.
	30º s. sh.	140º in. d.	40º s. sh.	160º W d.
	20º S sh.	100º W d.	0º	0º
	60º sh.	140 c. d.	40º s. sh.	80º in. d
	50º s. sh.	160º in. d.	20º s. sh.	60º in. d.

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 (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 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 varies 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 compression, 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 size and degree of elongation of this ellipse in comparison with 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 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?