Maps with indicators of longitude and latitude. How to set geographic coordinates on the map? Using a geographic map to determine longitude and latitude

On Yandex Maps, geographic coordinates are recognized in degrees, represented as decimal fractions. At the same time, several more formats for recording coordinates are used in the world, for example, in degrees, minutes and seconds.

Coordinates are a pair of numbers that define the location of an object on a map.

The first digit in the format accepted on Yandex Maps is , or the angle between the local zenith direction (that is, the direction pointing directly up over a specific place) and the equatorial plane. Northern latitude is indicated by the letter N, southern - by the letter S.

The second digit is the longitude, or the angle between the meridian plane (the line of section of the surface of the globe by a plane passing through a given point and the axis of rotation of the Earth) and the plane of the initial zero (Greenwich) meridian. Longitudes from 0° to 180° east of the prime meridian are called east (E), to the west - west (W).

Entering coordinates on Yandex Maps

Open a browser and type maps.yandex.ru in the address bar, or open the Yandex Maps application on or . In the search box, enter the coordinates, for example: 55.751710,37.617019 - then click "Find". In the application, to call the search bar, you must first click on the magnifying glass icon (usually located at the bottom of the screen). Please note that the format for entering coordinates should be exactly this: first latitude, then longitude; the integer part of the coordinates is separated from the fractional part by a dot; numbers do not contain spaces; latitude and longitude are separated by a comma.

After clicking on the "Find" button, the marker on the map will move to the point described by the coordinates - now you can build a route.

To the left of the map, the address corresponding to the coordinates will be displayed, as well as their alternative representation - with degrees, minutes and seconds. In our case, it will look like this:
Latitude: 55°45′6.16″N (55.75171)
Longitude: 37°37′1.27″E (37.617019)

If you enter the coordinates in the wrong order - for example, first the longitude and then the latitude (some navigators and other electronic mapping services work with data in this sequence) - you can quickly change the order of numbers on Yandex Maps. To do this, click on the "Swap" link under the full description of the coordinates, and the marker will move to the correct point.

In Chapter 1, it was noted that the Earth has the shape of a spheroid, that is, an oblate ball. Since the terrestrial spheroid differs very little from a sphere, this spheroid is usually called the globe. The earth rotates around an imaginary axis. The points of intersection of an imaginary axis with the globe are called poles. north geographic pole (PN) is considered to be the one from which the Earth's own rotation is seen counterclockwise. south geographic pole (PS) is the pole opposite to the north.
If we mentally cut the globe with a plane passing through the axis (parallel to the axis) of the Earth's rotation, we get an imaginary plane, which is called meridian plane . The line of intersection of this plane with the earth's surface is called geographic (or true) meridian .
The plane perpendicular to the earth's axis and passing through the center of the earth is called equatorial plane , and the line of intersection of this plane with the earth's surface - equator .
If you mentally cross the globe with planes parallel to the equator, then circles are obtained on the surface of the Earth, which are called parallels .
Parallels and meridians plotted on globes and maps make up degree grid (Fig. 3.1). The degree grid makes it possible to determine the position of any point on the earth's surface.
For the initial meridian in the preparation of topographic maps taken Greenwich astronomical meridian passing through the former Greenwich Observatory (near London from 1675 - 1953). Currently, the buildings of the Greenwich Observatory house a museum of astronomical and navigational instruments. The modern Prime Meridian passes through Hirstmonceau Castle 102.5 meters (5.31 seconds) east of the Greenwich Astronomical Meridian. The modern prime meridian is used for satellite navigation.

Rice. 3.1. Degree grid of the earth's surface

Coordinates - angular or linear quantities that determine the position of a point on a plane, surface or in space. To determine coordinates on the earth's surface, a point is projected by a plumb line onto an ellipsoid. To determine the position of horizontal projections of a terrain point in topography, systems are used geographical , rectangular and polar coordinates .
Geographical coordinates determine the position of a point relative to the earth's equator and one of the meridians, taken as the initial one. Geographic coordinates may be derived from astronomical observations or geodetic measurements. In the first case they are called astronomical , in the second - geodetic . For astronomical observations, the projection of points onto the surface is carried out by plumb lines, for geodetic measurements - by normals, therefore the values of astronomical and geodetic geographical coordinates are somewhat different. To create small-scale geographical maps, the compression of the Earth is neglected, and the ellipsoid of revolution is taken as a sphere. In this case, the geographic coordinates will be spherical .
Latitude - angular value that determines the position of a point on Earth in the direction from the equator (0º) to the North Pole (+90º) or South Pole (-90º). Latitude is measured by the central angle in the meridian plane of a given point. On globes and maps, latitude is shown using parallels.

Rice. 3.2. Geographic latitude

Longitude - angular value that determines the position of a point on Earth in the West-East direction from the Greenwich meridian. Longitudes are counted from 0 to 180 °, to the east - with a plus sign, to the west - with a minus sign. On globes and maps, latitude is shown using meridians.

Rice. 3.3. Geographic longitude

3.1.1. Spherical coordinates

spherical geographic coordinates called the angular quantities (latitude and longitude) that determine the position of terrain points on the surface of the earth's sphere relative to the plane of the equator and the initial meridian.

spherical latitude (φ) call the angle between the radius vector (the line connecting the center of the sphere and a given point) and the equatorial plane.

spherical longitude (λ) is the angle between the zero meridian plane and the meridian plane of the given point (the plane passes through the given point and the axis of rotation).

Rice. 3.4. Geographic spherical coordinate system

In the practice of topography, a sphere with a radius R = 6371 is used km, whose surface is equal to the surface of the ellipsoid. On such a sphere, the arc length of the great circle is 1 minute (1852 m) called nautical mile.

3.1.2. Astronomical coordinates

Astronomical geographical coordinates are latitude and longitude, which determine the position of points on geoid surface relative to the plane of the equator and the plane of one of the meridians, taken as the initial one (Fig. 3.5).

Astronomical latitude (φ) called the angle formed by a plumb line passing through a given point and a plane perpendicular to the axis of rotation of the Earth.

Plane of the astronomical meridian - a plane passing through a plumb line at a given point and parallel to the axis of rotation of the Earth.
astronomical meridian - the line of intersection of the surface of the geoid with the plane of the astronomical meridian.

Astronomical longitude (λ) called the dihedral angle between the plane of the astronomical meridian passing through a given point, and the plane of the Greenwich meridian, taken as the initial one.

Rice. 3.5. Astronomical latitude (φ) and astronomical longitude (λ)

3.1.3. Geodetic coordinate system

AT geodetic geographic coordinate system for the surface on which the positions of the points are found, the surface is taken reference -ellipsoid . The position of a point on the surface of the reference ellipsoid is determined by two angular values - the geodetic latitude (AT) and geodetic longitude (L).
Plane of the geodesic meridian - a plane passing through the normal to the surface of the earth's ellipsoid at a given point and parallel to its minor axis.
geodetic meridian - the line along which the plane of the geodesic meridian intersects the surface of the ellipsoid.
Geodetic parallel - the line of intersection of the surface of an ellipsoid by a plane passing through a given point and perpendicular to the minor axis.

Geodetic latitude (AT)- the angle formed by the normal to the surface of the earth's ellipsoid at a given point and the plane of the equator.

Geodetic longitude (L)- dihedral angle between the plane of the geodesic meridian of the given point and the plane of the initial geodesic meridian.

Rice. 3.6. Geodetic latitude (B) and geodetic longitude (L)

3.2. DETERMINATION OF GEOGRAPHICAL COORDINATES OF POINTS ON THE MAP

Topographic maps are printed in separate sheets, the sizes of which are set for each scale. The side frames of the sheets are the meridians, and the upper and lower frames are the parallels. . (Fig. 3.7). Consequently, geographic coordinates can be determined by the side frames of the topographic map . On all maps, the top frame always faces north.
Geographic latitude and longitude are signed in the corners of each sheet of the map. On maps of the Western Hemisphere, in the northwestern corner of the frame of each sheet, to the right of the longitude of the meridian, the inscription is placed: "West of Greenwich."
On maps of scales 1: 25,000 - 1: 200,000, the sides of the frames are divided into segments equal to 1 ′ (one minute, Fig. 3.7). These segments are shaded through one and divided by points (except for the map of scale 1: 200,000) into parts of 10 "(ten seconds). On each sheet of maps of scales 1: 50,000 and 1: 100,000, in addition, they show the intersection of the middle meridian and the middle parallel with digitization in degrees and minutes, and along the inner frame - outputs of minute divisions with strokes 2 - 3 mm long.This allows, if necessary, to draw parallels and meridians on a map glued from several sheets.

Rice. 3.7. Side frames of the map

When compiling maps of scales 1: 500,000 and 1: 1,000,000, a cartographic grid of parallels and meridians is applied to them. Parallels are drawn, respectively, through 20′ and 40 "(minutes), and meridians - through 30" and 1 °.
The geographical coordinates of a point are determined from the nearest southern parallel and from the nearest western meridian, the latitude and longitude of which are known. For example, for a map with a scale of 1: 50,000 "ZAGORYANI", the nearest parallel located to the south of a given point will be the parallel 54º40′ N, and the nearest meridian located to the west of the point will be the meridian 18º00′ E. (Fig. 3.7).

Rice. 3.8. Determination of geographical coordinates

To determine the latitude of a given point, you must:

set one leg of the measuring compass to a given point, set the other leg along the shortest distance to the nearest parallel (for our map 54º40 ′);
without changing the solution of the measuring compass, install it on the side frame with minute and second divisions, one leg should be on the south parallel (for our map 54º40 ′), and the other between the 10-second points on the frame;
count the number of minutes and seconds from the south parallel to the second leg of the measuring compass;
add the result obtained to the south latitude (for our map 54º40 ′).

To determine the longitude of a given point, you must:

set one leg of the measuring compass to a given point, set the other leg along the shortest distance to the nearest meridian (for our map 18º00 ′);
without changing the solution of the measuring compass, set it to the nearest horizontal frame with minute and second divisions (for our map, the lower frame), one leg should be on the nearest meridian (for our map 18º00 ′), and the other between the 10-second points on horizontal frame;
count the number of minutes and seconds from the western (left) meridian to the second leg of the measuring compass;
add the result to the longitude of the western meridian (for our map 18º00′).

note to the fact that this method of determining the longitude of a given point for maps at a scale of 1:50,000 and smaller has an error due to the convergence of the meridians that limit the topographic map from the east and west. The north side of the frame will be shorter than the south side. Therefore, the discrepancies between the measurements of longitude on the northern and southern frames may differ by several seconds. To achieve high accuracy in the measurement results, it is necessary to determine the longitude on both the south and north sides of the frame, and then interpolate.
To improve the accuracy of determining geographic coordinates, you can use graphic method. To do this, it is necessary to connect with straight lines the nearest ten-second divisions of the same name to the point in latitude to the south of the point and in longitude to the west of it. Then determine the dimensions of the segments in latitude and longitude from the drawn lines to the position of the point and summarize them, respectively, with the latitude and longitude of the drawn lines.
The accuracy of determining geographical coordinates on maps of scales 1: 25,000 - 1: 200,000 is 2" and 10", respectively.

3.3. POLAR COORDINATE SYSTEM

polar coordinates are called the angular and linear quantities that determine the position of a point on the plane relative to the origin, taken as a pole ( O), and the polar axis ( OS) (Fig. 3.1).

The location of any point ( M) is determined by the position angle ( α ), counted from the polar axis to the direction to the determined point, and the distance (horizontal distance - the projection of the terrain line on the horizontal plane) from the pole to this point ( D). Polar angles are usually measured from the polar axis in a clockwise direction.

Rice. 3.9. Polar coordinate system

For the polar axis can be taken: the true meridian, the magnetic meridian, the vertical line of the grid, the direction to any landmark.

3.2. BIPOLAR COORDINATE SYSTEMS

Bipolar coordinates call two angular or two linear quantities that determine the location of a point on a plane relative to two starting points (poles O 1 and O 2 rice. 3.10).

The position of any point is determined by two coordinates. These coordinates can be either two position angles ( α 1 and α 2 rice. 3.10), or two distances from the poles to the determined point ( D 1 and D 2 rice. 3.11).

Rice. 3.10. Determining the location of a point at two angles (α 1 and α 2 )

Rice. 3.11. Determining the location of a point by two distances

In a bipolar coordinate system, the position of the poles is known, i.e. the distance between them is known.

3.3. POINT HEIGHT

Previously reviewed plan coordinate systems , defining the position of any point on the surface of the earth's ellipsoid, or the reference ellipsoid , or on the plane. However, these planned coordinate systems do not allow obtaining an unambiguous position of a point on the physical surface of the Earth. Geographical coordinates refer the position of the point to the surface of the reference ellipsoid, polar and bipolar coordinates refer the position of the point to the plane. And all these definitions have nothing to do with the physical surface of the Earth, which is more interesting for a geographer than a reference ellipsoid.
Thus, the planned coordinate systems do not make it possible to unambiguously determine the position of a given point. It is necessary to somehow define your position, at least with the words “above”, “below”. Just about what? To obtain complete information about the position of a point on the physical surface of the Earth, the third coordinate is used - height . Therefore, it becomes necessary to consider the third coordinate system - height system .

The distance along a plumb line from the level surface to a point on the physical surface of the Earth is called height.

There are heights absolute if they are counted from the level surface of the Earth, and relative (conditional ) if they are counted from an arbitrary level surface. Usually, the level of the ocean or the open sea in a calm state is taken as the origin of absolute heights. In Russia and Ukraine, the absolute heights are taken as the origin zero of the Kronstadt footstock.

Footstock- a rail with divisions, fixed vertically on the shore so that it is possible to determine the position of the water surface in a calm state by it.
Kronstadt footstock- a line on a copper plate (board) mounted in the granite abutment of the Blue Bridge of the Obvodny Canal in Kronstadt.
The first footstock was installed during the reign of Peter the Great, and since 1703 regular observations of the level of the Baltic Sea began. Soon the footstock was destroyed, and only from 1825 (and up to the present time) regular observations were resumed. In 1840, hydrographer M.F. Reinecke calculated the average height of the Baltic Sea and recorded it on the granite abutment of the bridge in the form of a deep horizontal line. Since 1872, this feature has been taken as a zero mark when calculating the heights of all points on the territory of the Russian state. The Kronstadt footstock was repeatedly modified, however, the position of its main mark was kept the same during design changes, i.e. determined in 1840
After the collapse of the Soviet Union, Ukrainian surveyors did not invent their own national system of heights, and currently in Ukraine it is still used Baltic height system.

It should be noted that, in every necessary case, measurements are not taken directly from the level of the Baltic Sea. There are special points on the ground, the heights of which were previously determined in the Baltic system of heights. These points are called benchmarks .
Absolute heights H can be positive (for points above the Baltic Sea level) and negative (for points below the Baltic Sea level).
The difference between the absolute heights of two points is called relative height or excess (h):
h = H BUT-H AT .
The excess of one point over another can also be positive and negative. If the absolute height of the point BUT greater than the absolute height of the point AT, i.e. is above the point AT, then the excess of the point BUT over the dot AT will be positive, and vice versa, exceeding the point AT over the dot BUT- negative.

Example. Absolute heights of points BUT and AT: H BUT = +124,78 m; H AT = +87,45 m. Find Mutual Exceedances of Points BUT and AT.

Solution. Exceeding point BUT over the dot AT
h A(B) = +124,78 - (+87,45) = +37,33 m.
Exceeding point AT over the dot BUT
h B(A) = +87,45 - (+124,78) = -37,33 m.

Example. Point absolute height BUT is equal to H BUT = +124,78 m. Exceeding point FROM over the dot BUT equals h C(A) = -165,06 m. Find the absolute height of a point FROM.

Solution. Point absolute height FROM is equal to
H FROM = H BUT + h C(A) = +124,78 + (-165,06) = - 40,28 m.

The numerical value of the height is called the elevation of the point (absolute or conditional).
For example, H BUT = 528.752 m - absolute mark of the point BUT; H" AT \u003d 28.752 m - conditional elevation of the point AT .

Rice. 3.12. Heights of points on the earth's surface

To move from conditional to absolute heights and vice versa, it is necessary to know the distance from the main level surface to the conditional one.

Video
Meridians, parallels, latitudes and longitudes
Determining the position of points on the earth's surface

Questions and tasks for self-control

Expand the concepts: pole, equatorial plane, equator, meridian plane, meridian, parallel, degree grid, coordinates.
Relative to what planes on the globe (ellipsoid of revolution) are geographic coordinates determined?
What is the difference between astronomical geographic coordinates and geodetic coordinates?
Using the drawing, expand the concepts of "spherical latitude" and "spherical longitude".
On what surface is the position of points in the astronomical coordinate system determined?
Using the drawing, expand the concepts of "astronomical latitude" and "astronomical longitude".
On what surface is the position of points in the geodetic coordinate system determined?
Using the drawing, expand the concepts of "geodesic latitude" and "geodesic longitude".
Why, in order to improve the accuracy of determining longitude, is it necessary to connect the nearest ten-second divisions of the same name to the point with straight lines?
How can you calculate the latitude of a point if you determine the number of minutes and seconds from the northern frame of a topographic map?
What are the polar coordinates?
What is the purpose of the polar axis in a polar coordinate system?
What coordinates are called bipolar?
What is the essence of the direct geodetic problem?

Geographical coordinates determine the position of a point on the earth's surface. Geographic coordinates are based on the principle of spherical and consist of latitude and longitude.

Latitude— the angle between the local direction of the zenith and the plane of the equator, measured from 0° to 90° on both sides of the equator. The geographical latitude of points lying in the northern hemisphere (northern latitude) is considered to be positive, the latitude of points in the southern hemisphere is negative. It is customary to speak of latitudes close to the poles as high, and about those close to the equator - as about low.

Longitude- the angle between the plane of the meridian passing through the given point, and the plane of the initial zero meridian, from which the longitude is counted. Longitudes from 0° to 180° east of the prime meridian are called east, to the west - west. Eastern longitudes are considered positive, western longitudes negative.

Format of recording geographic coordinates

Geographic coordinates of a single point can be expressed in different formats. Depending on whether minutes and seconds are represented as values from 0 to 60 or from 0 to 100 (decimal).

The coordinate format is usually written as follows: DD- degrees, MM- minutes, SS- seconds, if minutes and seconds are represented as decimal fractions, then simply write DD.DDDD. For example:

DD MM SS: 50° 40" 45"" E, 40 50" 30"" N - degrees, minutes, seconds
DD MM.MM: 50° 40.75" E, 40 50.5" N - degrees, decimal minutes
DD.DDDD: 50.67916 E, 40.841666 N - decimal degrees

Why do you need to know the coordinates of your house

Often, houses in holiday villages and many villages do not have a clear navigation consisting of signs with street names and house numbers, or even houses with signs with numbers can be scattered throughout the village in a random order (historically formed as the village was built up). There are times when everything is fine with navigation in a village, but not all car GPS navigators have such a house or street. Residents of such houses have to explain for a long time and, as a rule, intricately how to get to them using different landmarks. In this case, it is easier to give the coordinates of the house, because any car navigator can pave the way for the coordinates.

To work out the technical feasibility of connecting the Internet in a country house, we also ask our customers to provide the coordinates of the house, especially if it is not located at the address on any of the online mapping services.

Determination of coordinates using online mapping services

Currently, the most famous online mapping services with a search function are Google and Yandex maps. Consider how you can determine geographic coordinates from a map or a satellite image in the service Google Maps:

2. Locate the exact location on the map. For this map can be moved mouse, zoom in and out by scrolling the mouse wheel. You can also find the desired locality using name search using the locality, street and house. To find the place at home as accurately as possible, switch between display modes: Map, Hybrid or Satellite.

3. Click right click on the desired location on the map and select from the opened menu paragraph “What is here?” . A marker in the form of a green arrow will appear on the map. Repeat the operation if the marker is set inaccurately.

4. When you hover the mouse over the green arrow, the geographical coordinates of the place will appear, they will also appear in the search bar from where they can be copied to the clipboard.

Rice. 1. Determining the coordinates of a place using a pointer on a Google map

Now let's look at how you can determine geographic coordinates from a map or a satellite image in the service Yandex maps:

To search for a place, we use the same algorithm as for searching on Google maps. Open Yandex.Maps: http://maps.yandex.ru . To get coordinates on Yandex map, use tool"Get information"(button with an arrow and a question mark, in the upper left part of the map). When you click on the map with this tool, a marker appears on the map and the coordinates are displayed in the search bar.

Rice. 2. Determining the coordinates of the place according to the pointer on the Yandex map

By default, search engine maps show coordinates in degrees with a decimal fraction with "-" signs for negative longitude. On Google maps and Yandex maps, latitude first, then longitude (until October 2012, the reverse order was adopted on Yandex maps: first longitude, then latitude).

Video lesson “Geographical latitude and geographical longitude. Geographical coordinates will help you get an idea of the geographic latitude and geographic longitude. The teacher will tell you how to correctly determine the geographical coordinates.

Geographic latitude is the length of the arc in degrees from the equator to the given point.

To determine the latitude of an object, you need to find the parallel on which this object is located.

For example, the latitude of Moscow is 55 degrees and 45 minutes north latitude, it is written as follows: Moscow 55 ° 45 "N; New York latitude - 40 ° 43" N; Sydney - 33°52"S

Geographic longitude is determined by meridians. Longitude can be western (from 0 meridian west to 180 meridian) and eastern (from 0 meridian east to 180 meridian). Longitudes are measured in degrees and minutes. Geographic longitude can have values from 0 to 180 degrees.

Geographic longitude- length of the arc of the equator in degrees from the initial meridian (0 degrees) to the meridian of the given point.

The prime meridian is the Greenwich meridian (0 degrees).

Rice. 2. Definition of longitudes ()

To determine longitude, you need to find the meridian on which the given object is located.

For example, the longitude of Moscow is 37 degrees and 37 minutes of east longitude, it is written as follows: 37 ° 37 "E; the longitude of Mexico City is 99 ° 08" W.

Rice. 3. Geographic latitude and geographic longitude

To accurately determine the location of an object on the surface of the Earth, you need to know its geographic latitude and geographic longitude.

Geographical coordinates- quantities that determine the position of a point on the earth's surface using latitudes and longitudes.

For example, Moscow has the following geographic coordinates: 55°45" N and 37°37" E. The city of Beijing has the following coordinates: 39°56′ N 116°24′ E The latitude value is written first.

Sometimes you need to find an object by already given coordinates, for this you must first assume in which hemispheres this object is located.

Homework

Paragraphs 12, 13.

1. What is geographic latitude and longitude?

Bibliography

Main

1. Initial course of geography: Proc. for 6 cells. general education institutions / T.P. Gerasimova, N.P. Neklyukov. - 10th ed., stereotype. - M.: Bustard, 2010. - 176 p.

2. Geography. Grade 6: atlas. - 3rd ed., stereotype. - M.: Bustard, DIK, 2011. - 32 p.

3. Geography. Grade 6: atlas. - 4th ed., stereotype. - M.: Bustard, DIK, 2013. - 32 p.

4. Geography. 6 cells: cont. cards. - M.: DIK, Bustard, 2012. - 16 p.

Encyclopedias, dictionaries, reference books and statistical collections

1. Geography. Modern illustrated encyclopedia / A.P. Gorkin. - M.: Rosmen-Press, 2006. - 624 p.

Literature for preparing for the GIA and the Unified State Examination

1. Geography: an initial course. Tests. Proc. allowance for students 6 cells. - M.: Humanit. ed. center VLADOS, 2011. - 144 p.

2. Tests. Geography. Grades 6-10: Teaching aid / A.A. Letyagin. - M .: LLC "Agency" KRPA "Olimp": "Astrel", "AST", 2001. - 284 p.

Materials on the Internet

1. Federal Institute of Pedagogical Measurements ().

2. Russian Geographical Society ().

For determining latitude it is necessary, using a triangle, to lower the perpendicular from point A to the degree frame to the line of latitude and read to the right or left on the latitude scale, the corresponding degrees, minutes, seconds. φА= φ0+ Δφ

φА=54 0 36 / 00 // +0 0 01 / 40 //= 54 0 37 / 40 //

For determining longitude it is necessary, using a triangle, to lower the perpendicular from point A to the degree frame of the line of longitude and read the corresponding degrees, minutes, seconds from above or below.

Determination of rectangular coordinates of a point on the map

The rectangular coordinates of the point (X, Y) on the map are determined in the square of the kilometer grid as follows:

1. Using a triangle, perpendiculars are lowered from point A to the kilometer grid line X and Y, values are taken XA=X0+Δ X; UA=U0+Δ At

For example, the coordinates of point A are: XA \u003d 6065 km + 0.55 km \u003d 6065.55 km;

UA \u003d 4311 km + 0.535 km \u003d 4311.535 km. (coordinate is reduced);

Point A is located in the 4th zone, as indicated by the first digit of the coordinate at given.

9. Measurement of line lengths, directional angles and azimuths on the map, determination of the angle of inclination of the line specified on the map.

Length measurement

To determine the distance between points of the terrain (objects, objects) on the map, using a numerical scale, it is necessary to measure the distance between these points in centimeters on the map and multiply the resulting number by the scale value.

A small distance is easier to determine using a linear scale. To do this, it is enough to apply a compass-meter, the solution of which is equal to the distance between given points on the map, to a linear scale and take a reading in meters or kilometers.

To measure the curves, the “step” solution of the measuring compass is set so that it corresponds to an integer number of kilometers, and an integer number of “steps” is set aside on the segment measured on the map. The distance that does not fit into an integer number of “steps” of the measuring compass is determined using a linear scale and added to the resulting number of kilometers.

Measurement of directional angles and azimuths on the map

We connect point 1 and 2. We measure the angle. The measurement takes place with the help of a protractor, it is located parallel to the median, then the angle of inclination is reported clockwise.

Determining the slope angle of a line defined on the map.

The definition occurs exactly according to the same principle as finding the directional angle.

10. Direct and inverse geodesic problem on the plane. In the computational processing of measurements made on the ground, as well as in the design of engineering structures and calculations for transferring projects to nature, it becomes necessary to solve direct and inverse geodetic problems. Direct geodetic problem . Known coordinates X 1 and at 1 point 1, directional angle 1-2 and distance d 1-2 to point 2 you need to calculate its coordinates X 2 ,at 2 .

Rice. 3.5. To the solution of direct and inverse geodetic problems

The coordinates of point 2 are calculated by the formulas (Fig. 3.5): (3.4) where X,atincrements of coordinates equal to

(3.5)

Inverse geodesic problem . Known coordinates X 1 ,at 1 point 1 and X 2 ,at 2 points 2 need to calculate the distance between them d 1-2 and directional angle  1-2 . From formulas (3.5) and fig. 3.5 shows that. (3.6) To determine the directional angle  1-2, we use the function of the arc tangent. At the same time, we take into account that computer programs and microcalculators give the main value of the arc tangent  = , lying in the range 90+90, while the desired directional angle  can have any value in the range 0360.

The formula for the transition from  to  depends on the coordinate quarter in which the given direction is located or, in other words, on the signs of the differences y=y 2 y 1 and  x=X 2 X 1 (see table 3.1 and fig. 3.6). Table 3.1

Rice. 3.6. Directional angles and main values of the arc tangent in I, II, III and IV quarters

The distance between points is calculated by the formula

(3.6) or in another way - according to the formulas (3.7)

In particular, electronic tacheometers are equipped with programs for solving direct and inverse geodetic problems, which makes it possible to determine the coordinates of observed points directly in the course of field measurements, calculate angles and distances for marking work.