Degree network and its elements. Degree network and its elements What is the value of the arc 1 meridians

The spherical shape of the Earth and the daily rotation determine the existence of two fixed points on the earth's surface - poles. An imaginary earth's axis passes through the poles, around which the earth rotates.

On maps and globes, the largest circle is drawn - the equator, the plane of which is perpendicular to the earth's axis. The equator divides the Earth into northern and southern hemispheres. The length of the arc 1° of the equator is 40075.7 km: 360° = 111.3 km.

Parallel to the plane of the equator, you can conditionally arrange a lot of planes. When they intersect with the surface of the globe, small circles are formed - parallels. They are held on a globe or map at a certain distance from the equator and are oriented from west to east. The length of the circles of parallels decreases uniformly from the equator to the poles. Recall that it is greatest at the equator and zero at the poles.

The globe can also be crossed by imaginary planes passing through the Earth's axis perpendicular to the plane of the equator. When these planes intersect with the surface of the Earth, large circles are formed - meridians. Meridians can be drawn through any point of the globe. All of them intersect at the points of the poles and are oriented from north to south. The average arc length of the 1st meridian is 40008.5 km: 360° = 111 km. The direction of the local meridian at any point can be determined at noon in the direction of the shadow from the gnomon or other object. In the northern hemisphere, the end of the shadow from the object shows the direction to the north, in the southern hemisphere - to the south.

To calculate distances on a map or globe, the following values ​​can be used: the length of the arc is 1º of the meridian and 1º of the equator, which is approximately 111 km.

To determine the distance in kilometers on a map or globe between two points located on the same meridian, the number of degrees between the points is multiplied by 111 km. To determine the distance in kilometers between points lying on the same parallel, the number of degrees is multiplied by the length of an arc of 1 ° parallel indicated on the map or determined from the tables.

The length of the arcs of parallels and meridians on the Krasovsky ellipsoid

For example, the distance between Kyiv and St. Petersburg, located approximately on the 30° meridian, is 111 km *9.5° = 1054 km; the distance between Kyiv and Kharkov (approximately 50° parallel) is 71 km * 6° = 426 km.

Parallels and meridians form degree network. The most accurate representation of the degree network can be obtained from the globe. On geographical maps, the location of parallels and meridians depends on the map projection. To verify this, you can compare different maps, such as maps of hemispheres, continents, Russia, Russian regions, etc.

The position of any point on the globe is determined using geographic coordinates: latitude and longitude.

Geographic latitude- distance along the meridian in degrees from the equator to any point on the globe. The equator is taken as the origin of the latitude reference - the zero parallel. Latitude varies from 0° at the equator to 90° at the pole. To the north of the equator, the northern latitude (north latitude) is counted, to the south of the equator - the southern latitude (south latitude). On the maps, the parallels are inscribed on the side frames, and on the globe - on the 0° and 180° meridians. For example, Kharkiv is located at 50° parallel north of the equator - its geographic latitude is 50° N. sh.; Kermadec Islands - in the Pacific Ocean at 30 ° south of the Equator, their latitude is approximately 30 ° S. sh.

If a point on a map or globe is located between two designated parallels, then its geographical latitude is additionally determined by the distance between these parallels. For example, to calculate the latitude of Irkutsk, located on the map of Russia between 50° and 60° N. sh., through the point draw a straight line connecting both parallels. Then it is conditionally divided into 10 equal parts - degrees, since the distance between the parallels is 10 °. Irkutsk is closer to the 50° parallel.

In practice, the geographic latitude is determined by the height of the North Star using a sextant device; at school, a vertical protractor, or eclimeter, is used for this purpose.

Geographic longitude- distance along the parallel in degrees from the prime meridian to any point on the globe. The Greenwich meridian, zero, which passes near London (where the Greenwich Observatory is located), is taken as the origin of longitude. To the east of the zero meridian to 180 °, eastern longitude (east longitude) is counted, to the west - western (west longitude). On maps, meridians are inscribed on the equator or the upper and lower frames of the map, and on the globe - on the equator. Meridians, like parallels, pass through the same number of degrees. For example, St. Petersburg is located on the 30th meridian east of the prime meridian, its geographic longitude is 30°E. d.; Mexico City - 100 meridian west of the zero meridian, its longitude is 100 ° W. d.

If the point is located between two meridians, then its longitude is specified by the distance between them. For example, Irkutsk is located between 100° and 110° E. but closer to 100°. A line is drawn through the point connecting both meridians, it is conditionally divided by 10 ° and the number of degrees is counted from 100 ° of the meridian to Irkutsk. Therefore, the geographical longitude of Irkutsk is approximately 104°.

Geographic longitude in practice is determined by the difference in time between a given point and the zero meridian or other known meridian. Geographical coordinates are recorded in whole degrees and minutes with latitude and longitude. In this case, 1º \u003d 60 min (60 "), a0.1 ° \u003d 6", 0.2 ° \u003d 12 ", etc.

Literature.

1. Geography / Ed. P.P. Vashchenko, E.I. Shipovich. - 2nd ed., revised and additional. - K .: Vishcha school. Head publishing house, 1986. - 503 p.

The length of the arc of the meridian and parallel. Sizes of trapezium frames for topographic maps

Kherson-2005

Meridian arc length S M between latitudes B1 and B2 is determined from the solution of an elliptic integral of the form:

(1.1)

which, as is known, is not taken in elementary functions. Numerical integration is used to solve this integral. According to Simpson's formula, we have:

(1.2)

(1.3)

where B1 and B2 are the latitudes of the ends of the meridian arc; M 1, M 2, Msr are the values ​​of the radii of curvature of the meridian at points with latitudes B1 and B2 and Bcp=(B 1 +B 2)/2; a is the semi-major axis of the ellipsoid, e 2 is the first eccentricity.

Parallel arc length S P is the length of a part of the circle, so it is obtained directly as the product of the radius of the given parallel r=NcosB for the difference in longitude l extreme points of the desired arc, i.e.

where l \u003d L 2 -L 1

The value of the radius of curvature of the first vertical N calculated by the formula

(1.5)

Filming trapezoid is a part of the surface of an ellipsoid bounded by meridians and parallels. Therefore, the sides of the trapezoid are equal to the lengths of the arcs of the meridians and parallels. Moreover, the northern and southern frames are arcs of parallels a 1 and a 2, and eastern and western - arcs of meridians With, equal to each other. Trapezium Diagonal d. To obtain specific dimensions of the trapezoid, it is necessary to divide the mentioned arcs by the scale denominator m and, to obtain dimensions in centimeters, multiply by 100. Thus, the working formulas are:

(1.6)

where m- denominator of the survey scale; N 1, N 2, are the radii of curvature of the first vertical at points with latitudes B1 and B2; M m- radius of curvature of the meridian at a point with latitude B m=(B1+B2)/2; ΔB \u003d (B 2 -B 1).

Task and initial data

1) Calculate the length of the meridian arc between two points with latitudes B 1 =30°00"00.000"" and B 2 \u003d 35 ° 00 "12.345" "+1" No., where № is the number of the variant.

2) Calculate the length of the arc of the parallel between the points lying on this parallel, with longitudes L1 = 0°00"00.000"" and L 2 \u003d 0 ° 45 "00.123" "+ 1" "No., where № is the number of the variant. Latitude of the parallel B=52°00"00.000""

3) Calculate the dimensions of the trapezoid frame at a scale of 1:100,000 for the N-35-№ map sheet, where № is the trapezoid number given by the teacher.

Solution scheme

The spherical shape of the Earth and the daily rotation determine the existence of two fixed points on the earth's surface - poles. An imaginary earth's axis passes through the poles, around which the earth rotates.

On maps and globes, the largest circle is drawn - the equator, the plane of which is perpendicular to the earth's axis. The equator divides the Earth into northern and southern hemispheres. The length of the arc 1° of the equator is 40075.7 km: 360° = 111.3 km.

Parallel to the plane of the equator, you can conditionally arrange a lot of planes. When they intersect with the surface of the globe, small circles are formed - parallels. They are held on a globe or map at a certain distance from the equator and are oriented from west to east. The length of the circles of parallels decreases uniformly from the equator to the poles. Recall that it is greatest at the equator and zero at the poles.

The globe can also be crossed by imaginary planes passing through the Earth's axis perpendicular to the plane of the equator. When these planes intersect with the surface of the Earth, large circles are formed - meridians. Meridians can be drawn through any point of the globe. All of them intersect at the points of the poles and are oriented from north to south. The average arc length of the 1st meridian is 40008.5 km: 360° = 111 km. The direction of the local meridian at any point can be determined at noon in the direction of the shadow from the gnomon or other object. In the northern hemisphere, the end of the shadow from the object shows the direction to the north, in the southern hemisphere - to the south.

To calculate distances on a map or globe, the following values ​​can be used: the length of the arc is 1º of the meridian and 1º of the equator, which is approximately 111 km.

To determine the distance in kilometers on a map or globe between two points located on the same meridian, the number of degrees between the points is multiplied by 111 km. To determine the distance in kilometers between points lying on the same parallel, the number of degrees is multiplied by the length of an arc of 1 ° parallel indicated on the map or determined from the tables.

The length of the arcs of parallels and meridians on the Krasovsky ellipsoid

For example, the distance between Kyiv and St. Petersburg, located approximately on the 30° meridian, is 111 km *9.5° = 1054 km; the distance between Kyiv and Kharkov (approximately 50° parallel) is 71 km * 6° = 426 km.

Parallels and meridians form degree network. The most accurate representation of the degree network can be obtained from the globe. On geographical maps, the location of parallels and meridians depends on the map projection. To verify this, you can compare different maps, such as maps of hemispheres, continents, Russia, Russian regions, etc.

The position of any point on the globe is determined using geographic coordinates: latitude and longitude.

Geographic latitude- distance along the meridian in degrees from the equator to any point on the globe. The equator is taken as the origin of the latitude reference - the zero parallel. Latitude varies from 0° at the equator to 90° at the pole. To the north of the equator, the northern latitude (north latitude) is counted, to the south of the equator - the southern latitude (south latitude). On the maps, the parallels are inscribed on the side frames, and on the globe - on the 0° and 180° meridians. For example, Kharkiv is located at 50° parallel north of the equator - its geographic latitude is 50° N. sh.; Kermadec Islands - in the Pacific Ocean at 30 ° south of the Equator, their latitude is approximately 30 ° S. sh.

If a point on a map or globe is located between two designated parallels, then its geographical latitude is additionally determined by the distance between these parallels. For example, to calculate the latitude of Irkutsk, located on the map of Russia between 50° and 60° N. sh., through the point draw a straight line connecting both parallels. Then it is conditionally divided into 10 equal parts - degrees, since the distance between the parallels is 10 °. Irkutsk is closer to the 50° parallel.

In practice, the geographic latitude is determined by the height of the North Star using a sextant device; at school, a vertical protractor, or eclimeter, is used for this purpose.

Geographic longitude- distance along the parallel in degrees from the prime meridian to any point on the globe. The Greenwich meridian, zero, which passes near London (where the Greenwich Observatory is located), is taken as the origin of longitude. To the east of the zero meridian to 180 °, eastern longitude (east longitude) is counted, to the west - western (west longitude). On maps, meridians are inscribed on the equator or the upper and lower frames of the map, and on the globe - on the equator. Meridians, like parallels, pass through the same number of degrees. For example, St. Petersburg is located on the 30th meridian east of the prime meridian, its geographic longitude is 30°E. d.; Mexico City - 100 meridian west of the zero meridian, its longitude is 100 ° W. d.

If the point is located between two meridians, then its longitude is specified by the distance between them. For example, Irkutsk is located between 100° and 110° E. but closer to 100°. A line is drawn through the point connecting both meridians, it is conditionally divided by 10 ° and the number of degrees is counted from 100 ° of the meridian to Irkutsk. Therefore, the geographical longitude of Irkutsk is approximately 104°.

Geographic longitude in practice is determined by the difference in time between a given point and the zero meridian or other known meridian. Geographical coordinates are recorded in whole degrees and minutes with latitude and longitude. In this case, 1º \u003d 60 min (60 "), a0.1 ° \u003d 6", 0.2 ° \u003d 12 ", etc.

Literature.

1. Geography / Ed. P.P. Vashchenko, E.I. Shipovich. - 2nd ed., revised and additional. - K .: Vishcha school. Head publishing house, 1986. - 503 p.

The meridian of the earth's ellipsoid is an ellipse, the radius of curvature of which is determined by the value M latitude dependent. The arc length of any curve of variable radius can be calculated by the well-known formula of differential geometry, which, as applied to the meridian, has the expression

Here IN 1 and IN 2 latitudes for which the length of the meridian is determined. The integral is not taken in closed form in elementary functions. Only approximate methods of integration are possible for its calculation. When choosing the method of approximate integration, we pay attention to the fact that the value of the eccentricity of the meridian ellipse is a small value, so here it is possible to apply a method based on the expansion in a series in powers of a small value ( e /2 cos 2 B < 7*10 -3) биномиального выражения, стоящего под знаком интеграла. Число членов разложения будет зависеть от необходимой точности вычисления длины дуги меридиана, а также от разности широт ее конечных точек.

In geodetic practice, various cases may arise, more often it is necessary to perform calculations for small lengths (up to 60 km), but for special purposes it may be necessary to calculate arcs of long meridians: from the equator to the current point (up to 10,000 km), between the poles (up to 20,000 km). The required accuracy of calculations can reach a value of 0.001 m. Therefore, we will first consider the general case, when the difference in latitudes can reach 180 0, and the length of the arc is 20,000 km.

To expand a binomial expression into a series, we use a formula known from mathematics.

Hold Calculation Error m it is sufficient here to determine the terms of the expansion using the remainder term in the Lagrange form, which is not less in absolute value than the sum of all the discarded terms of the expansion and is calculated by the formula

, (4. 27)

as the first of the discarded terms of the expansion, calculated at the maximum possible value of the quantity x.

In our case we have

Substituting the resulting expression into equation (4. 25), we obtain

, (4. 28)

which allows term-by-term integration with retention of the required number of expansion terms. Let us assume that the length of the meridian arc can reach a value of 10,000 km (from the equator to the pole), which corresponds to the difference in latitudes DB = p / 2, while it is required to calculate it with an accuracy of 0.001 m, which will correspond to a relative value of 10–10. The value of cosB will not exceed one in any case. If in the calculations we keep the third degrees of expansion, then the remainder term in the Lagrange form has the expression

As you can see, to achieve the required accuracy, such a number of expansion terms is not enough, it is necessary to keep four expansion terms and the residual term in the Lagrange form will have the expression

Therefore, when integrating, it is necessary to keep in this case four degrees of decomposition.

Term-by-term integration (4.28) is easy if you convert even powers to multiple arcs ( cos 2 n B in Cos(2nB)) using the well-known double argument cosine formula

; cos2 B = (1 + cos2B)/2,

successively applying which, we get

Acting in this way until cos 8 B, we obtain after simple transformations and integration

Here, the latitude difference is taken in radian measure and the following designations are used for coefficients that have constant values ​​for an ellipsoid with given parameters.

;

.

It is useful to remember that the length of the meridian arc with a latitude difference of one degree is approximately equal to 111 km, one minute - 1.8 km, one second - 0.031 km.

In geodetic practice, very often there is a need to calculate the meridian arc of small length (on the order of the length of the side of the triangulation triangle), in the conditions of Belarus this value will not exceed 30 km. In this case, there is no need to apply the cumbersome formula (4.29), but you can get a simpler one, but providing the same accuracy of calculations (up to 0.001 m).

Let the latitudes of the end points on the meridian be B1 and B2 respectively. For distances up to 30 km, this will correspond to the difference in latitude in radian measure, not more than 0. 27. Calculating the average latitude B m meridian arcs according to the formula B m = (B 1 + B 2) / 2, we take the arc of the meridian for the arc of a circle with a radius

(4. 30)

and its length is calculated by the formula for the length of the arc of a circle

, (4. 31)

where the difference in latitude is taken in radians.

The length of the arc of parallels and meridians on the Krasovsky ellipsoid,
taking into account distortions from the polar compression of the Earth

To determine the distance on a tourist map, in kilometers between points, the number of degrees is multiplied by the arc length of 1 ° of the parallel and meridian (in longitude and latitude, in the geographic coordinate system), the exact calculated values ​​​​of which are taken from the tables. Approximately, with a certain error, they can be calculated by the formula on the calculator.

An example of converting numerical values ​​of geographical coordinates from tenths to degrees and minutes.

The approximate longitude of the city of Sverdlovsk is 60.8° (sixty point and eight tenths of a degree) east longitude.
8 / 10 = X / 60
X \u003d (8 * 60) / 10 \u003d 48 (from the proportion we find the numerator of the right fraction).
Result: 60.8° = 60° 48" (sixty degrees and forty-eight minutes).

To add a degree symbol (°) - press Alt + 248 (with numbers in the right numeric keypad of the keyboard; in a laptop - with the special Fn button pressed or by turning on NumLk). This is done in Windows and Linux operating systems, and in Mac OS - using the Shift + Option + 8 keys

Latitude coordinates are always indicated before longitude coordinates (whether printed on a computer or written down on paper).

In the maps.google.ru service, supported formats are determined by the rules

Examples of how it would be correct:

The full form of the angle (degrees, minutes, seconds with fractions):
41° 24" 12.1674", 2° 10" 26.508"

Abbreviated forms of writing an angle:
Degrees and minutes with decimals - 41 24.2028, 2 10.4418
Decimal degrees (DDD) - 41.40338, 2.17403

The Google map service has an online converter for converting coordinates and converting them to the desired format.

As a decimal separator of numerical values, on Internet sites and in computer programs, it is recommended to use a dot.

tables

The length of the parallel arc in 1°, 1" and 1" in longitude, meters

A simplified formula for calculating the arcs of parallels (without taking into account distortions from polar compression):

L pairs \u003d l equiv * cos (Latitude).

The length of the meridian arc in 1 °, 1 "and 1" in latitude, meters

Picture. 1-second arcs of meridians and parallels (simplified formula).

A practical example of using tables. For example, if the map does not indicate a numerical scale and there is no scale bar, but there are lines of a degree cartographic grid, you can graphically determine the distances, based on the fact that one degree of the arc corresponds to the numerical value obtained from the table. In the "north-south" directions (between the horizontal lines of the geographical grid on the map) - the values ​​of the lengths of the arcs change, from the equator to the poles of the Earth, insignificantly and amount to approximately 111 kilometers.

Andreev N.V. Topography and Cartography: Optional course. M., Enlightenment, 1985

Mathematics textbook.

Http://ru.wikipedia.org/wiki/Geographic_coordinates