The figure of the earth. earth ellipsoid

The study of the figure of the Earth is one of the most ancient scientific problems of natural science, determined by the needs of practical activities.
human factors: land surveys, the construction of irrigation systems in the Nile Valley, the construction of a canal between the Nile and the Red Sea, etc. (X, IV centuries BC), which could not be carried out without the appropriate topographic and geodetic support.
Assumptions about the sphericity of the earth appeared in the 6th century BC, and from the 4th century BC some of the evidence known to us that the Earth is spherical (Pythagoras, Eratosthenes) were expressed. Ancient scientists proved the sphericity of the Earth based on the following phenomena:
- circular view of the horizon in open spaces, plains, seas, etc.;
- the circular shadow of the Earth on the surface of the Moon during lunar eclipses;
- change in the height of the stars when moving from north (N) to south (S) and back, due to the convexity of the noon line, etc.
In the essay "On the Sky" Aristotle (384 - 322 BC) pointed out that the Earth is not only spherical in shape, but also has finite dimensions; Archimedes (287 - 212 BC) argued that the surface of water in a calm state is a spherical surface. They also introduced the concept of the spheroid of the Earth, as geometric figure close in shape to a sphere.
The modern theory of studying the figure of the Earth originates from Newton (1643 - 1727), who discovered the law of universal gravitation and applied it to study the figure of the Earth.
By the end of the 80s of the 17th century, the laws of planetary motion around the Sun were known, the very precise dimensions of the globe determined by Picard from degree measurements (1670), the fact that the acceleration of gravity on the Earth's surface decreases from north (N) to south (S ), Galileo's laws of mechanics and Huygens' research on the motion of bodies along a curvilinear trajectory. The generalization of these phenomena and facts led scientists to a reasonable view of the spheroidity of the Earth, i.e. its deformation in the direction of the poles (oblateness).
Newton's famous work, "The Mathematical Principles of Natural Philosophy" (1867), sets out a new doctrine of the figure of the Earth. Newton came to the conclusion that the figure of the Earth should be in the form of an ellipsoid of revolution with a slight polar contraction (this fact was substantiated by him by a decrease in the length of the second pendulum with a decrease in latitude and a decrease in gravity from the pole to the equator due to the fact that "the Earth slightly higher at the equator).
Based on the hypothesis that the Earth consists of a homogeneous mass of density, Newton theoretically determined the polar compression of the Earth (α) in the first approximation to be approximately 1:230.
In fact, the Earth is not homogeneous: the crust has a density of 2.6 g/cm3, while the average density of the Earth is 5.52 g/cm3.
The uneven distribution of the Earth's masses produces vast, gently sloping convexities and concavities, which combine to form elevations, depressions, depressions, and other forms. Note that individual elevations above the Earth reach heights of more than 8000 meters above the ocean surface. It is known that the surface of the World Ocean (MO) occupies 71%, land - 29%; the average depth of the MO (World Ocean) is 3800 m, and the average land height is 875 m. The total area earth's surface equals 510 x 106 km2.
It follows from the above data that most of the Earth is covered with water, which gives reason to take it as a level surface (LE) and, ultimately, for the general figure of the Earth. The figure of the Earth can be represented by imagining a surface, at each point of which the force of gravity is directed along the normal to it (along a plumb line).
The complex figure of the Earth, bounded by a level surface, which is the beginning of the height report, is commonly called the geoid. Otherwise, the surface of the geoid, as an equipotential surface, is fixed by the surface of the oceans and seas, which are in a calm state. Beneath the continents, the geoid surface is defined as the surface perpendicular to the lines of force (Figure 3-1).
P.S. The name of the figure of the Earth - the geoid - was proposed by the German physicist I.B. Listig (1808 - 1882).
When mapping the earth's surface, based on many years of research by scientists, the complex figure of the geoid, without compromising accuracy, is replaced by a mathematically simpler one - an ellipsoid of revolution.
An ellipsoid of revolution is a geometric body formed as a result of the rotation of an ellipse around a minor axis.
The ellipsoid of revolution comes close to the body of the geoid (the deviation does not exceed 150 meters in some places). The dimensions of the earth's ellipsoid were determined by many scientists of the world.
Fundamental studies of the figure of the Earth, carried out by Russian scientists F.N. Krasovsky and A.A. Izotov, made it possible to develop the idea of a triaxial terrestrial ellipsoid, taking into account large waves of the geoid; as a result, its main parameters were obtained:
a \u003d 6 379 245 m, c \u003d 6 356 863, α \u003d 1: 298.3 (α \u003d (a - c) / a)
AT last years(the end of the 20th and the beginning of the 21st centuries) the parameters of the figure of the Earth and the external gravitational potential were determined using space objects and using astronomical-geodesic and gravimetric research methods so reliably that now we are talking about estimating their measurements over time.
The triaxial earth ellipsoid, which characterizes the figure of the Earth, is divided into a general earth ellipsoid (planetary), suitable for solving global problems of cartography and geodesy, and a reference ellipsoid, which is used in certain regions, countries of the world and their parts.
P.S. The reference - an ellipsoid - is oriented in a certain way in the body of the Earth and is adopted for topographic, geodetic and cartographic work.
An ellipsoid of revolution is unambiguously characterized by two parameters, namely: the major (equatorial) semi-axis - "a" and the polar compression - "α". For accurate calculations, other parameters are also used, such as the minor (polar) semi-axis - "b" and the first eccentricity of the meridional ellipse - "e". The above parameters are interconnected with each other as follows:
α \u003d (a - c) / a (11)
e2 \u003d (a2 - b2) / a2 (12)
c = a (1 - α) = a√1 - e2 (13)
α = 1 - √1 - e2 (14)
e2 = α (2 - α) (15)

The earth ellipsoid has three main parameters, any two of which uniquely determine its shape:

semi-major axis (equatorial radius) of the ellipsoid, a;
semi-minor axis (polar radius), b;
geometric (polar) compression, $f=\frac(a-b)(a)$ .

There are also other parameters of the ellipsoid:

first eccentricity, $e=\sqrt(\frac(a^2-b^2)(a^2))=\frac(\sqrt(a^2-b^2))(a)$ ;
second eccentricity, $e"=\sqrt(\frac(a^2-b^2)(b^2))=\frac(\sqrt(a^2-b^2))(b)$ .

For the practical implementation of the earth ellipsoid, it is necessary orientate in the body of the earth. In this case, a general condition is put forward: orientation must be carried out in such a way that the differences in astronomical and geodetic coordinates are minimal.

Reference ellipsoid

The figure of the reference ellipsoid is best suited for the territory of a single country or several countries. As a rule, reference ellipsoids are accepted for processing geodetic measurements. by law. In Russia/USSR, the Krasovsky ellipsoid has been used since 1946.

Orientation of the reference ellipsoid in the body of the Earth is subject to the following requirements:

The minor semiaxis of the ellipsoid ( b) must be parallel to the Earth's axis of rotation.
The surface of the ellipsoid should be as close as possible to the surface of the geoid within the given region.

To fix the reference ellipsoid in the body of the Earth, it is necessary to set the geodetic coordinates B0, L0, H0 the starting point of the geodetic network and the initial azimuth A0 to an adjacent point. The totality of these quantities is called original geodetic dates.

Basic reference ellipsoids and their parameters

Scientist	Year	The country	a, m	1/f
Delambre	1800	France	6 375 653	334,0
Delambre	1810	France	6 376 985	308,6465
Walbeck	1819	Finland, Russian Empire	6 376 896	302,8
Airy	1830		6 377 563,4	299.324 964 6
Everest	1830	India, Pakistan, Nepal, Sri Lanka	6 377 276,345	300.801 7
Bessel	1841	Germany, Russia (until 1942)	6 377 397,155	299.152 815 4
Tenner	1844	Russia	6 377 096	302.5
Clark	1866	USA, Canada, Lat. and Center. America	6 378 206,4	294.978 698 2
Clark	1880	France, South Africa	6 377 365	289.0
Listing	1880		6 378 249	293.5
Helmert	1907		6 378 200	298,3
hayford	1910	Europe, Asia, South America, Antarctica	6 378 388	297,0
Heiskanen	1929		6 378 400	298,2
Krasovsky	1936	the USSR	6 378 210	298,6
Krasovsky	1942	THE USSR, Soviet republics, east. Euro, Antarctica	6 378 245	298.3
Everest	1956	India, Nepal	6 377 301,243	300.801 7
IAG-67	1967		6 378 160	298.247 167
WGS-72	1972		6 378 135	298.26
IAU-76	1976		6 378 140	298.257
PZ-90	1990	Russia	6 378 136	298.258

General earth ellipsoid

The general earth ellipsoid must be oriented in the body of the Earth according to the following requirements:

The semi-minor axis must coincide with the axis of rotation of the Earth.
The center of the ellipsoid must coincide with the center of mass of the Earth.
Heights of the geoid above the ellipsoid h i(so-called height anomalies) must obey the least squares condition: $\sum_(n=0)^\infty h_i^2 = \min$ .

When orienting the general earth ellipsoid in the body of the Earth (unlike the reference ellipsoid), there is no need to enter initial geodetic dates.

Since the requirements for general-earth ellipsoids are satisfied in practice with some tolerances, and the fulfillment of the latter (3) in full is impossible, then in geodesy and related sciences, various implementations of the ellipsoid can be used, the parameters of which are very close, but do not coincide (see below).

Modern general earth ellipsoids and their parameters

Name	Year	Country/Organization	a, m	accuracy m a , m	1/f	accuracy m f	Note
GRS80	1980	MAGG (IUGG)	6 378 137	±2	298,257 222 101	±0.001	(English) Geodetic Reference System 1980) was developed by the International Geodetic and Geophysical Union (eng. International Union of Geodesy and Geophysics ) and is recommended for geodetic works
WGS84	1984	USA	6 378 137	±2	298,257 223 563	±0.001	(English) World Geodetic System 1984) used in the GPS satellite navigation system
PZ-90	1990	the USSR	6 378 136	± 1	298,257 839 303	±0.001	(Parameters of the Earth 1990) is used on the territory of Russia for geodetic support of orbital flights. This ellipsoid is used in the GLONASS satellite navigation system
IERS (IERS)	1996	IERS	6 378 136,49	-	298,256 45	-	(English) International Earth Rotation Service 1996 ) recommended by the International Earth Rotation Service for processing VLBI observations

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An excerpt characterizing the Earth's ellipsoid

“Well, she shuddered in exactly the same way, came up in the same way and smiled timidly when it was already,” Natasha thought, “and in exactly the same way ... I thought that something was missing in her.”
- No, this is the choir from the Water Carrier, do you hear! - And Natasha finished singing the motive of the choir in order to make Sonya understand it.
– Where did you go? Natasha asked.
- Change the water in the glass. I'm painting the pattern now.
“You are always busy, but I don’t know how,” said Natasha. - Where is Nikolai?
Sleeping, it seems.
“Sonya, you go wake him up,” said Natasha. - Say that I call him to sing. - She sat, thought about what it meant, that it all happened, and, without resolving this issue and not at all regretting it, she was again transported in her imagination to the time when she was with him, and he, with loving eyes looked at her.
“Oh, I wish he would come soon. I'm so afraid it won't! And most importantly: I'm getting old, that's what! There will be no more what is now in me. Or maybe he will come today, he will come now. Maybe he came and sits there in the living room. Maybe he arrived yesterday and I forgot. She got up, put down her guitar and went into the living room. All the household, teachers, governesses and guests were already sitting at the tea table. People stood around the table - but Prince Andrei was not there, and there was still the old life.
“Ah, here she is,” said Ilya Andreevich, seeing Natasha come in. - Well, sit down with me. But Natasha stopped beside her mother, looking around, as if she was looking for something.
- Mum! she said. “Give it to me, give it to me, mother, hurry, hurry,” and again she could hardly restrain her sobs.
She sat down at the table and listened to the conversations of the elders and Nikolai, who also came to the table. “My God, my God, the same faces, the same conversations, the same dad holds a cup and blows the same way!” thought Natasha, feeling with horror the disgust that rose in her against all the household because they were still the same.
After tea, Nikolai, Sonya and Natasha went to the sofa room, to their favorite corner, in which their most intimate conversations always began.

“It happens to you,” Natasha said to her brother when they sat down in the sofa room, “it happens to you that it seems to you that nothing will happen - nothing; that all that was good was? And not just boring, but sad?
- And how! - he said. - It happened to me that everything was fine, everyone was cheerful, but it would occur to me that all this was already tired and that everyone needed to die. Once I didn’t go to the regiment for a walk, and there was music playing ... and I suddenly became bored ...
“Ah, I know that. I know, I know, - Natasha picked up. “I was still little, so it happened to me. Do you remember, since they punished me for plums and you all danced, and I sat in the classroom and sobbed, I will never forget: I was sad and felt sorry for everyone, and myself, and I felt sorry for everyone. And, most importantly, I was not to blame, - said Natasha, - do you remember?
“I remember,” Nikolai said. - I remember that I came to you later and I wanted to console you and, you know, I was ashamed. We were awfully funny. I had a bobblehead toy then and I wanted to give it to you. Do you remember?
“Do you remember,” Natasha said with a thoughtful smile, how long, long ago, we were still very young, our uncle called us into the office, back in the old house, and it was dark - we came and suddenly it was standing there ...
“Arap,” Nikolai finished with a joyful smile, “how can you not remember? Even now I don’t know that it was a black man, or we saw it in a dream, or we were told.
- He was gray, remember, and white teeth - he stands and looks at us ...
Do you remember Sonya? Nicholas asked...
“Yes, yes, I also remember something,” Sonya answered timidly ...
“I asked my father and mother about this arap,” said Natasha. “They say there was no arap. But you do remember!
- How, as now I remember his teeth.
How strange, it was like a dream. I like it.
- Do you remember how we rolled eggs in the hall and suddenly two old women began to spin on the carpet. Was it or not? Do you remember how good it was?
- Yes. Do you remember how daddy in a blue coat on the porch fired a gun. - They sorted through, smiling with pleasure, memories, not sad senile, but poetic youthful memories, those impressions from the most distant past, where the dream merges with reality, and laughed quietly, rejoicing at something.
Sonya, as always, lagged behind them, although their memories were common.
Sonya did not remember much of what they remembered, and what she remembered did not arouse in her that poetic feeling that they experienced. She only enjoyed their joy, trying to imitate it.
She took part only when they recalled Sonya's first visit. Sonya told how she was afraid of Nikolai, because he had cords on his jacket, and her nanny told her that they would sew her into cords too.
“But I remember: they told me that you were born under cabbage,” said Natasha, “and I remember that then I did not dare not to believe, but I knew that this was not true, and I was so embarrassed.
During this conversation, the maid's head poked out of the back door of the divan. - Young lady, they brought a rooster, - the girl said in a whisper.
“Don’t, Polya, tell them to take it,” said Natasha.
In the middle of conversations going on in the sofa room, Dimmler entered the room and approached the harp in the corner. He took off the cloth, and the harp made a false sound.

Knowledge of the shape and dimensions of the Earth is necessary in many areas of science and technology, and above all for the correct representation of the earth's surface in the form of plans and maps.

The physical surface of the Earth consists of a land surface of 24.4% and of the water surface, considered in a calm state, 70.6%.

The earth is not a regular geometric body. Its surface, and in particular the surface of the land, is very complex and cannot be expressed by any mathematical formula.

An idea of the figure of the Earth as a whole can be obtained by imagining that the entire planet is limited by the mentally extended surface of the oceans in a calm state. Such a closed surface at each of its points is perpendicular to the plumb line, i.e., to the direction of gravity. They call her level surface.

A level surface is a convex surface perpendicular to the direction of gravity (plumb line).

Level surfaces enveloping the Earth, you can imagine a lot. The one that coincides with the average water level of the World Ocean, mentally continued under land, is called geoid surface, and the body limited by it - geoid.

The mathematical surface of the Earth is considered to be a level surface, at each point of which the direction of the plumb line (gravity) and the normal coincide.

Due to the uneven distribution of masses inside the Earth, the geoid does not have a regular geometric shape and its surface cannot be expressed mathematically, therefore, for practical calculations, it is replaced by simpler geometric models. Of these, it is closest to the geoid spheroid or ellipsoid of revolution obtained by rotating the ellipse around its minor (polar) axis.

The dimensions of an ellipsoid are characterized by the lengths of its major semiaxis a and minor semiaxis b, as well as compression, determined by the formula:

Over the past two centuries, scientists have repeatedly determined the size of the earth's ellipsoid. The mathematical model of the Earth, the most successful, was proposed in 1946. prof. Krasovsky as reference ellipsoid.

Major axis a= 6 378 245 m;

Semi-minor axis b=6 356 863 m.

Compression = 1:298,3=0,0033523299.

Krasovsky's ellipsoid is a figure obtained by rotating an ellipse around its minor axis. The earth is flattened at the poles under the action of the centrifugal force that occurs when the earth rotates around its axis.

In practical calculations, the Earth is taken as a sphere with an average radius R=6371.11 km. A small area of the Earth's surface can practically be considered a horizontal plane, a larger area - as part of a sphere.

In Russia, the Baltic system of heights, measured from the level of the Baltic Sea (Kronstadt footstock), is taken as a level surface.

§ 1. The figure and dimensions of the Earth

Numerous studies and measurements have made it possible to establish that the Earth has the shape of a mathematically irregular body called the geoid. The surface forming the geoid, in contrast to the physical surface of the Earth with its irregularities (mountains, depressions, etc.), is horizontal at all its points, that is, it coincides with the normal to the direction of gravity and is defined as a level surface. In nature, such a level surface coincides with the average water level of the oceans and open seas in a calm state (in the absence of waves, currents, tides and other disturbing factors), mentally continued under all continents. The irregularity of the geoid is due to the uneven distribution of masses in the thickness of the Earth, from the attracting action of which the direction of gravity depends.
Theoretical studies and the results of processing astronomical-geodesic and gravimetric measurements, as well as the results of observations of artificial satellites of the Earth, show that the geoid is close to a mathematically correct figure - an ellipsoid of revolution formed by the rotation of an ellipse around its minor axis. Therefore, in the production of geodetic, cartographic and other works that require high accuracy, an ellipsoid of revolution is taken as the figure of the Earth.
The height deviation of the geoid surface from the surface of the earth's ellipsoid, adopted in the USSR and properly sized and oriented in the body of the Earth, does not exceed 100-150 m. An ellipsoid of revolution is practically identified with a spheroid, representing the figure of equilibrium of a rotating homogeneous liquid mass. The height deviation of the surfaces of the ellipsoid of revolution and the spheroid does not exceed 2–3 m.

Determining the dimensions of the earth's ellipsoid, which has the closest proximity to the figure of the Earth as a whole, continues to be one of the main tasks of higher geodesy. Therefore, in different countries, the processing of the results of geodetic and topographic work is referred to as an auxiliary mathematical surface, representing the earth's ellipsoid with the dimensions adopted for a given country. An ellipsoid with a certain size, to the surface of which all the results of geodetic and topographic works in the state are referred, is called a reference ellipsoid.
The main elements that determine the dimensions of the earth's ellipsoid are its semiaxes: major a and minor b. In addition, to characterize the earth's ellipsoid, as well as for some calculations, the following concepts are used: polar compression α of the earth's ellipsoid, expressed by the formula
α \u003d a - b / a, (1 formula)
and its eccentricity (e), determined by the expression
e \u003d √ a 2 - b 2 / a (2 formula)
Since 1946, for all geodetic and cartographic works on the territory of the USSR, the reference ellipsoid of F. N. Krasovsky has been adopted with dimensions:
- semi-major axis a = 6 378 245 m;
- semi-minor axis b = 6 356 863 m;
- polar compression α = 1:298.3;
- square of eccentricity e 2 =1:149.15.

When deriving the dimensions of the reference ellipsoid, a group of scientists, geodesists, topographers and calculators under the guidance of Professor F.N. countries. The dimensions of Krasovsky's reference ellipsoid are also confirmed by the results of processing observations of artificial Earth satellites made in recent years.
Orientation in the body of the Earth of the earth's ellipsoid with the corresponding dimensions of the semi-axes and compression is characterized by the so-called initial geodetic dates. Initial geodetic dates are the coordinates of the starting point of triangulation, which determine its Latitude B 0 , longitude L 0 , azimuth A 0 to any adjacent point and height h 0 of the geoid surface relative to the surface of the reference ellipsoid.
These dates are taken as starting dates when calculating the coordinates of all other points on the earth's surface.
When using foreign maps, it should be remembered that different countries have adopted different initial geodetic dates. Therefore, the same points on maps published in different countries may have different coordinates. Although this difference may be small, it must be taken into account in navigation and the transfer of the ship’s place from one map to another when sailing near the coast should be carried out not according to geographical coordinates, but according to the direction and distance to the nearest stronghold placed on both maps.
The adoption of the Earth as an ellipsoid of revolution is, in essence, the second approximation in determining the figure of the Earth. When solving some problems of practical navigation that do not require high accuracy, it turns out to be possible to limit ourselves to the first approximation in determining the shape of the Earth - to take the Earth for a ball. Such tasks include calculations of the visibility range of landmarks in the sea, calculations for navigation along the shortest distance, analytical calculations when determining a position using radio bearings, calculations using analytical calculus formulas, and some others.
To determine the radius of the Earth - the ball usually proceed from some additional conditions.
One of them is the condition that the length of one minute of the meridian arc (or any great circle on the ball) be equal to 1852 m, i.e., the length of a standard nautical mile. In this case, the radius of the ball that meets the stated condition will be equal to
R \u003d 1852 * 60 * 360 / 2 π \u003d 6 366 707 m.
When solving a number of cartography problems, the condition is set that the volume of the globe is equal to the volume of the earth's ellipsoid or that the surface of the ball is equal to the surface of the ellipsoid. The length of the radius R of the ball, the same volume as the earth's ellipsoid, is equal to
R = cube root √ (a 2 * b) = 6371109.7 m.
If the condition is set that the surface of the ball is equal to the surface of the ellipsoid, then the radius of such a ball is taken equal to

where M is the radius of curvature of the meridian; N is the radius of curvature of the first vertical at a given point.

§ 2. Geographic coordinate system

The position of a point on any surface or in space is determined by a set of specific quantities called coordinates. Coordinates can be expressed both in linear and in angular measure; they determine the position of the coordinate lines relative to the coordinates taken as the origin of the axes. To determine the position of points on the earth's surface, various coordinate systems can be used: geographical, rectangular, polar, etc. The most common is the system of geographical coordinates.
The minor axis of the ellipsoid intersects the surface of the latter at two points, which are called the north and south poles. The planes passing through the Earth's axis of rotation are called the planes of the Earth's meridians, which, in cross section with the Earth's surface, form great circles called meridians. Plane perpendicular to earth's axis and passing through the center of the ellipsoid is called the plane of the equator. The great circle formed from the intersection of this plane with the surface of the ellipsoid is called the earth's equator. Planes parallel to the plane of the earth's equator in cross section with the surface of the earth form small circles called earth's parallels.

The coordinate axes of the geographic coordinate system are: the equator and one of the meridians, taken as the initial one; the coordinate lines are the earth's parallels and meridians, and the quantities that determine the position of the points, i.e., the coordinates, geographic latitude and geographic longitude.
The geographic latitude of a point on the Earth's surface is the angle between the normal to the surface of the ellipsoid at that point and the plane of the equator. Geographic latitude in navigation is indicated by the Greek letter φ (phi). Latitudes are counted from the equator to the poles from 0 to 90°. Latitudes northern hemisphere are considered positive and in analytical calculations they are taken with a plus sign. Northern latitudes are denoted by the letter N. Point latitudes southern hemisphere, denoted by the letter S, are considered negative and are assigned a minus sign.
Geographic latitude determines the position of the parallel on which the point being determined is located.
The geographical longitude of a point is the dihedral angle formed by the plane of the initial meridian and the plane of the meridian passing through this point. The dihedral angle is measured by the spherical angle at the pole between the initial meridian and the meridian of the determined point or the arc of the equator, numerically equal to it, enclosed between the named meridians.
In principle, any terrestrial meridian can be taken as the initial meridian. According to the international agreement of 1884, most countries of the world, including Soviet Union, the meridian passing through the Greenwich Observatory, located near London, is taken as the initial one.
Geographic longitudes are counted east and west of the Greenwich meridian from 0 to 180°. Geographic longitude in navigation is denoted by the Greek letter λ (lambda). Longitudes of points located in the Eastern Hemisphere are considered to be positive (plus sign), western longitudes are considered negative (minus sign). When determining the longitude of a particular point on the earth's surface, it is necessary to indicate its name: eastern - Ost or, as is now accepted, E, western - W. Depending on the method of calculating geographic coordinates, geodetic and astronomical coordinates are distinguished.
In the geometric definition of geodetic coordinates, which are obtained as a result of geodetic measurements (triangulation, polygonometry), there is no difference from the general formulation of geographical coordinates. The locations of points fixed by geodetic latitude and geodetic longitude also refer to a mathematically correct ellipsoid figure of revolution.
When determining a place by astronomical means, the observer is dealing with a plumb line that coincides with the direction of gravity, and not with the normal to the surface of the ellipsoid. Therefore, in the astronomical coordinate system, latitude is defined as the angle between the plane of the equator and the direction of the plumb line at a given point. The longitude of a place determined by an astronomical method is the dihedral angle between the plane of the prime meridian (Greenwich meridian) and the plane of the astronomical meridian of the given point. The applied term - astronomical meridian - must be understood as a trace from the section of the earth's surface by a plane passing through plumb line at a given point and parallel to the axis of the world. From the definition of astronomical coordinates, it can be seen that, unlike geodetic coordinates, they fix the position of points relative to the surface of the real figure of the Earth-geoid.

The normal to the surface of the earth's ellipsoid generally does not pass through the center of the earth. At the same time, when solving astronomical problems, as well as a number of special problems of mathematical cartography, it becomes necessary to determine the position of points on the earth's surface relative to the center of the earth. In this case, the longitude of an arbitrary point K will be determined in the same way as in the geographic coordinate system, and the latitude will be obtained as the angle between the equatorial plane and the straight line connecting this point with the center of the ellipsoid. Such a latitude is called geocentric latitude and is denoted by φ". The figure shows that the geocentric latitude is generally less than the geographic latitude by the reduction r of the latitude, which can be calculated by the formula
r "" \u003d φ - φ" \u003d α sin 2 φ / arc 1 "" (3rd formula)
For points located on the equator and at the pole, the latitude reduction is zero. Greatest value(11.5") reduction reaches 45° in latitude.
In cases where the shape of the Earth is taken as a sphere, the position of points on the Earth-ball is determined in the same way as on the surface of an ellipsoid, by their geographical coordinates, i.e., latitude and longitude. But the normal on the Earth-ball coincides with its radius.
Therefore, the geographical latitude φ of some point M on the globe will be the angle at the center of the sphere between the equatorial plane and the radius passing through the point being determined. From a comparison of the definitions of latitude, it can be seen that geocentric latitude is only a special case of spherical latitude.

Chapter 1

§ 3. Latitude difference and longitude difference

Geographic coordinates - latitude and longitude - uniquely determine the position of a particular point on the earth's surface. The transition from one point on the earth's surface to another is accompanied by a change in their geographical coordinates. Points lying on the same parallel have the same latitude and different longitudes. Points located on the same meridian have the same longitude and different latitudes. In general, two points that are not on the same meridian or on the same parallel have different latitudes and different longitudes. In the practice of navigation, it is often necessary to know how geographical coordinates have changed or will change when moving from one point on the earth's surface to another, and to be able to calculate these changes. The values characterizing the change in geographical coordinates during the transition from one point on the earth's surface to another are the difference in latitudes and the difference in longitudes.
The difference in latitude (RS) of two points on the surface of the Earth is the meridian arc enclosed between the parallels of these points.
To calculate the difference in latitude, use the formula
RSh \u003d φ 2 - φ 1,
taking into account the signs + and -, respectively, their name. Indeed, the figure shows that the change in latitude (RL) during the transition of the ship from point A to point B is characterized by the arc A "B, numerically equal to the difference between the arcs of the meridians of the points of arrival B and departure A, determined respectively by the latitudes φ B and φ A.
The latitude difference calculated by formula (4) is assigned a plus sign if it is made to N, and a minus sign if the latitude difference is made to S. The latitude difference can vary from 0 to ±180°.
The longitude difference (RD), which characterizes the change in longitude, as can be seen from the figure, is the central angle between the meridians of two points. This angle is measured by the arc of the equator between the indicated meridians. On this basis, the difference between the longitudes of two points on the surface of the Earth is the smallest of the arcs of the equator enclosed between the meridians of these points. It follows from this definition that the difference in longitudes can have values from 0 to ±180°. Taking into account the previously accepted notation (plus sign for east longitude and minus sign for west longitude), we can write a formula for calculating the RD of two points:
RD \u003d λ 2 - λ 1
The longitude difference will have a plus sign if it is made to Ost, and a minus sign if it is made to W. This rule has the following geometric meaning: if the meridian of the point of arrival λ 2 is located east of the meridian of the point of departure λ 1, then the difference in longitudes is made to Оst and a plus sign is assigned to it. Conversely, when the meridian of the point of arrival is located to the west of the meridian of the point of departure, the difference in longitudes is made to W and a minus sign is assigned to it.

When solving the problem of calculating the RD using the formula, a result exceeding 180° can be obtained. In these cases, to find the smaller of the equatorial arcs, the result obtained should be subtracted from 360 ° and its sign (name) reversed.

Planet Earth does not have a regular geometric shape. The figure of the Earth is called the geoid. It is generally accepted that the shape of the Earth is close to an ellipsoid, resulting from the rotation of an ellipse around a minor axis (Fig. 1).

The length of the major semi-axis of the Earth's ellipsoid is a = 6 378 245 m, the minor one is b = 6 356 863 m. The difference between the semi-axes is 21.4 km. Attitude

called the contraction of the earth. Such dimensions of the earth's ellipse were established by prof. N. F. Krasovsky. By Decree of the Council of Ministers of the USSR No. 760 of April 7, 1946, the dimensions of N. F. Krasovsky's ellipsoid were adopted for all geodesic, topographic and cartographic works of the USSR.

When solving most problems in navigation, the magnitude of the Earth's compression, which is 0.3%, is neglected and the Earth is taken as a ball, the volume of which is equal to the volume of the Earth's ellipsoid. Based on this convention, i.e., that

and substituting the values a and 6 into this formula, we determine the radius of such a ball R = 6 371 110 m.

Basic dots, lines and circles

The imaginary points PN and PS of the intersection of the Earth's axis of rotation with its surface are called the poles of the earth : northern(nordic) and southern(south), while the north pole is considered, from the side of which the rotation of the Earth is directed counterclockwise.

The circumference of the great circle EABQ (Fig. 2), which is a trace of the intersection of the surface of the globe with a plane perpendicular to the axis of rotation PNPS and passing through its center 0, is called equator. The plane of the equator divides Earth into two hemispheres: northern and southern.

Circles of small circles, for example eabq, e1a1b1q1, which are the trace of the intersection of the surface of the globe with planes parallel to the plane of the equator, are called parallels.

The circles of great circles, for example PN aAa1PS and PNbBb1PS, which are traces of the intersection of the surface of the globe with planes passing through the axis of rotation of the Earth (meridial planes), are called meridians.

An unlimited number of parallels and meridians can be drawn, but only one parallel and one meridian can be drawn through one point, which are called, respectively, the parallel of a given point or place and the meridian of a given point or place.

Rice. 2

By international agreement, it is considered zero or prime meridian meridian passing through the astronomical observatory in Greenwich (near London). He and his opposite divide the globe into two hemispheres: eastern and western.