Every modern person must know what a coordinate system is. Every day we come across such systems without even thinking about what they are. Once upon a time at school we learned basic concepts, we roughly know that there is an X-axis, an Y-axis and a reference point equal to zero. In fact, everything is much more complicated; there are several types of coordinate systems. In the article we will look at each of them in detail, and also give a detailed description of where and why they are used.

Definition and scope

A coordinate system is a set of definitions that specifies the position of a body or point using numbers or other symbols. The set of numbers that determine the location of a specific point is called the coordinates of that point. Coordinate systems are used in many fields of science, for example, in mathematics, coordinates are a set of numbers that are associated with points in some map of a predetermined atlas. In geometry, coordinates are quantities that determine the location of a point in space and on a plane. In geography, coordinates indicate latitude, longitude, and altitude above the general level of the sea, ocean, or other predetermined value. In astronomy, coordinates are quantities that make it possible to determine the position of a star, such as declination and right ascension. This is not a complete list of where coordinate systems are used. If you think that these concepts are far from people who are not interested in science, then believe that in everyday life they are found much more often than you think. Take at least a map of the city, why not a coordinate system?

Having dealt with the definition, let's look at what types of coordinate systems exist and what they are.

Zonal coordinate system

This coordinate system is used mainly for various horizontal surveys and drawing up reliable terrain plans. It is based on the equiangular transverse cylindrical Gaussian projection. In this projection, the entire surface of the earth's geoid is divided by meridians into 6-degree zones and numbered from 1st to 60th east of the Greenwich meridian. In this case, the middle meridian of this hexagonal zone is called the axial meridian. It is customary to combine it with the inner surface of the cylinder and consider it the abscissa axis. In order to avoid negative ordinate values (y), the ordinate on the axial meridian (the initial reference point) is taken not as zero, but as 500 km, that is, it is moved 500 km to the west. The zone number must be indicated before the ordinate.

Gauss-Kruger coordinate system

This coordinate system is based on the projection proposed by the famous German scientist Gauss and developed for use in geodesy by Kruger. The essence of this projection is that the earthly sphere is conventionally divided by meridians into six-degree zones. Zones are numbered from the Greenwich meridian from west to east. Knowing the zone number, you can easily determine the middle meridian, called the axial, using the formula Z = 60(n) – 3, where (n) is the zone number. For each zone, a flat image is made by projecting it onto the side surface of a cylinder, the axis of which is perpendicular to the earth's axis. Then this cylinder is gradually unfolded onto the plane. The equator and the axial meridian are depicted by straight lines. The abscissa axis in each zone is the axial meridian, and the equator serves as the ordinate axis. The starting point is the intersection of the equator and the axial meridian. Abscissas are counted north of the equator only with a plus sign and south of the equator only with a minus sign.

Polar coordinate system on a plane

This is a two-dimensional coordinate system, each point in which is defined on the plane by two numbers - the polar radius and the polar angle. The polar coordinate system is useful in cases where the relationship between points is easier to represent in the form of angles and radii. The polar coordinate system is defined by a ray called the polar or zero axis. The point from which a given ray emerges is called the pole or origin. An arbitrary point on a plane is determined by only two polar coordinates: angular and radial. The radial coordinate is equal to the distance from the point to the origin of the coordinate system. The angular coordinate is equal to the angle by which the polar axis must be rotated counterclockwise to get to the point.

Rectangular coordinate system

You probably know what a rectangular coordinate system is from school, but still, let’s remember one more time. A rectangular coordinate system is a rectilinear system in which the axes are located in space or on a plane and are mutually perpendicular to each other. This is the simplest and most commonly used coordinate system. It is directly and quite easily generalized to spaces of any dimension, which also contributes to its widest application. The position of a point on a plane is determined by two coordinates - x and y, respectively, there is an abscissa and ordinate axis.

Cartesian coordinate system

Explaining what a Cartesian coordinate system is, first of all it must be said that this is a special case of a rectangular coordinate system, in which the axes have the same scales. In mathematics, one most often considers a two-dimensional or three-dimensional Cartesian coordinate system. The coordinates are denoted by the Latin letters x, y, z and are called abscissa, ordinate and applicate, respectively. The coordinate axis (OX) is usually called the abscissa axis, the (OY) axis is the ordinate axis, and the (OZ) axis is the applicate axis.

Now you know what a coordinate system is, what they are and where they are used.

Coordinates

Coordinates pl.
1.

Data about the location of someone or something, determined on the basis of such quantities.

2. trans. decomposition

Information about the location or whereabouts of someone.

Explanatory Dictionary by Efremova. T. F. Efremova. 2000.

Synonyms:

See what “Coordinates” are in other dictionaries:

Coordinates of a quantity that determines the position of a point (body) in space (on a plane, on a straight line). The set of coordinates of all points in space is a coordinate system. Wiktionary has an article “coordinate” Concept and word... ... Wikipedia

- (from the Latin co prefix meaning compatibility, and ordinatus ordered, defined * a. coordinates; n. Koordinaten; f. coordonnees; i. coordenadas) numbers, quantities that determine the position of a point in space. In geodesy, topography... Geological encyclopedia

- (from the Latin co together and ordinatus ordered specific), numbers, the assignment of which determines the position of a point on a plane, on a surface or in space. Rectangular (Cartesian) coordinates of a point on a plane are equipped with signs + ...

- (from the Latin co together and ordinatus ordered), numbers that determine the position of a point on a straight line, plane, surface, in space. Coordinates are the distances to coordinate lines chosen in some way. For example,… … Modern encyclopedia

Spherical. If the origin of polar coordinates is taken at the center of the sphere, then all points of the spheres have the same radius vector and only the angles q and l remain changeable. Usually, instead of q, another coordinate j = 90 q is taken, which is called latitude, while the angle ...

- (cf. century lat., from lat. cum s, and ordinare to put in order). In analyt. geometry: quantities that serve to determine the position of a point. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910.… … Dictionary of foreign words of the Russian language

Position, location, position, location, location, location Dictionary of Russian synonyms. coordinates see location 1 Dictionary of synonyms of the Russian language. Practical guide. M.: Russ... Synonym dictionary

coordinates- COORDINATES, coordinates, plural. Address, telephone. He got married, changed his coordinates... Dictionary of Russian argot

In geodesy, quantities that determine the position of a point on the earth’s surface relative to the surface of the earth’s ellipsoid: latitude, longitude, height. Determined by geodetic methods... Big Encyclopedic Dictionary

- (from Latin co - together and ordinatus - ordered) basic. moments that define the given. In mathematics, quantities that determine the position of a point; They are often visually depicted using segments. If straight lines departing from a point (origin of coordinates) ... Philosophical Encyclopedia

Quantities that determine the position of a point. In Cartesian rectangular frames, the position of a point is determined by its three distances from three mutually perpendicular planes; the intersections of these planes are three straight lines emanating from one point... Encyclopedia of Brockhaus and Efron

Books

Coordinates of populated areas, time zones and changes in time calculation, Editor V. Fedorov. Compiled by I. Bariev, p. 71 Directory Coordinates of settlements, time zones and changes in time calculation. Format: 145 x 200 mm ISBN:5-87160-026-3… Category: Scientific and technical literature Publisher: Starklight, Manufacturer: Starklight,
Coordinates of Wonders, Robert Sheckley, American science fiction writer Robert Sheckley is popular all over the world. He graduated from a technical college, but since 1952 he decided to devote himself entirely to literature. I took a literature course from... Category: Science Fiction Series: Science Fiction Publisher: North-West, Manufacturer:

1.10. RECTANGULAR COORDINATES ON MAPS

Rectangular coordinates (flat) - linear quantities: abscissa X and ordinateYdefining the position of points on a plane (on a map) relative to two mutually perpendicular axes X AndY(Fig. 14). Abscissa X and ordinateYpoints A- distances from the origin to the bases of the perpendiculars dropped from the point A on the corresponding axes, indicating the sign.

Rice. 14.Rectangular coordinates

In topography and geodesy, as well as on topographic maps, orientation is carried out in the north with angles counted clockwise, therefore, to preserve the signs of trigonometric functions, the position of the coordinate axes, accepted in mathematics, is rotated by 90°.

Rectangular coordinates on topographic maps of the USSR are applied by coordinate zones. Coordinate zones are parts of the earth's surface bounded by meridians with longitude divisible by 6°. The first zone is limited by meridians 0° and 6°, the second by b" and 12°, the third by 12° and 18°, etc.

The zones are counted from the Greenwich meridian from west to east. The territory of the USSR is located in 29 zones: from the 4th to the 32nd inclusive. The length of each zone from north to south is about 20,000 km. The width of the zone at the equator is about 670 km, at latitude 40° - 510 km, t latitude 50°-430 km, at latitude 60°-340 km.

All topographic maps within a given zone have a common rectangular coordinate system. The origin of coordinates in each zone is the point of intersection of the average (axial) meridian of the zone with the equator (Fig. 15), the average meridian of the zone corresponds to

Rice. 15.System of rectangular coordinates on topographic maps: a-one zone; b-parts of the zone

the abscissa axes, and the equator the ordinate axes. With this arrangement of coordinate axes, the abscissa of points located south of the equator and the ordinate of points located west of the middle meridian will have negative values. For the convenience of using coordinates on topographic maps, a conditional count of ordinates has been adopted, excluding negative ordinate values. This is achieved by the fact that the ordinates are counted not from zero, but from the value 500 km, That is, the origin of coordinates in each zone is, as it were, moved to 500 km left along the axisY.In addition, to unambiguously determine the position of a point using rectangular coordinates on the globe to the coordinate valueYThe zone number (single or double digit number) is assigned to the left.

The relationship between conditional coordinates and their real values is expressed by the formulas:

X " = X-, Y = U-500,000,

Where X" And Y"-real ordinate values;X,Y-conditional values of ordinates. For example, if a point has coordinates

X = 5 650 450: Y= 3 620 840,

then this means that the point is located in the third zone at a distance of 120 km 840 m from the middle meridian of the zone (620840-500000) and north of the equator at a distance of 5650 km 450 m.

Full coordinates - rectangular coordinates, written (named) in full, without any abbreviations. In the example above, the full coordinates of the object are given:

X = 5 650 450; Y= 3620 840.

Abbreviated coordinates are used to speed up target designation on a topographic map; in this case, only tens and units of kilometers and meters are indicated. For example, the abbreviated coordinates of this object would be:

X = 50 450; Y = 20 840.

Abbreviated coordinates cannot be used for target designation at the junction of coordinate zones and if the area of operation covers a space of more than 100 km by latitude or longitude.

Coordinate (kilometer) grid - a grid of squares on topographic maps, formed by horizontal and vertical lines drawn parallel to the axes of rectangular coordinates at certain intervals (Table 5). These lines are called kilometer lines. The coordinate grid is intended for determining the coordinates of objects and plotting objects on a map according to their coordinates, for target designation, map orientation, measuring directional angles and for approximate determination of distances and areas.

Table 5 Coordinate grids on maps

Map scales	Dimensions of the sides of the squares		Areas of squares, sq. km
Map scales	on the map, cm	on the ground, km	Areas of squares, sq. km
1:25 000		1
1:50 000
1:100 000
1:200 000

On a map at a scale of 1:500,000, the coordinate grid is not completely shown; only the outputs of kilometer lines are plotted on the sides of the frame (after 2 cm). If necessary, a coordinate grid can be drawn on the map along these outputs.

Kilometer lines on maps are marked at their boundary exits and at several intersections inside the sheet (Fig. 16). The outermost kilometer lines on the map sheet are signed in full, the rest are abbreviated with two numbers (i.e., only tens and units of kilometers are indicated). The labels on the horizontal lines correspond to the distances from the ordinate axis (equator) in kilometers. For example, the signature 6082 in the upper right corner shows that this line is located at a distance of 6082 from the equator km.

The labels of the vertical lines indicate the zone number (one or two first digits) and the distance in kilometers (always three digits) from the origin of coordinates, conventionally moved west of the middle meridian by 500 km. For example, the signature 4308 in the lower left corner means: 4 - zone number, 308 - distance from the conditional origin in kilometers.

An additional coordinate (kilometer) grid can be plotted on topographic maps at scales of 1:25,000, 1:50,000, 1:100,000 and 1:200,000 along the exits of kilometer lines in the adjacent western or eastern zone. Outputs of kilometer lines in the form of dashes with corresponding signatures are given on maps located 2° east and west of the boundary meridians of the zone.

rice. 16.Coordinate (kilometer) grid on a map sheet

An additional coordinate grid is intended to transform the coordinates of one zone into the coordinate system of another, neighboring zone.

In Fig. 17 lines on the outside of the western frame with signatures 81,6082 and on the northern side of the frame with signatures 3693, 94, 95, etc. indicate the outputs of kilometer lines in the coordinate system of the adjacent (third) zone. If necessary, an additional coordinate grid is drawn on a sheet of map by connecting lines of the same name on opposite sides of the frame. The newly constructed grid is a continuation of the kilometer grid of the map sheet of the adjacent zone and must completely coincide (close) with it when gluing the map.

Western (3rd) zone coordinate grid

Rice. 17. Additional grid

For determining The positions of points in geodesy use spatial rectangular, geodetic and plane rectangular coordinates.

Spatial rectangular coordinates. The origin of the coordinate system is located at the center O earth's ellipsoid(Fig. 2.2).

Axis Z directed along the axis of rotation of the ellipsoid to the north. Axis X lies at the intersection of the equatorial plane with the initial Greenwich meridian. Axis Y directed perpendicular to the axes Z And X to the East.

Geodetic coordinates. The geodetic coordinates of a point are its latitude, longitude and height (Fig. 2.2).

Geodetic latitude points M called an angle IN, formed by the normal to the surface of the ellipsoid passing through a given point and the equatorial plane.

Latitude is measured from the equator north and south from 0° to 90° and is called north or south. Northern latitude is considered positive, and southern latitude negative.

Sectional planes of an ellipsoid passing through the axis OZ, are called geodetic meridians.

Geodetic longitude points M called dihedral angle L, formed by the planes of the initial (Greenwich) geodesic meridian and the geodesic meridian of a given point.

Longitudes are measured from the prime meridian in the range from 0° to 360° east, or from 0° to 180° east (positive) and from 0° to 180° west (negative).

Geodetic height points M is its height N above the surface of the earth's ellipsoid.

Geodetic coordinates and spatial rectangular coordinates are related by the formulas

X =(N+H)cos B cos L,

Y=(N+H)cos B sin L,

Z=[(1- e 2)N+H] sin B,

Where e- first eccentricity of the meridian ellipse and N-radius of curvature of the first vertical. In this case N=a/(1 - e 2 sin 2 B) 1/2 .

Geodetic and spatial rectangular coordinates of points are determined using satellite measurements, as well as by linking them with geodetic measurements to points with known coordinates.

Note that along with Along with geodesics, there are also astronomical latitude and longitude. Astronomical latitude j is the angle made by the plumb line at a given point with the plane of the equator. Astronomical longitude l is the angle between the planes of the Greenwich meridian and the astronomical meridian passing through the plumb line at a given point. Astronomical coordinates are determined on the ground from astronomical observations.

Astronomical coordinates differ from geodesics because the directions of the plumb lines do not coincide with the directions of the normals to the surface of the ellipsoid. The angle between the direction of the normal to the surface of the ellipsoid and the plumb line at a given point on the earth's surface is called deviation of the plumb line.

A generalization of geodetic and astronomical coordinates is the term - geographical coordinates.

Plane rectangular coordinates. To solve problems of engineering geodesy, they move from spatial and geodetic coordinates to simpler ones - flat coordinates, which make it possible to depict the terrain on a plane and determine the position of points using two coordinates X And at.

Since the convex surface of the Earth cannot be depicted on a plane without distortion; the introduction of plane coordinates is possible only in limited areas where the distortions are so small that they can be neglected. In Russia, a system of rectangular coordinates has been adopted, the basis of which is an equiangular transverse-cylindrical Gaussian projection. The surface of an ellipsoid is depicted on a plane in parts called zones. The zones are spherical triangles, bounded by meridians, and extending from the north pole to the south (Fig. 2.3). The size of the zone in longitude is 6°. The central meridian of each zone is called the axial meridian. The zones are numbered from Greenwich to the east.

The longitude of the axial meridian of the zone with number N is equal to:

l 0 = 6°× N - 3°.

The axial meridian of the zone and the equator are depicted on the plane by straight lines (Fig. 2.4). The axial meridian is taken as the abscissa axis x, and the equator is behind the ordinate axis y. Their intersection (point O) serves as the origin of coordinates for this zone.

To avoid negative ordinate values, the intersection coordinates are taken equal x 0 = 0, y 0 = 500 km, which is equivalent to axis displacement X 500 km west.

So that by the rectangular coordinates of a point one can judge in which zone it is located, to the ordinate y the number of the coordinate zone is assigned to the left.

Let, for example, the coordinates of a point A have the form:

x A= 6,276,427 m

y A= 12,428,566 m

These coordinates indicate that's the point A is located at a distance of 6276427 m from the equator, in the western part ( y < 500 км) 12-ой координатной зоны, на расстоянии 500000 - 428566 = 71434 м от осевого меридиана.

For spatial rectangular, geodetic and flat rectangular coordinates in Russia, a unified coordinate system SK-95 has been adopted, fixed on the ground by points of the state geodetic network and built according to satellite and ground-based measurements as of 1995.

Local rectangular coordinate systems. During the construction of various objects, local (conditional) coordinate systems are often used, in which the directions of the axes and the origin of coordinates are assigned based on the convenience of their use during the construction and subsequent operation of the object.

So, when shooting railway station axis at are directed along the axis of the main railway track in the direction of increasing picketage, and the axis X- along the axis of the passenger station building.

During construction bridge crossing axis X usually combined with the axis of the bridge, and the axis y goes in a perpendicular direction.

During construction large industrial and civil Axis facilities x And y directed parallel to the axes of buildings under construction.

To solve most problems in applied sciences, it is necessary to know the location of an object or point, which is determined using one of the accepted coordinate systems. In addition, there are height systems that also determine the altitude location of a point on

What are coordinates

Coordinates are numerical or alphabetic values that can be used to determine the location of a point on the ground. As a consequence, a coordinate system is a set of values of the same type that have the same principle for finding a point or object.

Finding the location of a point is required to solve many practical problems. In a science such as geodesy, determining the location of a point in a given space is the main goal, on the achievement of which all subsequent work is based.

Most coordinate systems typically define the location of a point on a plane limited by only two axes. In order to determine the position of a point in three-dimensional space, a height system is also used. With its help you can find out the exact location of the desired object.

Briefly about coordinate systems used in geodesy

Coordinate systems determine the location of a point on a territory by giving it three values. The principles of their calculation are different for each coordinate system.

The main spatial coordinate systems used in geodesy:

Geodetic.
Geographical.
Polar.
Rectangular.
Zonal Gauss-Kruger coordinates.

All systems have their own starting point, values for the location of the object and area of application.

Geodetic coordinates

The main figure used to measure geodetic coordinates is the earth's ellipsoid.

An ellipsoid is a three-dimensional compressed figure that best represents the shape of the globe. Due to the fact that the globe is a mathematically irregular figure, an ellipsoid is used instead to determine geodetic coordinates. This makes it easier to carry out many calculations to determine the position of a body on the surface.

Geodetic coordinates are defined by three values: geodetic latitude, longitude, and altitude.

Geodetic latitude is an angle whose beginning lies on the plane of the equator, and its end lies at the perpendicular drawn to the desired point.
Geodetic longitude is the angle measured from the prime meridian to the meridian on which the desired point is located.
Geodetic height is the value of the normal drawn to the surface of the Earth's ellipsoid of rotation from a given point.

Geographical coordinates

To solve high-precision problems of higher geodesy, it is necessary to distinguish between geodetic and geographic coordinates. In the system used in engineering geodesy, such differences are usually not made due to the small space covered by the work.

To determine geodetic coordinates, an ellipsoid is used as a reference plane, and a geoid is used to determine geographic coordinates. The geoid is a mathematically irregular figure that is closer to the actual shape of the Earth. Its leveled surface is taken to be that which continues under sea level in its calm state.

The geographic coordinate system used in geodesy describes the position of a point in space with three values. longitude coincides with the geodetic, since the reference point will also be called Greenwich. It passes through the observatory of the same name in London. determined from the equator drawn on the surface of the geoid.

Height in the local coordinate system used in geodesy is measured from sea level in its calm state. On the territory of Russia and the countries of the former Union, the mark from which heights are determined is the Kronstadt footpole. It is located at the level of the Baltic Sea.

Polar coordinates

The polar coordinate system used in geodesy has other nuances of making measurements. It is used over small areas of terrain to determine the relative location of a point. The origin can be any object marked as the initial one. Thus, using polar coordinates it is impossible to determine the unambiguous location of a point on the territory of the globe.

Polar coordinates are determined by two quantities: angle and distance. The angle is measured from the northern direction of the meridian to a given point, determining its position in space. But one angle will not be enough, so a radius vector is introduced - the distance from the standing point to the desired object. Using these two parameters, you can determine the location of the point in the local system.

As a rule, this coordinate system is used to perform engineering work carried out on a small area of terrain.

Rectangular coordinates

The rectangular coordinate system used in geodesy is also used in small areas of terrain. The main element of the system is the coordinate axis from which the counting occurs. The coordinates of a point are found as the length of perpendiculars drawn from the abscissa and ordinate axes to the desired point.

The northern direction of the X-axis and the eastern direction of the Y-axis are considered positive, and the southern and western directions are considered negative. Depending on the signs and quarters, the location of a point in space is determined.

Gauss-Kruger coordinates

The Gauss-Kruger coordinate zonal system is similar to the rectangular one. The difference is that it can be applied to the entire globe, not just small areas.

The rectangular coordinates of the Gauss-Kruger zones are essentially a projection of the globe onto a plane. It arose for practical purposes to depict large areas of the Earth on paper. Distortions arising during transfer are considered to be insignificant.

According to this system, the globe is divided by longitude into six-degree zones with an axial meridian in the middle. The equator is in the center along a horizontal line. As a result, there are 60 such zones.

Each of the sixty zones has its own system of rectangular coordinates, measured along the ordinate axis from X, and along the abscissa axis from the section of the earth's equator Y. To unambiguously determine the location on the territory of the entire globe, the zone number is placed in front of the X and Y values.

The X-axis values on the territory of Russia, as a rule, are positive, while the Y values can also be negative. In order to avoid a minus sign in the x-axis values, the axial meridian of each zone is conditionally moved 500 meters to the west. Then all coordinates become positive.

The coordinate system was proposed as a possibility by Gauss and calculated mathematically by Kruger in the mid-twentieth century. Since then, it has been used in geodesy as one of the main ones.

Height system

Coordinate and elevation systems used in geodesy are used to accurately determine the position of a point on the Earth. Absolute heights are measured from sea level or other surface taken as the source. In addition, there are relative heights. The latter are counted as the excess from the desired point to any other. They are convenient to use for working in a local coordinate system in order to simplify subsequent processing of the results.

Application of coordinate systems in geodesy

In addition to the above, there are other coordinate systems used in geodesy. Each of them has its own advantages and disadvantages. There are also areas of work for which one or another method of determining location is relevant.

It is the purpose of the work that determines which coordinate systems used in geodesy are best used. To work in small areas, it is convenient to use rectangular and polar coordinate systems, but to solve large-scale problems, systems are needed that allow covering the entire territory of the earth's surface.

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