What does coordinate system mean? Coordinate systems used in geodesy and topography. Relationship between flat rectangular and polar coordinate systems

Every modern man 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 consider each of them in detail, and also give detailed description, 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 inner surface cylinder and consider it as the abscissa axis. In order to avoid negative values ordinate (y), the ordinate on the axial meridian (the starting point of reference) 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. A flat image is made for each zone by projecting it onto lateral surface a cylinder whose axis 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 is necessary to say that it is special case 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.

4.1. RECTANGULAR COORDINATES

In topography, rectangular coordinates are most widely used. Let's take two mutually perpendicular lines on the plane - OX And OY. These lines are called coordinate axes, and their intersection point ( O) - the origin of coordinates.

Rice. 4.1. Rectangular coordinates

The position of any point on the plane can be easily determined by specifying the shortest distances from the coordinate axes to the given point. The shortest distances are perpendiculars. The perpendicular distances from the coordinate axes to a given point are called the rectangular coordinates of this point. Lines parallel to the axis X, are called coordinates XA , and parallel axes Y- coordinates atA .
The quarters of the rectangular coordinate system are numbered. They are counted clockwise from the positive direction of the abscissa axis - I, II, III, IV (Fig. 4.1).
The rectangular coordinates discussed are used on a plane. This is where they got their name flat rectangular coordinates. This coordinate system is used in small areas of terrain taken as a plane.

4.2. ZONAL SYSTEM OF RECTANGULAR GAUSSian COORDINATES

When considering the issue of “Projection of topographic maps,” it was noted that the surface of the Earth is projected onto the surface of a cylinder, which touches the surface of the Earth along the axial meridian. In this case, not the entire surface of the Earth is projected onto the cylinder, but only a part of it, limited by 3° of longitude to the west and 3° to the east from the axial meridian. Since each of the Gaussian projections transfers onto the plane only a fragment of the Earth’s surface, limited by meridians through 6° of longitude, a total of 60 projections (60 zones) must be compiled onto the Earth’s surface. In each of the 60 projections, a separate rectangular coordinate system.
In each zone the axis X is the average (axial) meridian of the zone, located 500 km to the west from its actual position, and the axis Y- equator (Fig. 4.2).

Rice. 4.2. Rectangular coordinate system
on topographic maps

The intersection of the extended axial meridian with the equator will be the origin of coordinates: x = 0, y = 0. The point of intersection of the equator and the actual central meridian has the coordinates : x = 0, y = 500 km.
Each zone has its own origin. The zones are counted from the Greenwich meridian to the east. The first six-degree zone is located between the Greenwich meridian and the meridian with eastern longitude 6º (axial meridian 3º). The second zone is 6º east. - 12º E (axial meridian 9º). Third zone - 12º east. - 18º east (axial meridian 15º). Fourth zone - 18º east. - 24º east (axial meridian 21º), etc.
The zone number is indicated in the coordinate at first digit. For example, record at = 4 525 340 means that the given point is in the fourth zone (first digit) at a distance 525 340 m from the axial meridian of the zone, located west of 500 km.

To determine the zone number by geographic coordinates, you need to add 6 to the longitude expressed in integer degrees and divide the resulting amount by 6. As a result of the division, we leave only an integer.

Example. Determine the number of the Gaussian zone for a point having an eastern longitude of 18º10".
Solution. To the whole number of degrees of longitude 18 we add 6 and divide the sum by 6
(18 + 6) / 6 = 4.
Our map is in the fourth zone.

Difficulties when using the zonal coordinate system arise in cases where topographic and geodetic work is carried out in border areas located in two adjacent (adjacent) zones. The coordinate lines of such zones are located at an angle to each other (Figure 4.3).

To eliminate emerging complications, a zone overlap strip , in which the coordinates of points can be calculated in two adjacent systems. The width of the overlap strip is 4°, 2° in each zone.

An additional grid on the map is applied only in the form of outputs of its lines between the minute and outer frames. Its digitization is a continuation of the digitization of the grid lines of the adjacent zone. Additional grid lines are signed outside the outer frame of the sheet. Consequently, on a map sheet located in the eastern zone, when connecting the same-name outputs of the additional grid, a kilometer grid of the western zone is obtained. Using this grid, you can determine, for example, the rectangular coordinates of a point IN in the rectangular coordinate system of the western zone, i.e. rectangular coordinates of points A And IN will be obtained in one coordinate system of the western zone.

Rice. 4.3. Additional kilometer lines at the borders of zones

On a 1:10,000 scale map, the additional grid is divided only on those sheets in which the eastern or western meridian of the inner frame (trapezoid frame) is the boundary of the zone. An additional grid is not applied to topographic plans.

4.3. DETERMINING RECTANGULAR COORDINATES USING A COMPASS METER

An important element topographic map(plan) is a rectangular grid. On all sheets of this 6-degree zone, the grid is applied in the form of rows of lines, parallel to the axial meridian and equator(Fig. 4.2). The vertical grid lines are parallel to the axial meridian of the zone, and the horizontal lines are parallel to the equator. Horizontal kilometer lines are counted from bottom to top, and vertical ones - from left to right. .

The intervals between lines on maps of scales 1:200,000 - 1:50,000 are 2 cm, 1:25,000 - 4 cm, 1:10,000 - 10 cm, which corresponds to an integer number of kilometers on the ground. Therefore, a rectangular mesh is also called kilometer, and its lines are kilometer.
The kilometer lines closest to the corners of the frame of the map sheet are signed with the full number of kilometers, the rest - with the last two digits. Inscription 60 65 (see Fig. 4.4) on one of the horizontal lines means that this line is 6065 km away from the equator (north): inscription 43 07 at the vertical line means that it is in the fourth zone and is 307 km east from the beginning of ordinate counting. If a three-digit number is written in small numbers near the vertical kilometer line, the first two indicate the zone number.

Example. It is necessary to determine from the map the rectangular coordinates of a terrain point, for example, a point of the state geodetic network (GGS) with mark 214.3 (Fig. 4.4). First, write down (in kilometers) the abscissa of the south side of the square in which this point is located (i.e. 6065). Then, using a measuring compass and a linear scale, determine the length of the perpendicular Δx= 550 m, descending from a given point onto this line. The resulting value (in this case 550 m) is added to the abscissa of the line. The number 6,065,550 is the abscissa X GGS point.
The ordinate of the GGS point is equal to the ordinate of the western side of the same square (4307 km), added to the length of the perpendicular Δу= 250 m, measured on the map. The number 4,307,250 is the ordinate of the same point.
In the absence of a measuring compass, distances are measured with a ruler or strip of paper.

X = 6065550, at= 4307250
Rice. 4.4. Defining rectangular coordinates using a linear scale

4.4. DETERMINING RECTANGULAR COORDINATES USING A COORDINATOMETER

Coordinator - a small square with two perpendicular sides. Along the internal edges of the rulers are scales, the lengths of which are equal to the length of the side of the coordinate cells of the map of a given scale. The divisions on the coordinate meter are transferred from the linear scale of the map.
The horizontal scale is aligned with the bottom line of the square (in which the point is located), and the vertical scale must pass through this point. The scales determine the distances from the point to the kilometer lines.

x A = 6135,350 y A = 5577,710
Rice. 4.5. Determining rectangular coordinates using a coordinate meter

4.5. PLACING POINTS ON THE MAP AT SPECIFIED RECTANGULAR COORDINATES

To plot a point on a map according to given rectangular coordinates, proceed as follows: in the coordinate record, two-digit numbers are found that abbreviate the lines of the rectangular grid. Using the first number, a horizontal grid line is found on the map, and a vertical grid line is found using the second number. Their intersection forms the southwestern corner of the square in which the desired point lies. On the eastern and western sides of the square, two equal segments are laid from its southern side, corresponding on the map scale to the number of meters in the abscissa X . The ends of the segments are connected by a straight line and on it, from the western side of the square, a segment corresponding to the number of meters in the ordinate is plotted on the map scale; the end of this segment is the desired point.

4.6. CALCULATION OF FLAT RECTANGULAR GAUSSian COORDINATES BY GEOGRAPHICAL COORDINATES

Plane rectangular Gaussian coordinates X And at very difficult to relate to geographic coordinates φ (latitude) and λ (longitude) points on the earth's surface. Suppose that some point A has geographic coordinates φ And λ . Since the difference in the longitudes of the boundary meridians of the zone is 6°, then, accordingly, for each of the zones it is possible to obtain the longitudes of the extreme meridians: 1st zone (0° - 6°), 2nd zone (6° - 12°), 3rd zone (12° - 18°), etc. Thus, according to the geographical longitude of the point A you can determine the number of the zone in which this point is located. At the same time, longitude λ The axis of the axial meridian of the zone is determined by the formula
λ OS = (6°n - 3°),
wherein n- zone number.

To define plane rectangular coordinates X And at by geographic coordinates φ And λ Let’s use the formulas derived for Krasovsky’s reference ellipsoid (the reference ellipsoid is a figure that is as close as possible to the figure of the Earth in the part on which a given state or group of states is located):

X = 6367558,4969 (φ glad ) − (a 0 − l 2 N)sinφ cosφ (4.1)
at(l) = lNcosφ (4.2)

Formulas (4.1) and (4.2) use the following notation:
y(l) - distance from the point to the axial meridian of the zone;
l= (λ - λ OS ) - the difference between the longitudes of the determined point and the axial meridian of the zone);
φ glad - latitude of a point, expressed in radian measure;
N = 6399698,902 - cos 2φ;
A 0 = 32140,404 - cos 2 φ;
A 3 = (0,3333333 + 0,001123 cos 2 φ) cos 2φ - 0.1666667;
A 4 = (0,25 + 0,00252 cos 2φ) cos 2φ - 0.04166;
A 5 = 0,0083 - cos 2φ;
A 6 = (0.166 cos 2 φ - 0.084) cos 2 φ.
y" is the distance from the axial meridian located west of 500 km.

According to formula (4.1), the coordinate value y(l) are obtained relative to the axial meridian of the zone, i.e. it can turn out with “plus” signs for the eastern part of the zone or “minus” signs for the western part of the zone. To record coordinates y in the zonal coordinate system, it is necessary to calculate the distance to a point from the axial meridian of the zone, located 500 km to the west (y"in the table ) , and write the zone number in front of the resulting value. For example, the value received is
y(l)= -303678.774 m in zone 47.
Then
at= 47 (500000.000 - 303678.774) = 47196321.226 m.
We use spreadsheets for calculations Microsoft XL .

Example. Calculate the rectangular coordinates of a point having geographic coordinates:
φ = 47º02"15.0543"N; λ = 65º01"38.2456" east.

To the table Microsoft XL enter the initial data and formulas (Table 4.1).

Table 4.1.

View of the table after calculations (Table 4.2).

Table 4.2.

4.7. CALCULATION OF GEOGRAPHICAL COORDINATES USING FLAT RECTANGULAR GAUSSian COORDINATES

To solve this problem, recalculation formulas obtained for Krasovsky’s reference ellipsoid are also used.
Suppose we need to calculate geographic coordinates φ And λ points A by its flat rectangular coordinates X And at, specified in the zonal coordinate system. In this case, the coordinate value at written down indicating the zone number and taking into account the transfer of the axial meridian of the zone to the west by 500 km.
Pre-by value at find the number of the zone in which the point being determined is located, and use the zone number to determine the longitude λ o the axial meridian and by the distance from the point to the axial meridian referred to the west, find the distance y(l) from a point to the axial meridian of the zone (the latter can have a plus or minus sign).
Geographic coordinate values φ And λ on flat rectangular coordinates X And at found using the formulas:
φ = φ X - z 2 b 2 ρ″ (4.3)
λ = λ 0 + l (4.4)
l = zρ″ (4.5)

In formulas (4.3) and (4.5):
φ x ″= β″ +(50221746 + cos 2 β)10-10sinβcosβ ρ″;
β″ = (X / 6367558.4969) ρ″; ρ″ = 206264.8062″ - number of seconds in one radian
z = У(L) / (Nx сos φx);
N x = 6399698.902 - cos 2 φ x;
b 2 = (0.5 + 0.003369 cos 2 φ x) sin φ x cos φ x;
b 3 = 0.333333 - (0.166667 - 0.001123 cos2 φ x) cos2 φ x;
b 4 = 0.25 + (0.16161 + 0.00562 cos 2 φ x) cos 2 φ x;
b 5 = 0.2 - (0.1667 - 0.0088 cos 2 φ x) cos 2 φ x.

We use spreadsheets for calculations Microsoft XL .
Example. Calculate the geographic coordinates of a point using rectangular coordinates:
x = 5213504.619; y = 11654079.966.

To the table Microsoft XL enter the initial data and formulas (Table 4.3).

Table 4.3.

View of the table after calculations (Table 4.4).

Table 4.4.

If the calculations are made correctly, copy both tables onto one sheet, hide the lines of intermediate calculations and the column No., and leave only the lines for entering the initial data and calculation results. We format the table and adjust the names of columns and columns at your discretion.

Table 4.5.

Notes.
1. Depending on the required accuracy, you can increase or decrease the bit depth.
2. The number of rows in the table can be reduced by combining calculations. For example, do not calculate the radians of an angle separately, but immediately write them in the formula =SIN(RADIANS(C3)).
3. Rounding in paragraph 23 of the table. 4.1. We produce for “clutch”. Number of digits in rounding 3.
4. If you do not change the format of the cells in the “Grad” and “Min” columns, then there will be no zeros before the numbers. The format change here is made only for visual perception (at the author's discretion) and does not affect the calculation results.
5. To avoid accidentally damaging formulas, you should protect the table: Service / Protect sheet. Before protecting, select the cells for entering the original data, and then: Cell format / Protection / Protected cell - uncheck the box.

4.8. RELATIONSHIP OF FLAT RECTANGULAR AND POLAR COORDINATE SYSTEMS

The simplicity of the polar coordinate system and the possibility of constructing it relative to any point in the terrain taken as a pole led to its widespread use in topography. In order to connect together the polar systems of individual terrain points, it is necessary to move on to determining the position of the latter in a rectangular coordinate system, which can be extended to a much larger area. The connection between the two systems is established by solving direct and inverse geodetic problems.
Direct geodetic problem consists in determining the coordinates of the end point IN (Fig. 4.4) lines AB along its length G horizontal layoutd , directionα and coordinates of the starting point XA , atA .

Rice. 4.6. Solving direct and inverse geodetic problems

So, if we accept the point A(Fig. 4.4) beyond the pole of the polar coordinate system, and the straight line AB- beyond the polar axis parallel to the axis OH, then the polar coordinates of the point IN will d And α . It is necessary to calculate the rectangular coordinates of this point in the system HOU.

From Fig. 3.4 it is clear that XIN differs from XA by the amount ( XIN - XA ) = Δ XAB , A atIN differs from atA by the amount ( atIN - atA ) = Δ atAB . Final coordinate differences IN and primary A line points AB Δ X and Δ at called coordinate increments . Coordinate increments are orthogonal projections of the line AB on the coordinate axis. Coordinates XIN And atIN can be calculated using the formulas:

XIN = XA + Δ XAB (4.1)
atIN = atA + Δ atAB (4.2)

Δ XAB =dcos α (4.3)
Δ atAB = dsin α (4.4)

The sign of the coordinate increments depends on the position angle.

Table 4.1.

Substituting the value of increments Δ XAB and Δ atAB into formulas (3.1 and 3.2), we obtain formulas for solving the direct geodetic problem:

XIN = XA + dcos α (4.5)
atIN = atA + dsin α (4.6)

Inverse geodetic problem consists in determining the length of the horizontal spacedand the direction α of line AB according to the given coordinates of its starting point A (xA, yA) and final point B (xB, yB). The direction angle is calculated using the legs of a right triangle:

tan α = (4.7)

Horizontal layout d, determined by the formula:

d = (4.8)

To solve direct and inverse geodetic problems, you can use spreadsheets Microsoft Excel .

Example.
Point given A with coordinates: XA = 6068318,25; atA = 4313450.37. Horizontal layout (d) between point A and dot IN equals 5248.36 m. The angle between the north direction of the axis OH and direction to the point IN(position angle - α ) is equal to 30º.

Entering source data and formulas into spreadsheets Microsoft Excel (Table 4.2).

Table 4.2.

Table 4.3.

Example.
Points specified A And IN with coordinates:
XA = 6068318,25; atA = 4313450,37;
XIN = 6072863,46; atIN = 4313450,37.
Calculate horizontal distance d between point A and dot IN, and also the angle α between the north direction of the axis OH and direction to the point IN.
Entering source data and formulas into spreadsheets Microsoft Excel (Table 4.4).

Table 4.4.

Table 4.5.

If your calculations match those in the tutorial, hide the intermediate calculations, format and protect the table.

Video
Rectangular coordinates

Questions and tasks for self-control

1. What quantities are called rectangular coordinates?
2. On what surface are rectangular coordinates used?
3. What is the essence of the zonal rectangular coordinate system?
4. What is the number of the six-degree zone in which the city of Lugansk is located with coordinates: 48°35′ N. 39°20′ E
5. Calculate the longitude of the axial meridian of the six-degree zone in which Lugansk is located.
6. How are x and y coordinates calculated in the rectangular Gaussian coordinate system?
7. Explain the procedure for determining rectangular coordinates on a topographic map using a measuring compass.
8. Explain the procedure for determining rectangular coordinates on a topographic map using a coordinate meter.
9. What is the essence of the direct geodetic problem?
10. What is the essence of the inverse geodetic problem?
11. What quantity is called the coordinate increment?
12. Define sine, cosine, tangent and cotangent of an angle.
13. How can we apply the Pythagorean theorem on the relationship between the sides of a right triangle in topography?

To specify a Cartesian rectangular coordinate system, you need to select several mutually perpendicular lines, called axes. The point where the O axes intersect is called the origin.

On each axis you need to set a positive direction and select a scale unit. The coordinates of point P are considered positive or negative depending on which semi-axis the projection of point P falls on.

Cartesian rectangular coordinates of point P on surface two mutually perpendicular lines - coordinate axes or, what is the same, projections of the radius vector r point P on two

When talking about a two-dimensional coordinate system, the horizontal axis is called the axis abscissa(axis Ox), vertical axis - axis ordinate(Oy axis). Positive directions are chosen on the Ox axis - to the right, on the Oy axis - up. The x and y coordinates are called the abscissa and ordinate of a point, respectively.

The notation P(a,b) means that a point P on the plane has an abscissa a and an ordinate b.

Cartesian rectangular coordinates points P in three-dimensional space are called distances taken with a certain sign (expressed in scale units) of this point to three mutually perpendicular coordinate planes or, what is the same, projections of the radius vector r point P on three mutually perpendicular coordinate axes.

Depending on the relative position of the positive directions of the coordinate axes, it is possible left And right coordinate systems.

Rice. 3a	Rice. 3b

As a rule, a right-handed coordinate system is used. Positive directions are chosen: on the Ox axis - towards the observer; on the Oy axis - to the right; on the Oz axis - up. The coordinates x, y, z are called abscissa, ordinate and applicate, respectively.

Coordinate surfaces for which one of the coordinates remains constant are planes parallel to the coordinate planes, and coordinate lines along which only one coordinate changes are straight lines parallel to the coordinate axes. Coordinate surfaces intersect along coordinate lines.

The notation P(a,b,c) means that the point Q has an abscissa a, an ordinate b and an applicate c.

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 crossings 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:

1. Geodetic.
2. Geographical.
3. Polar.
4. Rectangular.
5. 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.

1. 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.
2. Geodetic longitude is the angle measured from the prime meridian to the meridian on which the desired point is located.
3. 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 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.