Coordinates - these are quantities that determine the position of any point on the surface or in space in the accepted coordinate system. The coordinate system sets the initial (original) points, lines or planes for reading the required quantities - the origin of the coordinates and the units of their calculation. In topography and geodesy, systems of geographic, rectangular, polar and bipolar coordinates have received the greatest application.
Geographic coordinates (Fig. 2.8) are used to determine the position of points on the Earth's surface on an ellipsoid (ball). In this coordinate system, the initial meridian plane and the equatorial plane are the initial ones. A meridian is a line of section of an ellipsoid by a plane passing through given point and the Earth's axis of rotation.

A parallel is a line of section of an ellipsoid by a plane passing through a given point and perpendicular to the earth's axis. The parallel whose plane passes through the center of the ellipsoid is called the equator. Through every point on the surface the globe, you can draw only one meridian and only one parallel.
Geographical coordinates are angular quantities: longitude l and latitude j.
Geographic longitude l is called dihedral angle, enclosed between the plane of the given meridian (passing through point B) and the plane of the initial meridian. The meridian passing through the center of the main hall of the Greenwich Observatory within the city of London was taken as the initial (zero) meridian. For point B, longitude is determined by the angle l = WCD. Longitudes are counted from the prime meridian in both directions - east and west. In this regard, we distinguish between western and eastern longitudes, which vary from 0° to 180°.
Geographic latitude j is the angle formed by the plane of the equator and the plumb line passing through the given point. If the Earth is taken as a ball, then for point B (Fig. 2.8) latitude j is determined by the angle DCB. The latitudes measured from the equator to the north are called northern, and to the south - southern, they vary from 0 ° at the equator to 90 ° at the poles.
Geographic coordinates may be derived from astronomical observations or geodetic measurements. In the first case, they are called astronomical, and in the second - geodetic (L - longitude, B - latitude). In astronomical observations, the projection of points onto the reference surface is carried out by plumb lines, in geodetic measurements - by normals. Therefore, the values ​​of astronomical and geodetic coordinates differ by the amount of deviation of the plumb line.
The use of different reference ellipsoids by different states leads to differences in the coordinates of the same points calculated relative to different initial surfaces. In practice, this is expressed in the general displacement of the cartographic image relative to the meridians and parallels on maps of large and medium scales.
Rectangular coordinates linear quantities are called - the abscissa and the ordinate, which determine the position of a point on the plane relative to the original directions.

(Fig. 2.9)
In geodesy and topography, it is accepted right system rectangular coordinates. This distinguishes it from the left coordinate system used in mathematics. The initial directions are two mutually perpendicular lines with the origin at the point of their intersection O.
The XX straight line (abscissa axis) is aligned with the direction of the meridian passing through the origin, or with the direction parallel to some meridian. The straight line YY (y-axis) passes through the point O perpendicular to the x-axis. In such a system, the position of a point on a plane is determined by the shortest distance to it from the coordinate axes. The position of point A is determined by the length of the perpendiculars Xa and Ya. The segment Xa is called the abscissa of the point A, and Yа is the ordinate of this point. Rectangular coordinates are usually expressed in meters. The abscissa and ordinate axes divide the terrain at point O into four quarters (Fig. 2.9). The name of the quarters is determined by the accepted designations of the countries of the world. Quarters are numbered clockwise: I - SV; II - SE; III - SW; IV - NW.
In table. 2.3 shows the signs of abscissas X and ordinates Y for points located in different quarters and their names are given.

Table 2.3
The abscissas of points located up from the origin are considered positive, and down from it - negative, the ordinates of points located to the right - positive, to the left - negative. The system of flat rectangular coordinates is used in limited areas earth's surface, which can be taken as flat.
Coordinates, the origin of which is any point in the terrain, are called polar. In this coordinate system, orientation angles are measured. On a horizontal plane (Fig. 2.10), through an arbitrarily chosen point O, called the pole, a straight line OX is drawn - the polar axis.

Then the position of any point, for example, M will be determined by the radius - the vector r1 and the direction angle a1, and the point N - respectively r2 and a2. Angles a1 and a2 are measured from the polar axis clockwise to the radius vector. The polar axis can be located arbitrarily or combined with the direction of any meridian passing through the pole O.
The bipolar coordinate system (Fig. 2.11) represents two selected fixed poles O1 and O2, connected by a straight line - the polar axis. This coordinate system allows you to determine the position of the point M relative to the polar axis on the plane using two angles b1 and b2, two radius vectors r1 and r2, or combinations thereof. If the rectangular coordinates of the points O1 and O2 are known, then the position of the point M can be calculated in an analytical way.


Rice. 2.11

Rice. 2.12
Heights of points on the earth's surface. To determine the position of the points of the physical surface of the Earth, it is not enough to know only the planned coordinates X, Y or l, j, a third coordinate is needed - the height of the point H. The height of the point H (Fig. 2.12) is the distance along the vertical direction from a given point (A´; B´ ´) to the accepted main level surface MN. The numerical value of the height of a point is called elevation. Heights measured from the main level surface MN are called absolute heights (AA´; BB´´), and those determined relative to an arbitrarily chosen level surface are called conditional heights (В´В´´). The height difference of two points or the distance along the vertical direction between the level surfaces passing through any two points of the Earth is called the relative height (В´В´´) or the excess of these points h.
In the Republic of Belarus, the Baltic system of heights of 1977 was adopted. The heights are counted from the level surface, which coincides with the average water level in the Gulf of Finland, from the zero of the Kronstadt footstock.

Here's another

4.1. RECTANGULAR COORDINATES

In topography, rectangular coordinates are most widely used. Take on a plane two mutually perpendicular lines - OX and OY. These lines are called coordinate axes, and the point of their intersection ( O) is 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. Distances along perpendiculars from the coordinate axes to a given point are called rectangular coordinates of this point. Line segments parallel to the axis X, are called coordinates XBUT , and the parallel axes Y- coordinates atBUT .
The quarters of the rectangular coordinate system are numbered. Their count goes clockwise from the positive direction of the x-axis - I, II, III, IV (Fig. 4.1).
The rectangular coordinates discussed above are used on a plane. Hence they got the name flat rectangular coordinates. This coordinate system is used in small areas of the terrain, taken as a plane.

4.2. ZONAL GAUSSIAN RECTANGULAR COORDINATE SYSTEM

When considering the issue of "Projections of topographic maps", it was noted that the Earth's surface is projected onto the surface of a cylinder that touches the Earth's surface 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 of the axial meridian. Since each of the Gaussian projections transmits to the plane only a fragment of the Earth's surface, limited by meridians through 6 ° of longitude, then a total of 60 projections (60 zones) should be made on the Earth's surface. In each of the 60 projections, a separate system rectangular coordinates.
In each zone, the axis X is the middle (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 axial 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 east longitude 6º (axial meridian 3º). The second zone is 6º E. - 12º E (axial meridian 9º). The third zone - 12º E - 18º E (axial meridian 15º). The fourth zone - 18º E - 24º E (axial meridian 21º), etc.
The zone number is indicated in the coordinate at the first digit. For example, the entry at = 4 525 340 means that the specified point is in the fourth zone (first digit) at a distance 525 340 m from the axial meridian of the zone, which is located west of 500 km.

To determine the zone number by geographical coordinates, it is necessary to add 6 to the longitude expressed in whole numbers of degrees and divide the resulting amount by 6. As a result of division, we leave only an integer.

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

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

In order to eliminate the arising complications, zone overlap band , in which the coordinates of the points can be calculated in two adjacent systems. Overlap width 4°, 2° in each zone.

An additional grid on the map is applied only in the form of outlets 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 outputs of the additional grid of the same name, a kilometer grid of the western zone is obtained. Using this grid, you can determine, for example, the rectangular coordinates of a point AT in the system of rectangular coordinates of the western zone, i.e., the rectangular coordinates of the points BUT and AT will be obtained in the same coordinate system of the western zone.

Rice. 4.3. Additional kilometer lines on the border of zones

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

4.3. DETERMINATION OF RECTANGULAR COORDINATES WITH THE HELP OF A COMPASS-MEASURER

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 central meridian and the equator(Fig. 4.2). The vertical lines of the grid are parallel to the axial meridian of the zone, and the horizontal lines are parallel to the equator. The horizontal kilometer lines are counted from the bottom up, and the 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 a whole number of kilometers on the ground. Therefore, a rectangular grid is also called kilometer, and its lines are kilometer.
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 (to the north): the inscription 43 07 at the vertical line means that it is in the fourth zone and is 307 km away from the beginning of the calculation of ordinates to the east. 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 the rectangular coordinates of a point on the map, for example, a point of the state geodetic network (GGS) with a mark of 214.3 (Fig. 4.4). First, write down (in kilometers) the abscissa of the southern 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 Δх= 550 m pubescent from given point to this line. The resulting value (in this case 550 m) is added to the abscissa of the line. The number 6 065 550 is an abscissa X point GGS.
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. Determining Rectangular Coordinates Using a Linear Scale

4.4. DETERMINATION OF RECTANGULAR COORDINATES USING A COORDINATOMER

Coordinator - a small square with two perpendicular sides. Scales are marked along the inner edges of the rulers, the lengths of which are equal to the length of the side of the coordinate cells of the map of the given scale. 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 distance from the point to the kilometer lines.

x A = 6135 350 y A = 5577 710
Rice. 4.5. Determining Cartesian Coordinates Using the Coordinometer

4.5. APPLICATION OF POINTS ON A MAP BY GIVEN RECTANGULAR COORDINATES

To plot a point on a map at given rectangular coordinates, proceed as follows: in the coordinate record, two-digit numbers are found, which abbreviated the lines of the rectangular grid. According to the first number, a horizontal grid line is found on the map, according to the second - a vertical one. 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 off from its southern side, corresponding on the scale of the map 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 laid on the scale of the map; the end of this segment is the desired point.

4.6. CALCULATION OF FLAT RECTANGULAR GAUSS COORDINATES FROM GEOGRAPHICAL COORDINATES

Plane Gaussian Cartesian Coordinates X and at very difficult to relate to geographic coordinates φ (latitude) and λ (longitude) points on the earth's surface. Suppose some point BUT has geographic coordinates φ and λ . Since the difference in the longitudes of the boundary meridians of the zone is 6°, then, respectively, 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 geographical longitude points BUT you can determine the number of the zone in which this point is located. While the longitude λ os of the axial meridian of the zone is determined by the formula
λ os = (6°n - 3°),
wherein n- zone number.

To define planar rectangular coordinates X and at by geographic coordinates φ and λ we will 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 that part of it on which it is located given state, or a group of states):

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 the 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 \u003d (0.166 cos 2 φ - 0.084) cos 2 φ.
y" - the distance from the axial meridian referred to the west of 500 km.

According to formula (4.1), the value of the coordinate y(l) are obtained relative to the axial meridian of the zone, i.e. it can be obtained 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, 500 km to the west (at"in the table ) , and in front of the obtained value, assign the zone number. For example, given the value
y(l)= -303678.774 m in zone 47.
Then
at= 47 (500000.000 - 303678.774) = 47196321.226 m.
We use spreadsheets for calculations. MicrosoftXL .

Example. Calculate the rectangular coordinates of a point that has geographic coordinates:
φ \u003d 47º02 "15.0543" N; λ = 65º01"38.2456"E

To table MicrosoftXL enter the initial data and formulas (tab. 4.1).

Table 4.1.

				D	E	F
		Parameter	Computing	hail
		φ (deg)	D2+E2/60+F2/3600
		φ (rad)	RADIANS(C3)
		Cos 2 φ
		zone number	INTEGER((D8+6)/6)
		λos (deg)
		l (deg)	D11+E11/60+F11/3600
		l (rad)	RADIANS(C12)
			6399698,902-((21562,267- (108.973-0.612*C6^2)*C6^2))*C6^2
		a 0	32140,404-((135,3302- (0.7092-0.004*C6^2)*C6^2))*C6^2
		a 4	=(0.25+0.00252*C6^2)*C6^2-0.04166
		a 6	=(0.166*C6^2-0.084)*C6^2
		a 3	=(0.3333333+0.001123*C6^2)*C6^2-0.1666667
		a 5	0.0083-((0.1667-(0.1968+0.004*C6^2)*C6^2))*C6^2
			6367558.4969*C4-(((C15-(((0.5+(C16+C17*C20)*C20)) *C20*C14)))*C5*C6)
			=((1+(C18+C19*C20)*C20))*C13*C14*C6
			ROUND((500000+C23);3)
			CONCATENATE(C9;C24)

View of the table after calculations (tab. 4.2).

Table 4.2.

		Parameter	Computing	hail
		φ (deg, min, sec)
		φ (degrees)
		φ (radians)
		Cos 2 φ
		λ (deg, min, sec)
		Zone number
		λos (deg)
		l (min, sec)
		l (degrees)
		l (radians)
		a 0
		a 4
		a 6
		a 3
		a 5

4.7. CALCULATION OF GEOGRAPHICAL COORDINATES FROM FLAT RECTANGULAR GAUSS COORDINATES

To solve this problem, the recalculation formulas obtained for the Krasovsky reference ellipsoid are also used.
Suppose we need to calculate geographic coordinates φ and λ points BUT by its flat rectangular coordinates X and at given in the zonal coordinate system. In this case, the value of the coordinate at recorded with the indication of the zone number and taking into account the shift 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 determined point is located, determine the longitude by the zone number λ o axial meridian and the distance from the point to the westward axial meridian find the distance y(l) from the point to the axial meridian of the zone (the latter can be with a plus or minus sign).
Geographic coordinate values φ and λ in planar rectangular coordinates X and at are found by the formulas:
φ = φ X - z 2 b 2 p″ (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 = Y(L) / (Nx cos φx);
N x \u003d 6399698.902 - cos 2 φ x;
b 2 \u003d (0.5 + 0.003369 cos 2 φ x) sin φ x cos φ x;
b 3 \u003d 0.333333 - (0.166667 - 0.001123 cos2 φ x) cos2 φ x;
b 4 \u003d 0.25 + (0.16161 + 0.00562 cos 2 φ x) cos 2 φ x;
b 5 \u003d 0.2 - (0.1667 - 0.0088 cos 2 φ x) cos 2 φ x.

We use spreadsheets for calculations. MicrosoftXL .
Example. Calculate geographic coordinates of a point from rectangular:
x = 5213504.619; y = 11654079.966.

To table MicrosoftXL enter the initial data and formulas (tab. 4.3).

Table 4.3.

1 Parameter calculation Grad. Min. Sec. 2 1 X 5213504,619 2 at 11654079,966 4 3 №*zones IF(C3<1000000; C3/100000;C3/1000000) 5 4 zone number WHOLE(C4) 6 5 λoos C5*6-3 7 6 at" C3-C5*1000000 8 7 y(l) C7-500000 9 8 ρ″ 206264,8062 10 9 β" C2/6367558.4969*C9 11 10 β rad RADIANS(C10/3600) 12 11 β WHOLE (C10/3600) WHOLE ((C10-D12*3600)/60) C10-D12* 3600-E12*60 13 12 Sin β SIN(C11) 14 13 Cosβ COS(C11) 15 14 Cos 2 β C14^2 16 15 φ X " C10+(((50221746+((293622+ (2350+22*C14^2)*C14^2))*C14^2))) *10^-10*C13*C14*C9 17 16 φ X glad RADIANS(C16/3600) 18 17 φ X WHOLE (C16/3600) WHOLE ((C16-D18*3600)/60) C16-D18* 3600-E18*60 19 18 Sin phi. SIN(C17) 20 19 Cos φ X COS(C17) 21 20 Cos 2 φ X C20^2 22 21 N X 6399698,902-((21562,267- (108.973-0.612*C21)*C21))*C21 23 22 Ν X Cosφ X C22*C20 24 23 z C8/(C22*C20) 25 24 z 2 C24^2 26 25 b 4 0.25+(0.16161+0.00562*C21)*C21 27 26 b 2 =(0.5+0.003369*C21)*C19*C20 28 27 b 3 0.333333-(0.166667-0.001123*C21)*C21 29 28 b 5 0.2-(0.1667-0.0088*C21)*C21 30 29 C16-((1-(C26-0.12 *C25)*C25))*C25*C27*C9 31 30 φ =INTEGER (C30/3600) =INTEGER ((C30-D31*3600)/60) =C30-D31* 3600-E31*60 32 31 l" =((1-(C28-C29*C25)*C25))*C24*C9 33 32 l 0 =INTEGER (C32/3600) =INTEGER ((C32-D33*3600)/60) =C32-D33* 3600-E33*60 34 33 λ C6+D33

View of the table after calculations (tab. 4.4).

Table 4.4.

		Parameter	calculation	Grad.
		Zone number*
		Zone number
		λoos (deg)
		at"
		β rad
		Cos 2 β
		φ X "
		φ X glad
		φ X
		Cos φ X
		Cos 2 φ X
		N X
		Ν X Cos φ X
		z 2
		b 4
		b 2
		b 3
		b 5
		φ
		l 0
		λ

If the calculations are made correctly, copy both tables onto one sheet, hide the rows of intermediate calculations and the column No. p / p, and leave only the lines for entering the initial data and calculation results. We format the table and adjust the names of the columns and columns as you wish.

Worksheets might look like this

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 it into the formula =SIN(RADIANS(C3)).
3. Rounding in paragraph 23 of the table. 4.1. we produce for "clutch". Number of digits to round 3.
4. If you do not change the format of the cells in the columns "Grad" and "Min", then there will be no zeros in front of the numbers. The format change here is made only for visual perception (by the decision of the author) and does not affect the results of calculations.
5. In order not to accidentally damage the formulas, you should protect the table: Tools / Protect sheet. Before protection, select the cells for entering the initial data, and then: Format cells / Protection / Protected cell - uncheck.

4.8. RELATIONSHIP OF PLANE 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 link together the polar systems of individual points of the terrain, 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 AT (Fig. 4.4) lines AB along its length G horizontald , directionα and coordinates of the starting point XBUT , atBUT .

Rice. 4.6. Solution of direct and inverse geodetic problems

So, if we take the point BUT(Fig. 4.4) for the pole of the polar coordinate system, and the straight line AB- for the polar axis parallel to the axis OH, then the polar coordinates of the point AT will d and α . It is necessary to calculate the rectangular coordinates of this point in the system HOW.

From fig. 3.4 shows that XAT differs from XBUT by the value ( XAT - XBUT ) = Δ XAB , a atAT differs from atBUT by the value ( atAT - atBUT ) = Δ atAB . Differences in the coordinates of the final AT and primary BUT line dots AB Δ X and Δ at called coordinate increments . Coordinate increments are orthogonal projections of the line AB on the coordinate axis. Coordinates XAT and atAT can be calculated using the formulas:

XAT = XBUT + Δ XAB (4.1)
atAT = atBUT + Δ atAB (4.2)

The increment values ​​are determined from the right-angled triangle ASV according to the given d and α, since the increments Δ X and Δ at are the legs of this right triangle:

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

XAT = XBUT + dcos α (4.5)
atAT = atBUT + dsin α (4.6)

Inverse geodesic problem is to determine the length of the horizontal spandand the direction α of the line AB according to the given coordinates of its initial point A (xA, yA) and final point B (xB, yB). The direction angle is calculated from the legs of a right triangle:

tgα = (4.7)

Horizontal spacing d, determined by the formula:

d = (4.8)

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

Example.
Point given BUT with coordinates: XBUT = 6068318,25; atBUT = 4313450.37. Horizontal spacing (d) between point BUT and dot AT equal to 5248.36 m. The angle between the northern direction of the axis OH and direction to the point AT(position angle - α ) is equal to 30º.

Calculate rectangular coordinates of a point B(xAT ,atAT ).

Entering raw data and formulas into spreadsheets Microsoft Excel (tab. 4.2).

Table 4.2.

	Initial data
	XBUT
	atBUT
	Computing
	Δ XAB =d cos α	B4*COS(RADIANS(B5))
	Δ atAB = d sin α	B4*SIN(RADIANS(B5))
	XAT
	atAT

Table view after calculations (tab. 4.3).

Table 4.3.

	Initial data
	XBUT
	atBUT
	Computing
	Δ XAB =d cos α
	Δ atAB = d sin α
	XAT
	atAT

Example.
Points are given BUT and AT with coordinates:
XBUT = 6068318,25; atBUT = 4313450,37;
XAT = 6072863,46; atAT = 4313450,37.
Calculate horizontal distance d between point BUT and dot AT, and also the angle α between north axis OH and direction to the point AT.
Entering raw data and formulas into spreadsheets Microsoft Excel (tab. 4.4).

Table 4.4.

	Initial data
	XBUT
	atBUT
	XAT
	atAT
	Computing
	ΔхAB
	ΔуAB
		ROOT(B7^2+B8^2)
	Tangent
	Arctangent
	degrees	DEGREES(B11)
	Choice	IF(B12<0;B12+180;B12)
	Position angle (deg)	IF(B8<0;B13+180;B13)

View of the table after calculations (tab. 4.5).

Table 4.5.

	Initial data
	XBUT
	atBUT
	XAT
	atAT
	Computing
	ΔхAB
	ΔуAB
	Tangent
	Arctangent
	degrees
	Choice
	Position angle (deg)

If your calculations match those of the tutorial, hide intermediate calculations, format and protect the spreadsheet.

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 system of rectangular coordinates?
4. What is the number of the six-degree zone in which the city of Lugansk is located with coordinates: 48°35′ N.L. 39°20′ E
5. Calculate the longitude of the axial meridian of the six-degree zone in which the city of Lugansk is located.
6. How are x and y coordinates counted in a Gaussian rectangular 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 geodesic problem?
11. What is the increment of coordinates?
12. Define the sine, cosine, tangent and cotangent of an angle.
13. How can the Pythagorean theorem on the relationship between the sides of a right triangle be applied in topography?

1.10. RECTANGULAR COORDINATES ON MAP

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

Rice. fourteen.Rectangular coordinates

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

Rectangular coordinates on topographic maps of the USSR applied to coordinate zones. Coordinate zones - parts of the earth's surface, limited by meridians with a longitude that is a multiple of 6 °. The first zone is limited by the meridians 0° and 6°, the second - b "and 12°, the third - 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 system of rectangular coordinates. The origin of coordinates in each zone is the point of intersection of the middle (axial) meridian of the zone with the equator (Fig. 15), the middle meridian of the zone corresponds to

Rice. fifteen.The 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 such an arrangement of the coordinate axes, the abscissas of points located south of the equator and the ordinates of points located west of the middle meridian will have negative values. For the convenience of using coordinates on topographic maps, a conditional account of ordinates is adopted, excluding negative values ​​of ordinates. This is achieved by the fact that the ordinates are not counted from zero, but from the value 500 km, That is, the origin of coordinates in each zone is, as it were, moved by 500 km to the left along the axisY .In addition, to unambiguously determine the position of a point in rectangular coordinates on the globe to the coordinate valueYthe zone number is assigned to the left (one-digit or two-digit number).

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

X" \u003d X-, Y \u003d U- 500 000,

where x" and Y"-real values ​​of ordinates;X , Y -conditional values ​​of ordinates. For example, if the 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 accelerate target designation on a topographic map, in this case only tens and units of kilometers and meters are indicated. For example, the shortened coordinates of a given object would be:

X = 50 450; Y = 20 840.

Abbreviated coordinates cannot be used when targeting at the junction of coordinate zones and if the area of ​​​​action covers a space with a length 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 kilometers. The coordinate grid is intended for determining the coordinates of objects and drawing objects on the map by their coordinates, for target designation, map orientation, measurement of directional angles, and for approximate determination of distances and areas.

Table 5 Coordinate grids on maps

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

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

Kilometer lines on the maps are signed at their out-of-bounds exits and at several intersections inside the sheet (Fig. 16). The kilometer lines that are extreme on the map sheet are signed in full, the rest are abbreviated, with two digits (that is, only tens and units of kilometers are indicated). Signatures near the horizontal lines correspond to distances from the y-axis (equator) in kilometers. For example, the caption 6082 in the upper right corner shows that this line is 6082 from the equator km.

The vertical line captions indicate the zone number (one or two first digits) and the distance in kilometers (always three digits) from the origin of coordinates, conditionally 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 a scale of 1:25,000, 1:50,000, 1:100,000, and 1:200,000 at the exits of kilometer lines in the adjacent western or eastern zone. The exits of kilometer lines in the form of dashes with the corresponding signatures are given on maps located over a distance of 2 ° to the 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 convert the coordinates of one zone into the coordinate system of another, neighboring, zone.

On fig. 17 dashes on the outer side of the western frame with signatures 81.6082 and on the north side of the frame with signatures 3693, 94, 95, etc. denote the exits of kilometer lines in the coordinate system of the adjacent (third) zone. If necessary, an additional coordinate grid is drawn on the map sheet by connecting dashes 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 (merge) with it when gluing the map.

Coordinate grid of the western (3rd) zone

Rice. 17. Additional coordinate grid

Chapter I. Vectors on the plane and in space

§ 13. Transition from one rectangular Cartesian coordinate system to another

We offer you to consider this topic in two versions.

1) Based on the textbook by I.I. Privalov "Analytical Geometry" (textbook for higher technical educational institutions, 1966)

I.I. Privalov "Analytical geometry"

§ 1. The problem of coordinate transformation.

The position of a point on the plane is determined by two coordinates relative to some coordinate system. The point's coordinates will change if we choose a different coordinate system.

The task of transforming coordinates is to to, knowing the coordinates of a point in one coordinate system, find its coordinates in another system.

This problem will be solved if we establish formulas that relate the coordinates of an arbitrary point in two systems, and the coefficients of these formulas will include constant values ​​that determine the mutual position of the systems.

Let two Cartesian coordinate systems be given hoy and XO 1Y(Fig. 68).

Position of the new system XO 1Y relative to the old system hoy will be determined if the coordinates are known a and b new beginning O 1 according to the old system and the angle α between axles Oh and About 1 X. Denote by X and at coordinates of an arbitrary point M relative to the old system, through X and Y-coordinates of the same point relative to the new system. Our task is to make the old coordinates X and at expressed in terms of the new X and Y. The resulting transformation formulas must obviously include the constants a, b and α .

We will obtain the solution of this general problem by considering two special cases.

1. The origin of coordinates changes, while the directions of the axes remain unchanged ( α = 0).

2. The directions of the axes change, while the origin of coordinates remains unchanged ( a = b = 0).

§ 2. Transfer of the origin.

Let two systems of Cartesian coordinates with different origins be given O and O 1 and the same directions of the axes (Fig. 69).

Denote by a and b coordinates of a new beginning About 1 in the old system and through x, y and X, Y-coordinates of an arbitrary point M, respectively, in the old and new systems. Projecting point M on the axis About 1 X and Oh, as well as the point About 1 per axle Oh, we get on the axis Oh three dots Oh, a and R. Segment values OA, AR and OR are related by the following relation:

| OA| + | AR | = | OR |. (1)

Noticing that | | OA| = a , | OR | = X , | AR | = | O 1 R 1 | = X, we rewrite equality (1) in the form:

a + X = x or x = X + a . (2)

Similarly, projecting M and About 1 on the y-axis, we get:

y = Y + b (3)

So, the old coordinate is equal to the new one plus the coordinate of the new origin according to the old system.

From formulas (2) and (3), the new coordinates can be expressed in terms of the old ones:

X = x - a , (2")

Y = y-b . (3")

§ 3. Rotation of coordinate axes.

Let two Cartesian coordinate systems with the same origin be given O and different directions of the axes (Fig. 70).

Let α is the angle between the axes Oh and OH. Denote by x, y and X, Y coordinates of an arbitrary point M, respectively, in the old and new systems:

X = | OR | , at = | RM | ,

X= | OR 1 |, Y= | R 1 M |.

Consider a broken line OR 1 MP and take its projection onto the axis Oh. Noticing that the projection of the broken line is equal to the projection of the closing segment (Chapter I, § 8), we have:

OR 1 MP = | OR |. (4)

On the other hand, the projection of a broken line is equal to the sum of the projections of its links (Chapter I, § 8); therefore, equality (4) will be written as follows:

etc OR 1+ pr R 1 M+ pr MP= | OR | (4")

Since the projection of a directed segment is equal to its value multiplied by the cosine of the angle between the projection axis and the axis on which the segment lies (Chapter I, § 8), then

etc OR 1 = X cos α

etc R 1 M = Y cos (90° + α ) = - Y sin α ,

pr MP= 0.

Hence equality (4") gives us:

x = X cos α - Y sin α . (5)

Similarly, projecting the same broken line onto the axis OU, we get an expression for at. Indeed, we have:

etc OR 1+ pr R 1 M+ pr MP= pr OR = 0.

Noticing that

etc OR 1 = X cos ( α - 90°) = X sin α ,

etc R 1 M = Y cos α ,

pr MP = - y ,

will have:

X sin α + Y cos α - y = 0,

y = X sin α + Y cos α . (6)

From formulas (5) and (6) we obtain new coordinates X and Y expressed through old X and at , if we solve equations (5) and (6) with respect to X and Y.

Comment. Formulas (5) and (6) can be obtained differently.

From fig. 71 we have:

X = OP = OM cos ( α + φ ) = OM cos α cos φ - OM sin α sin φ ,

at = PM = OM sin ( α + φ ) = OM sin α cos φ + OM cos α sin φ .

Since (Ch. I, § 11) OM cos φ = X, OM sin φ =Y, then

x = X cos α - Y sin α , (5)

y = X sin α + Y cos α . (6)

§ 4. General case.

Let two Cartesian coordinate systems with different origins and different directions of the axes be given (Fig. 72).

Denote by a and b coordinates of a new beginning O, according to the old system, through α - the angle of rotation of the coordinate axes and, finally, through x, y and X, Y- coordinates of an arbitrary point M, respectively, according to the old and new systems.

To express X and at through X and Y, we introduce an auxiliary coordinate system x 1 O 1 y 1 , whose beginning we place at the new beginning O 1 , and take the directions of the axes to coincide with the directions of the old axes. Let x 1 and y 1 denote the coordinates of the point M relative to this auxiliary system. Passing from the old coordinate system to the auxiliary one, we have (§ 2):

X = X 1 + a , y = y 1 +b .

X 1 = X cos α - Y sin α , y 1 = X sin α + Y cos α .

Replacing X 1 and y 1 in the previous formulas by their expressions from the last formulas, we finally find:

x = X cos α - Y sin α + a

y = X sin α + Y cos α + b (I)

Formulas (I) contain, as a special case, the formulas of §§ 2 and 3. Thus, for α = 0 formulas (I) turn into

x = X + a , y = Y + b ,

and at a = b = 0 we have:

x = X cos α - Y sin α , y = X sin α + Y cos α .

From formulas (I) we obtain new coordinates X and Y expressed through old X and at if equations (I) are solvable with respect to X and Y.

We note a very important property of formulas (I): they are linear with respect to X and Y, i.e. of the form:

x = AX+BY+C, y = A 1 X+B 1 Y+C 1 .

It is easy to check that the new coordinates X and Y expressed through the old X and at also formulas of the first degree with respect to X and y.

G.N. Yakovlev "Geometry"

§ 13. Transition from one rectangular Cartesian coordinate system to another

By choosing a rectangular Cartesian coordinate system, a one-to-one correspondence is established between the points of the plane and ordered pairs of real numbers. This means that each point of the plane corresponds to a single pair of numbers, and each ordered pair of real numbers corresponds to a single point.

The choice of one or another coordinate system is not limited by anything and is determined in each particular case only by considerations of convenience. Often the same set has to be considered in different coordinate systems. One and the same point in different systems obviously has different coordinates. A set of points (in particular, a circle, a parabola, a straight line) in different coordinate systems is given by different equations.

Let us find out how the coordinates of the points of the plane are transformed in the transition from one coordinate system to another.

Let two rectangular coordinate systems be given on the plane: O, i, j and about", i",j" (Fig. 41).

The first system with origin at point O and basis vectors i and j we agree to call the old one, the second - with the beginning at the point O" and the basis vectors i" and j" - new.

We will consider the position of the new system relative to the old one to be known: let the point O" in the old system have coordinates ( a;b ), a vector i" forms with vector i corner α . Corner α counting in the opposite direction of the clockwise movement.

Consider an arbitrary point M. Denote its coordinates in the old system through ( x;y ), in the new one - through ( x"; y" ). Our task is to establish the relationship between the old and new coordinates of the point M.

Connect in pairs the points O and O", O" and M, O and M. According to the triangle rule, we obtain

OM > = OO" > + O"M > . (1)

Let's decompose the vectors OM> and OO"> by basis vectors i and j , and the vector O"M> by basis vectors i" and j" :

OM > = x i+y j , OO" > = a i+b j , O"M > = x" i"+y" j "

Now equality (1) can be written as follows:

x i+y j = (a i+b j ) + (x" i"+y" j "). (2)

New basis vectors i" and j" expanded over the old basis vectors i and j in the following way:

i" = cos α i + sin α j ,

j" = cos( π / 2 + α ) i + sin ( π / 2 + α ) j = - sin α i + cos α j .

Substituting the found expressions for i" and j" into formula (2), we obtain the vector equality

x i+y j = a i+b j + X"(cos α i + sin α j ) + at"(-sin α i + cos α j )

equivalent to two numerical equalities:

x = a + X" cos α - at" sin α , at = b+ X" sin α + at" cos α

Formulas (3) give the desired expressions for the old coordinates X and at points through its new coordinates X" and at". In order to find expressions for the new coordinates in terms of the old ones, it is sufficient to solve the system of equations (3) with respect to the unknowns X" and at".

So, the coordinates of the points when moving the origin to the point ( a; b ) and rotate the axes by an angle α are transformed by formulas (3).

If only the origin of coordinates changes, and the directions of the axes remain the same, then, assuming in formulas (3) α = 0, we get

Formulas (5) are called rotation formulas.

Task 1. Let the coordinates of the new beginning in the old system be (2; 3), and the coordinates of point A in the old system (4; -1). Find coordinates of point A in new system if the directions of the axes remain the same.

By formulas (4) we have

Answer. A(2;-4)

Task 2. Let the coordinates of the point P in the old system (-2; 1), and in the new system, the directions of the axes of which are the same, the coordinates of this point (5; 3). Find the coordinates of the new beginning in the old system.

And According to formulas (4), we obtain

- 2= a + 5 1 = b + 3

where a = - 7, b = - 2.

Answer. (-7; -2).

Task 3. Point A coordinates in the new system (4; 2). Find the coordinates of this point in the old system, if the origin remains the same, and the coordinate axes of the old system are rotated by an angle α = 45°.

By formulas (5) we find

Task 4. The coordinates of point A in the old system (2 √3 ; - √3 ). Find the coordinates of this point in the new system, if the origin of the old system is moved to the point (-1;-2), and the axes are rotated by an angle α = 30°.

By formulas (3) we have

Solving this system of equations for X" and at", we find: X" = 4, at" = -2.

Answer. A(4;-2).

Task 5. Given the equation of a straight line at = 2X - 6. Find the equation of the same line in the new coordinate system, which is obtained from the old system by rotating the axes by an angle α = 45°.

The rotation formulas in this case have the form

Replacing the straight line in the equation at = 2X - 6 old variables X and at new, we get the equation

√ 2 / 2 (x" + y") = 2 √ 2 / 2 (x" - y") - 6 ,

which, after simplifications, takes the form y" = x" / 3 - 2√2

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 elevation systems that also determine the altitude location of a point on

What are coordinates

Coordinates are numeric or literal values ​​that can be used to determine the location of a point on the terrain. 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, locating a point in a given space is the main objective upon which all subsequent work is based.

Most coordinate systems, as a rule, 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 system of heights 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. Geographic.
3. Polar.
4. Rectangular.
5. Zonal Gauss-Kruger coordinates.

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

Geodetic coordinates

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

An ellipsoid is a three-dimensional compressed figure that the best way represents the figure of the globe. Due to the fact that the globe is a mathematically incorrect figure, it is the ellipsoid that is used instead to determine geodetic coordinates. This facilitates the implementation of many calculations to determine the position of the 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 the end lies at the perpendicular drawn to the desired point.
2. Geodetic longitude is the angle that is measured from the zero meridian to the meridian on which the desired point is located.
3. Geodetic height - the value of the normal drawn to the surface of the ellipsoid of the Earth's rotation from a given point.

Geographical coordinates

To solve high-precision problems of higher geodesy, it is necessary to distinguish between geodetic and geographical coordinates. In the system used in engineering geodesy, such differences, due to the small space covered by the work, as a rule, do not.

An ellipsoid is used as a reference plane to determine geodetic coordinates, and a geoid is used to determine geographical coordinates. The geoid is a mathematically incorrect figure, closer to the actual figure of the Earth. For its leveled surface, they take that which is continued 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 geodesic, since the reference point will also be called Greenwich. It passes through the observatory of the same name in the city of 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 the heights are determined is the Kronstadt footstock. It is located at the level of the Baltic Sea.

Polar coordinates

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

Polar coordinates are defined by two quantities: angle and distance. The angle is measured from the north 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. With these two options, you can determine the location of the point in the local system.

Typically, this coordinate system is used to perform engineering work held in a small area.

Rectangular coordinates

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

The north direction of the x-axis and the east of the y-axis are considered positive, and the south and west are 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 territory of the globe, and not just to small areas.

The rectangular coordinates of the Gauss-Kruger zones, in fact, are the projection of the globe onto a plane. It arose for practical purposes to depict large areas of the Earth on paper. Transferring distortions are considered insignificant.

According to this system, the globe is divided by longitude into six-degree zones with the 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 area of ​​the earth's equator Y. To unambiguously determine the location on the territory of the entire globe, the zone number is put in front of the X and Y values.

The values ​​of the x-axis in Russia are usually positive, while the values ​​of y can be negative. In order to avoid the minus sign in the values ​​of the abscissa axis, the axial meridian of each zone is conditionally moved 500 meters to the west. Then all coordinates become positive.

The coordinate system was proposed by Gauss as possible and calculated mathematically by Krüger in the middle of the twentieth century. Since then, it has been used in geodesy as one of the main ones.

Height system

The systems of coordinates and heights 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 original. In addition, there are relative heights. The latter are counted as an excess from the desired point to any other. It is convenient to use them for working in the local coordinate system in order to simplify the 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 their own areas of work for which this or that method of determining the location is relevant.

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