A method for triangulating goals. Geodetic networks. Triangulation method. Angular measurements How it works in practice



Triangulation(from Latin triangulum - triangle) - one of the methods for creating a geodetic reference network.
Triangulation- a method of constructing horizontal structures on the ground in the form of triangles, in which all angles and basic output sides are measured (Fig. 14.1). The lengths of the remaining sides are calculated using trigonometric formulas (for example, a=c . sinA/sinC, b=c . sinA/sinB), then the directional angles (azimuths) of the sides are found and the coordinates are determined.

It is generally accepted that the triangulation method was invented and first used by W. Snell in 1615–17. when laying out a series of triangles in the Netherlands for degree measurements. Work on the use of the triangulation method for topographic surveys in pre-revolutionary Russia began at the turn of the 18th–19th centuries. By the beginning of the 20th century. The triangulation method has become widespread.
Triangulation is of great scientific and practical importance. It serves to: determine the shape and size of the Earth using the method of degree measurements; studying horizontal movements of the earth's crust; justification of topographic surveys at various scales and purposes; justification of various geodetic works in the survey, design and construction of large engineering structures, in the planning and construction of cities, etc.

In practice, it is allowed to use the polygonometry method instead of triangulation. In this case, the condition is set that when constructing a reference geodetic network using this and other methods, the same accuracy in determining the position of points on the earth’s surface is achieved.

The vertices of the triangulation triangles are marked on the ground by wooden or metal towers with a height of 6 to 55 m, depending on terrain conditions (see Geodetic signal). For the purpose of their long-term preservation on the ground, triangulation points are secured by placing special devices in the ground in the form of metal pipes or concrete monoliths with metal marks embedded in them (see Geodetic center), fixing the position of points for which coordinates are given in the corresponding catalogs.

3) Satellite topographic survey

Satellite photography is used to compile topographic maps of an overview or small scale. Satellite GPS measurements are very accurate. But in order to avoid the use of this system for military needs, the accuracy was reduced from
Topographic surveys using global navigation satellite systems make it possible to depict the following objects on topographic plans at scales of 1:5000, 1:2000, 1:1000 and 1:500 with the necessary reliability and accuracy:

1) points of triangulation, polygonometry, trilateration, ground benchmarks and survey justification points fixed on the ground (marked by coordinates);
2) industrial facilities - drilling and production wells, oil and gas rigs, above-ground pipelines, wells and underground communication networks (during as-built survey);
3) railways, highways and dirt roads of all types and some structures attached to them - crossings, crossings, etc.;
4) hydrography - rivers, lakes, reservoirs, spill areas, tidal strips, etc. Coastlines are drawn according to the actual state at the time of surveying or at low water;
5) hydraulic engineering and water transport facilities - canals, ditches, water conduits and water distribution devices, dams, piers, moorings, piers, locks, etc.;
6) water supply facilities - wells, standpipes, reservoirs, settling tanks, natural springs, etc.;
7) terrain using contours, elevation marks and symbols of cliffs, craters, screes, ravines, landslides, glaciers, etc. Microrelief forms are depicted by semi-horizontals or auxiliary contours with terrain elevation marks;
8) shrubby, herbaceous, cultivated vegetation (plantations, meadows, etc.), free-standing bushes;
9) soils and microforms of the earth’s surface: sands, pebbles, takyrs, clayey, crushed stone, monolithic, polygonal and other surfaces, swamps and salt marshes;
10) boundaries - political and administrative, land use and nature reserves, various fences.
The many GPS devices on the market today allow specialists to take careful measurements when laying roads, constructing various structures, measuring land area, creating terrain maps for oil production, etc.
The use of computer modeling methods and the perfection of calculations perfectly complement topographic survey.

The meaning of the word "Triangulation (in geodesy)" in the Great Soviet Encyclopedia

Triangulation(from Latin triangulum - triangle), one of the methods for creating a network of support geodetic points and the network itself created by this method; consists of constructing rows or networks of triangles adjacent to each other and determining the position of their vertices in a chosen coordinate system. In each triangle, all three angles are measured, and one of its sides is determined from calculations by sequentially solving previous triangles, starting from the one in which one of its sides is obtained from measurements. If the side of a triangle is obtained from direct measurements, then it is called the base side Triangulation (in geodesy) In the past, instead of the base side, a short line called the basis was directly measured, and from this, through trigonometric calculations, through a special network of triangles, they passed to the side of the triangle Triangulation (in geodesy) This side Triangulation (in geodesy) is usually called the output side, and the network of triangles through which it is calculated is called the basis network. In rows or networks Triangulation (in geodesy) to control and improve their accuracy, measure a larger number of bases or base sides than is minimally necessary.

It is generally accepted that the method Triangulation (in geodesy) invented and first used V. Snellius in 1615-17 when laying a series of triangles in the Netherlands for degree measurements . Work on the application of the method Triangulation (in geodesy) for topographic surveys in pre-revolutionary Russia began at the turn of the 18th–19th centuries. By the beginning of the 20th century. method Triangulation (in geodesy) has become widespread.

Triangulation (in geodesy) has great scientific and practical significance. It serves to: determine the shape and size of the Earth using the method of degree measurements; studying horizontal movements of the earth's crust; justification of topographic surveys at various scales and purposes; justification of various geodetic works in the survey, design and construction of large engineering structures, in the planning and construction of cities, etc.

When building Triangulation (in geodesy) They proceed from the principle of transition from the general to the specific, from large triangles to smaller ones. Due to this Triangulation (in geodesy) is divided into classes that differ in the accuracy of measurements and the sequence of their construction. In small countries Triangulation (in geodesy) the highest class are built in the form of continuous networks of triangles. In countries with a large territory (USSR, Canada, China, USA, etc.) Triangulation (in geodesy) built according to some scheme and program. The most harmonious scheme and construction program Triangulation (in geodesy) used in the USSR.

State Triangulation (in geodesy) in the USSR it is divided into 4 classes ( rice. ). State Triangulation (in geodesy) USSR 1st class is built in the form of rows of triangles with sides 20-25 km, located approximately along the meridians and parallels and forming polygons with a perimeter of 800-1000 km. The angles of the triangles in these rows are measured with high precision theodolites , with an error of no more than ± 0.7 " . At the intersection of rows Triangulation (in geodesy) 1st class measures bases using measuring wires (see. Basic device ), and the measurement error of the basis does not exceed 1: 1,000,000 of a fraction of its length, and the output sides of the basis networks are determined with an error of about 1: 300,000. After the invention of high-precision electro-optical rangefinders began to measure directly the base sides with an error of no more than 1: 400,000. Spaces inside polygons Triangulation (in geodesy) 1st class is covered with continuous networks of 2nd class triangles with sides of about 10-20 km, and the angles in them are measured with the same accuracy as in Triangulation (in geodesy) 1st class. In a continuous network Triangulation (in geodesy) 2nd class inside a 1st class polygon, the base side is also measured with the accuracy indicated above. At the ends of each base side in Triangulation (in geodesy) Classes 1 and 2 perform astronomical determinations of latitude and longitude with an error of no more than ± 0.4 " , as well as azimuth with an error of about ± 0.5 " . In addition, astronomical determinations of latitude and longitude are also performed at intermediate points of the rows Triangulation (in geodesy) 1st class every approximately 100 km, and in some specially selected rows and much more often.

Based on rows and networks Triangulation (in geodesy) 1st and 2nd classes are determined by points Triangulation (in geodesy) 3rd and 4th classes, and their density depends on the scale of the topographic survey. For example, with a shooting scale of 1: 5000, one point Triangulation (in geodesy) should occur every 20-30 km 2. IN Triangulation (in geodesy) Class 3 and 4 angle measurement errors do not exceed 1.5, respectively " and 2.0 " .

In USSR practice, it is allowed instead Triangulation (in geodesy) apply the method polygonometry . In this case, the condition is set that when constructing a reference geodetic network using this and other methods, the same accuracy in determining the position of points on the earth’s surface is achieved.

Vertices of triangles Triangulation (in geodesy) are marked on the ground by wooden or metal towers with a height of 6 to 55 m depending on terrain conditions (see Geodetic signal ). Items Triangulation (in geodesy) for the purpose of their long-term preservation on the ground, they are fixed by placing special devices in the ground in the form of metal pipes or concrete monoliths with metal marks embedded in them (see. Geodetic center ), fixing the position of points for which coordinates are given in the corresponding catalogs.

Point coordinates Triangulation (in geodesy) determined from mathematical processing of series or networks Triangulation (in geodesy) In this case, the real Earth is replaced by some reference ellipsoid , onto the surface of which the results of measuring angles and base sides are given Triangulation (in geodesy) In the USSR, Krasovsky’s reference ellipsoid was adopted (see. Krasovsky ellipsoid ). Construction Triangulation (in geodesy) and its mathematical processing lead to the creation of a unified coordinate system throughout the country, allowing topographic and geodetic work to be carried out in different parts of the country simultaneously and independently of each other. At the same time, it is ensured that these works are combined into one whole and the creation of a unified national topographic map of the country on the established scale.

Lit.: Krasovsky F.N., Danilov V.V., Guide to higher geodesy, 2nd ed., part 1, century. 1-2, M., 1938-39; Instructions on the construction of the state geodetic network of the USSR, 2nd ed., M., 1966.

L. A. Izotov.

Doug Struve, named after its creator - the Russian astronomer Friedrich Georg Wilhelm Struve (Vasily Yakovlevich Struve) - a network of 265 triangulation points, which were stone cubes embedded in the ground with an edge length of 2 meters, with a length of more than 2820 kilometers. It was created to determine the parameters of the Earth, its shape and size.

Geodetic point

Geodetic point- a point fixed in a special way on the ground (in the ground, less often on a building or other artificial structure), and is the carrier of coordinates determined by geodetic methods. A geodetic point is an element of a geodetic network, which serves as the geodetic basis for topographic surveying of the area and a number of other geodetic works, and according to its purpose is divided into planning (trigonometric), high-altitude (leveling) and gravimetric. A class 1 planned network, the elements of which are also determined by astronomical and gravimetric methods, is called astronomical-geodetic.

Recently, work has been carried out to create a new satellite geodetic network (primarily in industrialized and populated areas), with satellite geodetic network points fixed on the ground, the coordinates of which are determined by relative methods of space geodesy. Whenever possible, such points are combined with existing points of old geodetic networks, and the created satellite network is subject to rigid binding to existing geodetic points. In addition, geodetic points also include points for special purposes. These are points for laser ranging of satellites, ultra-long-baseline radio interferometry, points for the Earth's rotation service and some others.

Therefore, geodetic points belonging to these networks have different purposes.

Items planned geodetic network are carriers of plan coordinates that are defined in a known coordinate system with a given degree of accuracy, as a result of geodetic measurements. Traditional geodetic methods for determining the coordinates of planned (trigonometric) geodetic points are triangulation (then such a point is called a triangulation point or triangulation point), polygonometry (then such a point is called a polygonometry point or polygonometric point), trilateration (then such a point is called a trilateration point), or their combination (then it is called a point of the linear-angular network). They are located, if possible, on elevated places (tops of hills, hills, mountains) to ensure visibility to neighboring network points in all directions. The points of the planned geodetic network are also determined by height above sea level, but the accuracy of determination in height is lower than the accuracy of determination in plan, as a result of technological differences in determination methods.

Items high-altitude geodetic network are carriers of altitude coordinates determined with great accuracy by the method of geometric leveling. Therefore, such points are also called leveling points(the centers of the leveling points are called benchmarks) . In the plan they are defined only approximately. There is no need for mutual visibility between leveling points, and the measurement technology requires the location of these points, if possible, in flat places (most often along rivers), since with the presence of a height difference the accuracy of determination is lost. For this reason, as a rule, the points of the trigonometric network do not coincide with the leveling points (leveling points).

At points gravimetric network gravity deviations are determined. The parameters of such points are determined using a special device - a gravimeter. Gravimetric points are also determined in plan and height, with a certain degree of accuracy.

Each geodetic point is fixed by a special geodetic center, to which the coordinates of the geodetic point are given (at leveling points the geodetic centers are called benchmarks or marks). (Points of the satellite network and other special networks are assigned by centers or groups of centers of a special design). A geodetic sign is constructed above the center of a point of a trigonometric (planned) network - a ground structure (wooden, metal, stone or reinforced concrete), in the form of a tour, a tripod, a pyramid, a geodetic pyramid or a geodetic signal, which serves to secure the sighting target, install a geodetic instrument, and is platform for the observer to work. Also serves to identify a point on the ground. At a certain distance from the trigonometric point, reference points are laid with the front panel facing the geodetic point itself, and an astronomical pillar is also built (if astronomical determinations are carried out at the point). In addition, the geodetic point has a special external design. If it is economically beneficial, the sign at the point can be constructed temporary (dismountable or transportable).

At points of other geodetic networks (altitude and gravimetric) the sign is not constructed, since according to the definition technology it is not used. In this case, to secure and identify a point on the ground, an identification pole (metal, reinforced concrete) with a security sign is used, and special external design of the point, determined by the “Instructions for the construction of geodetic marks” (trenching with ditches, creating stone shafts, filling a mound, etc.).

Therefore, most often it is the planning (trigonometric) point with its large and noticeable sign located somewhere on a hill that the average person associates with the concept of “geodesic point”.

Each geodetic point of the State Geodetic Network has an individual number printed on the center stamp and entered into a special catalog. In addition, each point of the planned State network is assigned a name, which is entered into the appropriate catalogs indicating all the parameters of the point. The names of some trigopoints are shown on the topographic map next to their symbols.

Trigonometric point

Material from Wikipedia - the free encyclopedia


Japan First Class Geodetic Network Trigonometric Sign Element

Trigonometric point, trigopoint (triangulation point) is a geodetic point, the plan coordinates of which are determined by trigonometric methods.
This term is not official. This is a professional collective term in geodesy to separate the concept of a planned geodetic point, determined by trigonometric methods, from high-altitude, astronomical and others, since the purpose of the latter is different.
To determine coordinates, methods of triangulation, polygonometry,

When designing triangulation networks, the requirements given in Table 1 must be met.

Table 1

Index Class
Average length of a triangle side, km 20-25 7-20 5-8 2-5
Relative error of base output side 1:400000 1:300000 1:200000 1:100000
Approximate relative error of the party at the weak point 1:150000 1:200000 1:120000 1:70000
Smallest angle of a triangle, degree 40 20 20 20
Allowable triangle discrepancy, angle. With 3 4 6 6
Average square error of angle based on triangle residuals, ang. With 0,7 1 1,5 2,0
Mean square error of the relative position of adjacent points, m 0,15 0,06 0,06 0,06

3.1. Calculation of the number of characters

When designing a triangulation network of classes 3 and 4, it is necessary to calculate the number of points of a separate class.

The required density of geodetic points for national mapping of the country's territory depends on the scale of topographic survey, methods of its implementation, as well as on methods for creating survey geodetic justification.

table 2

The following approximate relationships must be observed between the lengths of the sides of triangles of different classes:

s 1= s 1 s 2 =0.58s 1 s 3 =0.33s 1 s 4 =0.19s 1. (1)

If we take the initial length of the side in the 1st class triangulation, equal on average to S 1 = 23 km, then using formulas (1) we obtain the following lengths of the sides of the triangles in the 2-4 class triangulation networks (Table 3).

Table 3

In real triangulation networks, triangles deviate somewhat from the equilateral shape. However, on average, for an extensive geodetic network, the ratios (1) of the lengths of the sides of the triangles must be more or less accurately observed, otherwise the total number of points in the network may turn out to be unjustifiably inflated. Average number of points of different classes in any area R mapped territory can be calculated using the formulas

where is the area served by one point of the th class ( i=1,2,3,4). Calculation results should be rounded to the nearest ten. As an example, using these formulas we will determine the number of class 3-4 points in the area P = 200 km 2 with n 1 = 0, n 2 = 2.

For class 3 triangulation:

For class 4 triangulation:

Consequently, on the area of ​​the surveyed territory P = 200 km 2, 11 points should be designed, that is, 2 points of class 2, 2 points of class 3 and 7 points of class 4.

3.2. Construction of a triangulation network

When developing a graphical network design, special attention should be paid to the choice of location for each individual point. All points of the state geodetic network must be located on the commanding peaks of the area. This is necessary in order, firstly, to ensure mutual visibility between adjacent points with minimal heights of geodetic signs, and secondly, the possibility of developing the network in any direction in the future. The lengths of the sides between adjacent points must comply with the requirements of the instructions. In all cases, geodetic points must be located in places where the safety of their position in plan and height will be ensured for a long time. Since on average 50-60% of all costs for creating a network are spent on the construction of geodetic signs, it is necessary to pay the most serious attention to the choice of locations for installing points on the ground in order to reduce their height.

When designing triangulation networks of different classes, it is important to ensure reliable connection of lower-class networks to higher-class networks.

Rice. 1. Schemes for linking geodetic networks to the sides (a) and points (b) of the highest class triangulation

Fig.2. Schemes for constructing triangulation networks

After all points are plotted on the map, they are connected by straight lines. On a separate sheet, a diagram of the designed network is drawn, on which the names of the points, the lengths of the sides in kilometers, the values ​​of angles in triangles accurate to a degree, and the height of the earth’s surface accurate to a meter. Angles are measured using a protractor using a topographic map. The sums of angles in triangles should be equal to 180º, and at the pole of the central system 360º. The lengths of the sides are measured with a ruler. Below the diagram are symbols of the source sides, triangulation sides and network points.

3.3. Calculation of sign heights

At points of the geodetic network, geodetic signs are built of such a height that the sighting rays during angular and linear measurements pass in each direction at a given minimum height above the obstacle without touching it. First, determine the approximate heights of the signs l 1' and l 2 ' for each pair of adjacent points, and then correct them and find the final height values l 1 and l 2 . Approximate sign heights l 1' and l 2 ' (Fig. 3) is calculated using the formulas

Where h 1 And h 2- the excess of the top of the obstacle at point C (taking into account the height of the forest) above the bases of the first and second signs, respectively; A- the permissible height of the origin of the targeting beam above the obstacle established by the current instructions; u 1 And u 2- corrections for the curvature of the Earth and refraction.

Signs when h 1 And h 2 determined by the signs of the differences

h 1=H c -H 1,

h 2 = Hc-H2,(5)

Where N s- height of the top of the obstacle at a point WITH; H 1 And H 2- the height of the earth's surface in the places where the first and second signs are installed.

Fig.3. Scheme for determining the height of geodetic signs

Corrections v for the curvature of the Earth and refraction are calculated using the formula

where k is the coefficient of terrestrial refraction; R is the radius of the Earth; s is the distance from the obstacle to the corresponding point. At k = 0.13 and R=6371 km, formula (6) will take the form

V=0.068s 2 , (7)

where v is in meters and s is in kilometers.

In the event that excess h 1 And h 2 have the same sign, but the distances s 1 and s 2 are significantly different, the heights of the signs l' 1 and l’ 2 calculated using formulas (4) will differ significantly from each other: one sign is low, and the other is excessively high (Fig. 4). It is not economically profitable to build tall signs. Therefore, the heights of the signs calculated using formulas (4) must be adjusted so that the sum of the squares of the final heights of the signs l 1 and l 2 was the smallest, i.e. = min. If this requirement is met, the cost of building a given pair of signs will, as a rule, be the least, since the cost of building each sign, other things being equal, is almost proportional to the square of its height.

The adjusted heights of each pair of signs at the ends of the side, subject to the condition = min and the requirement that the sighting beam passes at a given height a above the obstacle, are calculated using the formulas

Fig.4. Scheme for adjusting the height of a geodetic sign

At a point with n directions, n values ​​of sign height will be obtained, since calculations for each individual side (direction) will give different values ​​of sign height at a given point. The final height is taken to be the one at which visibility is ensured in all directions at the minimum (permissible) height of the passage of the sighting rays over obstacles. The results of calculating the heights of geodetic signs are presented in Table 4.

Table 4

Name of points Distances s 1 and s 2 Heights N,m Exceeding h 1 and h 2 v, m a,m Approximate heights l 1 ’ and l 2 ’ Corrected heights Standard sign heights
Liskino 2,4 137,5 3,5 0,4 1,0 4,9 6,2
WITH 141,0
Popovo 5,2 138,2 2,8 1,8 1,0 5,6 2,8

For the most difficult sides, construct profiles on which, in addition to the ground surface, show the new visibility after installing the geodetic sign with a red line.

3.4. Pre-calculation of the accuracy of triangulation network elements

To confidently use the final version of the geodetic network design, it is necessary to have reliable numerical characteristics of its weak elements. Using the diagram we have compiled, we find the weaknesses of the network. The weak side is located according to the principle equal to its distance from the original side.

The mean square error of the measured values ​​is taken as an accuracy criterion

where µ is the mean square error of a unit of weight;

Р F – weight of the function under consideration.

The error of the measured values ​​is taken as the error of the unit of weight. Since the network is still being designed, the angles and lengths involved in the pre-calculation are determined from the topographic map.

The mean square error of the weak side of an n-triangle included in the central system or geodesic quadrilateral is determined by the formula

where m lgb is the mean square error of the logarithm of the original side;

m β - root mean square error of angle measurement in the considered triangulation class;

R i is the error in the geometric connection of the triangle.

The mean square error of the weak side of an n-triangle, which is an element of a simple chain of triangles, is determined by the formula

The geometric connection error is calculated using the formula:

R i =δ 2 A i + δ 2 V i + δ A i * δ B i, (12)

where A i and B i are connecting angles in triangles;

δ A i, δ B i - increments of the logarithms of the sines of angles A and B when the angles change by 1" in units of the 6th sign of the logarithm. The value of δ can be determined by the formula

δ A i = МctgA i (1¤ρ")10 6 =2.11ctgA i . (13)

When pre-calculating the accuracy of the weak side from the mean square errors obtained from two moves, the average weight value is calculated using the formula:

where m logS 1 and m logS 2 mean square errors of determination from the basis for moves 1 and 2.

We find the relative error using the formula

Example. The designed class 3 triangulation network consists of a central system (Fig. 5). The weak side is “Klenovo-Zavikhrastovo”; we will perform a pre-calculation of its accuracy; the results of calculating the error of the geometric connection for the first and second moves will be presented in Table 5.

Fig. 5. Network fragment

Table 5

Move 1 Move 2
№ A IN R i № A IN R i
5,44 5,05
5,62 5,40
6,28 4,81
Sum 17,34 Sum 15,25

m logS1 =5.11; m logS2 =4.86; m Sn(avg) =3.52;

Conclusion: The obtained relative error of the weak side satisfies the requirements of the instructions for a class 3 triangulation network.

Precalculation of accuracy in class 4 triangulation is performed in a similar way.

3.5. Calculating network quality in a rigorous way

We will calculate the quality of the network in a strict way using the example of the network shown in Fig. 6. For this network we have 9 independent conditional equations: 7 figure equations, 1 horizon condition, 1 pole conditional equation. The initial data are given in table. 6

Table 6

Item name Angle no. Angle, º δ Item name Angle no. Angle, º δ
A 0.68 F 1.08
1.71 J 1.17
B 0.73 1.37
1.27 1.65
C 1.37 O 0.60
0.60 1.12
D 1.59 1.97
1.71 1.32
E 1.59 1.03
1.17 1.48
0.98

Fig.6. Class 3 triangulation network

Conditional equations of figures:

(1) + (2) + (3) + W1 = 0

(4) + (5) + (6) + W2 = 0

(7) + (8) + (9) + W3 = 0

(10) + (11) + (12) + W4 = 0

(13) + (14) + (15) + W5 = 0

(16) + (17) + (18) + W6 = 0

(19) + (20) + (21) + W7 = 0

Conditional horizon equations

(1) + (5) + (8) + (11) + (14) + (17)+ W8 = 0

Pole conditional equations.

After logarithm, bringing to linear form, we will have

δ 2 (2)-δ 3 (3)+δ 4 (4)-δ 6 (6)+δ 7 (7)-δ 9 (9)+δ 10 (10)-δ 12 (12)+δ 13 (13)-δ 15 (15)+δ 16 (16)-δ 18 (18)+W9=0

To compile the weight function, we determine the weak side using a known basis.

Based on the resulting system of equations, we will compile a table of coefficients of conditional equations and a weight function (Table 7). The values ​​of δ n are calculated using the formula δ=2.11ctgβ.

Table 7

Conditional equation coefficients

No. a b c d e g h i k f s
+1 +1 -0.60 +1.40
+1 +1.59 +1.59 +4.18
+1 -1.59 -0.59
+1 +1.37 +2.37
+1 +1 +2.00
+1 -1.17 -0.17
+0.68
+1 +0.68 +1.68
+1 +1 +2.00
+1 -1.17 -0.17
0.7
+1 +0.73 +1.73
+1 +1 +1.32 +3.32
+1 -1.71 -1.71 -2.42
+1 +1.37 +1.37 +3.74
+1 +1 +2.00
+1 -1.27 -1.27 -1.54
+1 +1.71 +1.71 +4.42
+1 +1 +2.00
+1 -0.60 -0.60 -0.20
+1.00
+1 +1.00
+1 +1.00
+1 +1.00
Σ -0.06 1.81 28.75

Since we have a large number of conditional equations, it is most appropriate to calculate the inverse weight of the function using the two-group adjustment method. The inverse weight is calculated using the formula

where f are the coefficients of a given function for which the mean square error is found; a, b, … - coefficients of primary, secondary, etc. transformed equations of the second group; , , … - the sum of the coefficients of a given function according to those corrections of the first, second, etc. equations of figures of the first group, which are included in the expression of the function;

n 1, n 2, ... - the number of amendments included in the first, second, etc., respectively. equations of figures of the first group.

When dividing the equations into two groups, the first group includes all the equations of the figures (for our network, since there are no overlapping triangles). The second group will include all other equations and the weight function, i.e. equation of horizon, pole and function equation.

Table 8

Coefficients of conditional equations of the first group

No. a b c d e g h f
-0.60
1.59
=0.99
=0
=0
1.32
-1.71
=-0.39
1.37
-1.27
=0.10
1.71
-0.60
=1.11
=0

I= 2 /n 1 + …+ 7 /n 7 = 0.33+0.05+0.003+0.41=0.79

The converted coefficients are calculated using the formula

A=a-[a]/n; B=b-[b]/n,

where A, B – converted coefficients; n – number of angles included in the triangle; [a]/n – average value of untransformed coefficients in the triangle; [a] is the sum of untransformed coefficients in the triangle.

Table 9

Table of transformed equations of the second group and determination of coefficients of normal equations

N amendments i k I K f s
0,67 -0,60 0,07
1,59 -0,33 1,59 1,59 2,85
-1,59 -0,34 -1,59 -1,93
0,33
1,37 -0,33 1,30 0,97
0,67 -0,06 0,61
-1,17 -0,34 -1,24 -1,58
0,33 0,07
0,68 -0,33 ,84 0,51
0,67 0,17 0,84
-1,17 -0,34 -1,01 -1,35
0,33 -0,16
0,73 -0,33 1,06 0,73
0,67 0,32 1,32 2,31
-1,71 -0,34 -1,38 -1,71 -3,43
0,33 -0,33
1,37 -0,33 1,34 1,37 2,38
0,67 -0,04 0,63
-1,27 -0,34 -1,30 -1,27 -2,91
0,33 0,03
1,71 -0,33 1,34 1,71 2,72
0,67 -0,37 0,30
-0,60 -0,34 -0,97 -0,60 -1,91
0,33 0,37
author Dawkins Clinton Richard

Triangulation Linguists often wish to trace the history of languages. Where written evidence has been preserved, this is quite easy. The linguistic historian can use the second of our two methods of reconstruction, tracing the past of the reconstructed relics, in this

Triangulation

From the book An Ancestor's Story [Pilgrimage to the Origins of Life] author Dawkins Clinton Richard

Triangulation Linguists often need to reconstruct the history of languages. In cases where written sources have been preserved, this is quite simple. The historical linguist can use the second method of reconstruction by studying “biography.”

"Triangulation of Desire" 1890s

From the book Erotic Utopia: New Religious Consciousness and the Fin de Siècle in Russia author Matic Olga

"Triangulation of Desire" 1890s Throughout the 1890s. Gippius combined a virgin marriage with numerous intersecting love triangles. Her “relations” with men outside of marriage apparently also did not include intercourse and were “fictitious”, like her marriage. Despite

Coordinates (in geodesy)

TSB

Space triangulation

From the book Great Soviet Encyclopedia (KO) by the author TSB

Rectangular coordinates (in geodesy)

From the book Great Soviet Encyclopedia (PR) by the author TSB

What is triangulation? It should be noted that this word has several meanings. Thus, it is used in geometry, geodesy and information technology. Within the framework of the article, attention will be paid to all topics, but the most popular area will receive the most attention - use in technical equipment.

In geometry

So, let's begin to understand what triangulation is. What is this in geometry? Let's say we have a non-developable surface. But at the same time it is necessary to have an idea of ​​​​its structure. And to do this you need to expand it. Sounds impossible? But no! And the triangulation method will help us with this. It should be noted that its use provides the opportunity to construct only an approximate scan. The triangulation method involves the use of triangles adjacent to one another, where all three angles can be measured. In this case, the coordinates of at least two points must be known. The rest are to be determined. In this case, either a continuous network or a chain of triangles is created.

To obtain more accurate data, electronic computers are used. Separately, mention should be made of such a point as Delaunay triangulation. Its essence is that given the set of points, with the exception of the vertices, they all lie outside the circle that is described around the triangle. This was first described by the Soviet mathematician Boris Delaunay in 1934. His developments are used in the Euclidean traveling salesman problem, bilinear interpolation and This is what Delaunay triangulation is.

In geodesy

In this case, it is envisaged that a triangulation point is created, which is subsequently included in the network. Moreover, the latter is built in such a way that it resembles a group of triangles on the ground. All angles of the resulting figures are measured, as well as some basic sides. How surface triangulation will be carried out depends on the geometry of the object, the qualifications of the performer, the available fleet of instruments and technical and economic conditions. All this decides the level of complexity of the work that can be carried out, as well as the quality of its implementation.

In information networks

And we are gradually approaching the most interesting interpretation of the word “triangulation”. What is this in information networks? It should be noted that there are a large number of different options for interpretation and use. But within the framework of the article, due to the limitation of its size, only GPS (global positioning system) will receive attention. Despite certain similarities, they are quite different. And now we will find out what exactly it is.

Global Positioning System

More than a decade has passed since GPS was launched and is functioning successfully. The Global Positioning System consists of a central control station located in Colorado and observation posts around the world. During its work, several generations of satellites have already changed.

GPS is now a worldwide radio navigation system that is based on a number of satellites and earth stations. Its advantage is the ability to calculate the coordinates of an object with an accuracy of a few meters. How can triangulation be represented? What is it and how does it work? Imagine that every meter on the planet has its own unique address. And if there is a user receiver, then you can request the coordinates of your location.

How does this work in practice?

Conventionally, four main stages can be distinguished here. Initially, triangulation of the satellites is carried out. Then the distance from them is measured. Absolute measurement of time and determination of satellites in space are carried out. And finally, differential correction is carried out. That's it in short. But it is not entirely clear how triangulation works in this case. It is clear that this is not good. Let's get into detail.

So, initially to the satellite. It was found that it is 17 thousand kilometers. And the search for our location is significantly narrowed. It is known for sure that we are at a specific distance, and we must be looked for in that part of the earth’s sphere that is located 17 thousand kilometers from the detected satellite. But that is not all. We measure the distance to the second satellite. And it turns out that we are 18 thousand kilometers away from him. So, we should be looked for at the place where the spheres of these satellites intersect at a set distance.

Contacting a third satellite will further reduce the search area. And so on. The location is determined by at least three satellites. The exact parameters are determined according to the provided data. Let's assume that the radio signal moves at a speed close to light (that is, a little less than 300 thousand kilometers per second). The time it takes for it to travel from the satellite to the receiver is determined. If the object is at an altitude of 17 thousand kilometers, then it will be about 0.06 seconds. Then the position in the space-time coordinate system is established. Thus, each satellite has a clearly defined rotation orbit. And knowing all this data, the technology calculates the person’s location.

Specifics of the global positioning system

According to the documentation, its accuracy ranges from 30 to 100 meters. In practice, the use of differential correction makes it possible to obtain data detail down to centimeters. Therefore, the scope of application of the global positioning system is simply enormous. It is used to track the transportation of expensive cargo, helps to accurately land planes, and navigate ships in foggy weather. Well, the most famous is its use in automobile

Triangulation algorithms, due to their versatility and coverage of the entire planet, allow you to freely travel even to unfamiliar places. At the same time, the system itself paves the way, indicates where it is necessary to turn in order to get to the established final goal. Thanks to the gradual reduction in the cost of GPS, there are even car alarms based on this technology, and now if a car is stolen, it will not be difficult to find and return it.

What about mobile communications?

Here, alas, not everything is so smooth. While GPS can determine coordinates with an accuracy of up to a meter, triangulation in cellular communications cannot provide such quality. Why? The fact is that in this case the base station acts as the base station. It is believed that if there are two BSs, then you can get one of the phone coordinates. And if there are three of them, then the exact location is not a problem. This is partly true. But mobile phone triangulation has its own characteristics. But here the question of accuracy arises. Before this, we looked at a global positioning system that can achieve phenomenal accuracy. But, despite the fact that mobile communications have significantly more equipment, there is no need to talk about any kind of qualitative correspondence. But first things first.

Looking for answers

But first, let's formulate questions. Is it possible to determine the distance from the base station to the phone using standard means? Yes. But will this be the shortest distance? Who does the measurements - the phone or the base station? What is the accuracy of the data obtained? While servicing a conversation, the base station measures the time it takes for the signal to travel from it to the phone. Only in this case it can be reflected, say, from buildings. It should be understood that the distance is calculated in a straight line. And remember - only during the call service process.

Another significant disadvantage is the rather significant level of error. So, it can reach a value of five hundred meters. Mobile phone triangulation is further complicated by the fact that base stations do not know what devices are in the territory under their control. The device catches their signals, but does not inform itself. In addition, the phone is able to measure the base station signal (which, however, it constantly does), but the amount of attenuation is unknown to it. And here comes an idea!

Base stations know their coordinates and transmitter power. The phone can determine how well it can hear them. In this case, it is necessary to detect all stations that are operating, exchange data (for this you will need a special program that sends out test packets), collect coordinates and, if necessary, transfer them to other systems. It would seem that everything is in the bag. But, alas, for this it is necessary to make a number of modifications, including the SIM card, access to which is not at all guaranteed. And in order to turn a theoretical opportunity into a practical one, it is necessary to work significantly.

Conclusion

Despite the fact that almost all people have phones, one should not say that a person can be easily tracked. After all, this is not as easy as it might seem at first glance. You can more or less confidently talk about luck only when using a global positioning system, but it requires a special transmitter. In general, after reading this article, we hope that the reader no longer has any questions regarding what triangulation is.


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