Scale and its application. Scales of topographic maps and plans Scale 1 200 or 1

Map scale is the ratio of the length of a segment on the map to its actual length on the ground.

Scale ( from German - measure and Stab - stick) is the ratio of the length of a segment on a map, plan, aerial or satellite image to its actual length on the ground.

Let's consider the types of scales.

Numerical scale

This is a scale expressed as a fraction, where the numerator is one and the denominator is a number indicating how many times the image is reduced.

Numerical scale is a scale expressed as a fraction in which:

• the numerator is equal to one,
• the denominator is equal to the number showing how many times the linear dimensions on the map are reduced.

Named (verbal) scale

This is a type of scale, a verbal indication of what distance on the ground corresponds to 1 cm on a map, plan, photograph.

A named scale is expressed by named numbers indicating the lengths of mutually corresponding segments on the map and in nature.

For example, there are 5 kilometers in 1 centimeter (5 kilometers in 1 cm).

Linear scale

This an auxiliary measuring ruler applied to maps to facilitate the measurement of distances.

Plan scale and map scale

The scale of the plan is the same at all its points.

The map scale at each point has its own particular value, depending on the latitude and longitude of the given point. Therefore, its strict numerical characteristic is the numerical scale - the ratio of the length of an infinitesimal segment D on the map to the length of the corresponding infinitesimal segment on the surface of the ellipsoid of the globe.

However, for practical measurements on a map, its main scale is used.

Forms of expression of scale

The designation of scale on maps and plans has three forms - numerical, named and linear scales.

The numerical scale is expressed as a fraction in which:

• numerator - unit,
• denominator M - a number showing how many times the dimensions on the map or plan are reduced (1:M)

In Russia, standard numerical scales have been adopted for topographic maps

• 1:1 000 000
• 1:500 000
• 1:300 000
• 1:200 000
• 1:100 000
• 1:50 000
• 1:25 000
• 1:10 000
• For special purposes, topographic maps are also created on scales 1:5 000 And 1:2 000

The main scales of topographic plans in Russia are

• 1:5000
• 1:2000
• 1:1000
• 1:500

In land management practice, land use plans are most often drawn up on a scale 1:10 000 And 1:25 000 , and sometimes - 1:50 000.

When comparing different numerical scales, the smaller one is the one with the larger denominator. M, and, conversely, the smaller the denominator M, the larger the scale of the plan or map.

Yes, scale 1:10000 larger than scale 1:100000 , and the scale 1:50000 smaller scale 1:10000 .

Note

The scales used in topographic maps are established by the Order of the Ministry of Economic Development of the Russian Federation “On approval of requirements for state topographic maps and state topographic plans, including requirements for the composition of information displayed on them, for the symbols of this information, requirements for the accuracy of state topographic maps and state topographic plans , to the format of their presentation in electronic form, requirements for the content of topographic maps, including relief maps" (No. 271 of June 6, 2017, as amended on December 11, 2017).

Named scale

Since the lengths of lines on the ground are usually measured in meters, and on maps and plans in centimeters, it is convenient to express the scales in verbal form, for example:

There are 50 m in one centimeter. This corresponds to the numerical scale 1:5000. Since 1 meter is equal to 100 centimeters, the number of meters of terrain contained in 1 cm of a map or plan is easily determined by dividing the denominator of the numerical scale by 100.

Linear scale

It is a graph in the form of a straight line segment, divided into equal parts with signed values ​​of the corresponding lengths of terrain lines. Linear scale allows you to measure or plot distances on maps and plans without calculations.

Scale accuracy

The maximum possibility of measuring and constructing segments on maps and plans is limited to 0.01 cm. The corresponding number of meters of terrain on the scale of a map or plan represents the maximum graphic accuracy of a given scale.

Since the accuracy of the scale expresses the length of the horizontal location of the terrain line in meters, to determine it, the denominator of the numerical scale should be divided by 10,000 (1 m contains 10,000 segments of 0.01 cm). So, for a scale map 1:25 000 scale accuracy is 2.5 m; for map 1:100 000 - 10 m, etc.

Scales of topographic maps

numerical scale cards	Name cards	1 cm on the map corresponds on the grounddistance	1 cm 2 on the map corresponds on the area area
	five thousandth
1:10 000	ten-thousandth
1:25 000	twenty-five thousandth
1:50 000	fifty thousandth
1:1100 000	hundred thousandth
1:200 000	two hundred thousandth
1:500 000	five hundred thousandth, or half a millionth
1:1000000	millionth

Below are the numerical scales of the maps and the corresponding named scales:

Scale 1:100,000

• 1 mm on the map - 100 m (0.1 km) on the ground
• 1 cm on the map - 1000 m (1 km) on the ground
• 10 cm on the map - 10,000 m (10 km) on the ground

Scale 1:10000

• 1 mm on the map - 10 m (0.01 km) on the ground
• 1 cm on the map - 100 m (0.1 km) on the ground
• 10 cm on the map - 1000m (1 km) on the ground

Scale 1:5000

• 1 mm on the map - 5 m (0.005 km) on the ground
• 1 cm on the map - 50 m (0.05 km) on the ground
• 10 cm on the map - 500 m (0.5 km) on the ground

Scale 1:2000

• 1 mm on the map - 2 m (0.002 km) on the ground
• 1 cm on the map - 20 m (0.02 km) on the ground
• 10 cm on the map - 200 m (0.2 km) on the ground

Scale 1:1000

• 1 mm on the map - 100 cm (1 m) on the ground
• 1 cm on the map - 1000 cm (10 m) on the ground
• 10 cm on the map - 100 m on the ground

Scale 1:500

• 1 mm on the map - 50 cm (0.5 meters) on the ground
• 1 cm on the map - 5 m on the ground
• 10 cm on the map - 50 m on the ground

Scale 1:200

• 1 mm on the map - 0.2 m (20 cm) on the ground
• 1 cm on the map - 2 m (200 cm) on the ground
• 10 cm on the map - 20 m (0.2 km) on the ground

Scale 1:100

• 1 mm on the map - 0.1 m (10 cm) on the ground
• 1 cm on the map - 1 m (100 cm) on the ground
• 10 cm on the map - 10 m (0.01 km) on the ground

Example 1

Convert the numerical scale of the map to a named one:

1. 1:200 000
2. 1:10 000 000
3. 1:25 000

Solution:

To more easily convert a numerical scale into a named one, you need to count how many zeros the number in the denominator ends with.

For example, on a scale of 1:500,000, there are five zeros in the denominator after the number 5.

If after the number in the denominator there are five more zeros, then by covering (with a finger, a pen or simply crossing out) the five zeros, we get the number of kilometers on the ground corresponding to 1 centimeter on the map.

Example for scale 1:500,000

The denominator after the number has five zeros. Closing them, we get for a named scale: 1 cm on the map is 5 kilometers on the ground.

If there are less than five zeros after the number in the denominator, then by closing two zeros, we get the number of meters on the ground corresponding to 1 centimeter on the map.

If, for example, in the denominator of the scale 1:10 000 cover two zeros, we get:

in 1 cm - 100 m.

Answers :

1. 1 cm - 2 km
2. 1 cm - 100 km
3. in 1 cm - 250 m

Use a ruler and place it on the maps to make it easier to measure distances.

Example 2

Convert the named scale to a numerical one:

1. in 1 cm - 500 m
2. 1 cm - 10 km
3. 1 cm - 250 km

Solution:

To more easily convert a named scale to a numerical one, you need to convert the distance on the ground indicated in the named scale into centimeters.

If the distance on the ground is expressed in meters, then to obtain the denominator of the numerical scale, you need to assign two zeros, if in kilometers, then five zeros.

For example, for a named scale of 1 cm - 100 m, the distance on the ground is expressed in meters, so for the numerical scale we assign two zeros and get: 1:10 000 .

For a scale of 1 cm - 5 km, we add five zeros to the five and get: 1:500 000 .

Answers :

1. 1:50 000;
2. 1:1 000 000;
3. 1:25 000 000.

Types of maps depending on scale

Depending on the scale, maps are conventionally divided into the following types:

• topographic plans - 1:400 - 1:5 000;
• large-scale topographic maps - 1:10,000 - 1:100,000;
• medium-scale topographic maps - from 1:200,000 - 1:1,000,000;
• small-scale topographic maps - less than 1:1,000,000.

Topographic map

Topographical maps are those whose content allows them to solve various technical problems.

Maps are either the result of direct topographic survey of the area, or are compiled from existing cartographic materials.

The terrain on the map is depicted at a certain scale.

The smaller the denominator of a numerical scale, the larger the scale. Plans are drawn up on a large scale, and maps are drawn up on a small scale.

Maps take into account the “spherical shape” of the earth, but plans do not. Because of this, plans are not drawn up for areas larger than 400 km² (that is, areas of land approximately 20 km × 20 km).

• Standard scales for topographic maps

The following scales of topographic maps are accepted in our country:

1. 1:1 000 000
2. 1:500 000
3. 1:200 000
4. 1:100 000
5. 1:50 000
6. 1:25 000
7. 1:10 000.

This series of scales is called standard. Previously, this series included scales of 1:300,000, 1:5000 and 1:2000.

• Large scale topographic maps

Scale maps:

1. 1:10,000 (1cm =100m)
2. 1:25,000 (1cm = 100m)
3. 1:50,000 (1cm = 500m)
4. 1:100,000 (1cm =1000m)

are called large-scale.

• Other scales and maps

Topographic maps of the territory of Russia up to a scale of 1:50,000 inclusive are classified, topographic maps of a scale of 1:100,000 are chipboard (for official use), and smaller ones are unclassified.

Currently, there is a technique for creating topographic maps and plans of any scale that are not classified and intended for public use.

A tale about a map on a scale of 1:1

Once upon a time there lived a Capricious King. One day he traveled around his kingdom and saw how large and beautiful his land was. He saw winding rivers, huge lakes, high mountains and wonderful cities. He became proud of his possessions and wanted the whole world to know about them.

And so, the Capricious King ordered cartographers to create a map of the kingdom. The cartographers worked for a whole year and finally presented the King with a wonderful map, on which all the mountain ranges, large cities and large lakes and rivers were marked.

However, the Capricious King was not satisfied. He wanted to see on the map not only the outlines of mountain ranges, but also an image of each mountain peak. Not only large cities, but also small ones and villages. He wanted to see small rivers flowing into rivers.

The cartographers set to work again, worked for many years and drew another map, twice the size of the previous one. But now the King wanted the map to show passes between mountain peaks, small lakes in the forests, streams, and peasant houses on the outskirts of villages. Cartographers drew more and more maps.

The Capricious King died before the work was completed. The heirs, one after another, ascended the throne and died in turn, and the map was drawn up and drawn up. Each king hired new cartographers to map the kingdom, but each time he was dissatisfied with the fruits of his labor, finding the map insufficiently detailed.

Finally, the cartographers drew the Incredible Map! It depicted the entire kingdom in great detail - and was exactly the same size as the kingdom itself. Now no one could tell the difference between the map and the kingdom.

Where were the Capricious Kings going to keep their wonderful map? The casket is not enough for such a map. You will need a huge room like a hangar, and in it the map will lie in many layers. But is such a card necessary? After all, a life-size map can be successfully replaced by the terrain itself))))

It is useful to familiarize yourself with this

• You can familiarize yourself with the units of measurement of land areas used in Russia.
• For those who are interested in the possibility of increasing the area of ​​land plots for individual housing construction, private household plots, gardening, vegetable farming, owned, it is useful to familiarize yourself with the procedure for registering additions.
  
• From January 1, 2018, the exact boundaries of the plot must be recorded in the cadastral passport, since it will simply be impossible to buy, sell, mortgage or donate land without an accurate description of the boundaries. This is regulated by amendments to the Land Code. A total revision of borders at the initiative of municipalities began on June 1, 2015.
• On March 1, 2015, the new Federal Law “On Amendments to the Land Code of the Russian Federation and certain legislative acts of the Russian Federation” (N 171-FZ dated June 23, 2014) came into force, according to which, in particular, the procedure for purchasing land plots has been simplified from municipalities.You can familiarize yourself with the main provisions of the law.
• With regard to the registration of houses, bathhouses, garages and other buildings on land plots owned by citizens, the new dacha amnesty will improve the situation.

Map scale is the ratio of the length of a segment on the map to its actual length on the ground.

Scale ( from German - measure and Stab - stick) - the ratio of the length of a segment on a map, plan, aerial or satellite image to its actual length on the ground.

Numerical scale - a scale expressed as a fraction, where the numerator is one, and the denominator is a number indicating how many times the image is reduced.

Named (verbal) scale - type of scale, verbal indication of what distance on the ground corresponds to 1 cm on a map, plan, photograph.

Linear scale - an auxiliary measuring ruler applied to maps to facilitate the measurement of distances.

A named scale is expressed by named numbers, indicating the lengths of mutually corresponding segments on the map and in nature.

For example, there are 5 kilometers in 1 centimeter (5 kilometers in 1 cm). Numerical scale is a scale expressed as a fraction in which: the numerator is equal to one, and the denominator is equal to a number showing how many times the linear dimensions on the map are reduced. Below are the numerical scales of the maps and the corresponding named scales.

The scale of the plan is the same at all its points.

The map scale at each point has its own particular value, depending on the latitude and longitude of the given point. Therefore, its strict numerical characteristic is a partial scale - the ratio of the length of an infinitesimal segment D/ on the map to the length of the corresponding infinitesimal segment on the surface of the ellipsoid of the globe. However, for practical measurements on a map, its main scale is used.

Forms of expression of scale

The designation of scale on maps and plans has three forms: numerical, named and linear scales.

The numerical scale is expressed as a fraction in which the numerator is one, and the denominator M is a number showing how many times the dimensions on the map or plan are reduced (1: M)

In Russia, standard numerical scales are adopted for topographic maps:

For special purposes, topographic maps are also created at scales of 1: 5,000 and 1: 2,000.

The main scales of topographic plans in Russia are:

1:5000, 1:2000, 1:1000 and 1:500.

However, in land management practice, land use plans are most often drawn up at scales of 1: 10,000 and 1:25,000, and sometimes 1: 50,000.

When comparing different numerical scales, the smaller one is the one with the larger denominator M, and, conversely, the smaller the denominator M, the larger the scale of the plan or map.

Thus, a scale of 1: 10,000 is larger than a scale of 1: 100,000, and a scale of 1: 50,000 is smaller than a scale of 1: 10,000.

Named scale

Since the lengths of lines on the ground are usually measured in meters, and on maps and plans - in centimeters, it is convenient to express the scales in verbal form, for example:

There are 50 meters in one centimeter. This corresponds to a numerical scale of 1: 5000. Since 1 meter is equal to 100 centimeters, the number of meters of terrain contained in 1 cm of a map or plan is easily determined by dividing the denominator of the numerical scale by 100.

Linear scale

It is a graph in the form of a straight line segment, divided into equal parts with signed values ​​of the corresponding lengths of terrain lines. Linear scale allows you to measure or plot distances on maps and plans without calculations.

Scale accuracy

The maximum possibility of measuring and constructing segments on maps and plans is limited to 0.01 cm. The corresponding number of meters of terrain on the scale of a map or plan represents the maximum graphic accuracy of a given scale. Since the accuracy of the scale expresses the length of the horizontal location of the terrain line in meters, then to determine it, the denominator of the numerical scale should be divided by 10,000 (1 m contains 10,000 segments of 0.01 cm). So, for a map of scale 1: 25,000, the scale accuracy is 2.5 m; for map 1: 100,000- 10 m, etc.

Scales of topographic maps

Numerical scale cards	Name cards	1 cm on the map corresponds on terrain distance	1 cm 2 on the map corresponds on the ground area
1: 5 000 1: 1 000 000	Five thousandth Ten-thousandth Twenty-five thousandth Fifty thousandth One hundred thousandth Two hundred thousandth Five hundred thousandth, Or half a millionth Millionth	50 m 10 km	0.25 ha 100 km 2

Below are the numerical scales of the maps and the corresponding named scales:

1. 
   Scale 1: 100,000

1 mm on the map - 100 m (0.1 km) on the ground

1 cm on the map - 1000 m (1 km) on the ground

10 cm on the map - 10,000 m (10 km) on the ground

1. 
   Scale 1:10000

1 mm on the map – 10 m (0.01 km) on the ground
1 cm on the map - 100 m (0.1 km) on the ground

10 cm on the map - 1000m (1 km) on the ground

1. 
   Scale 1:5000

1 mm on the map – 5 m (0.005 km) on the ground

1 cm on the map - 50 m (0.05 km) on the ground

10 cm on the map – 500 m (0.5 km) on the ground

1. 
   Scale 1:2000

1 mm on the map – 2 m (0.002 km) on the ground

1 cm on the map – 20 m (0.02 km) on the ground

10 cm on the map – 200 m (0.2 km) on the ground

1. 
   Scale 1:1000

1 mm on the map – 100 cm (1 m) on the ground

1 cm on the map – 1000 cm (10 m) on the ground

10 cm on the map – 100 m on the ground

1. 
   Scale 1:500

1 mm on the map – 50 cm (0.5 meters) on the ground

1 cm on the map – 5 m on the ground

10 cm on the map – 50 m on the ground

1. 
   Scale 1:200

1 mm on the map –0.2 m (20 cm) on the ground

1 cm on the map – 2 m (200 cm) on the ground

10 cm on the map – 20 m (0.2 km) on the ground

1. 
   Scale 1:100

1 mm on the map – 0.1 m (10 cm) on the ground

1 cm on the map – 1 m (100 cm) on the ground

10 cm on the map – 10 m (0.01 km) on the ground

Each card has scale– a number that shows how many centimeters on the ground correspond to one centimeter on the map.

Map scale usually indicated on it. Entry 1: 100,000,000 means that if the distance between two points on a map is 1 cm, then the distance between the corresponding points on its terrain is 100,000,000 cm.

May be specified in numerical form as a fraction– numerical scale (for example, 1: 200,000). Or may be designated in linear form: as a simple line or strip divided into units of length (usually kilometers or miles).

The larger the scale of the map, the more detailed the elements of its content can be depicted on it, and vice versa, the smaller the scale, the more extensive the space can be shown on the map sheet, but the terrain on it is depicted in less detail.

The scale is a fraction, the numerator of which is one. To determine which scale is larger and by how many times, remember the rule for comparing fractions with the same numerators: of two fractions with the same numerators, the one with the smaller denominator is larger.

The ratio of the distance on the map (in centimeters) to the corresponding distance on the ground (in centimeters) is equal to the map scale.

How will this knowledge help us when solving problems in mathematics?

Example 1.

Let's look at two cards. A distance of 900 km between points A and B corresponds to a distance of 3 cm on one map. A distance of 1,500 km between points C and D corresponds to a distance of 5 cm on another map. Let us prove that the scales of the maps are the same.

Solution.

Let's find the scale of each map.

900 km = 90,000,000 cm;

the scale of the first map is: 3: 90,000,000 = 1: 30,000,000.

1500 km = 150,000,000 cm;

the scale of the second map is: 5: 150,000,000 = 1: 30,000,000.

Answer. The scales of the maps are the same, i.e. equal to 1: 30,000,000.

Example 2.

Map scale – 1: 1,000,000. Let’s find the distance between points A and B on the ground, if on the map
AB = 3.42 cm?

Solution.

Let's create an equation: the ratio AB = 3.42 cm on the map to the unknown distance x (in centimeters) is equal to the ratio between the same points A and B on the ground to the map scale:

3.42: x = 1: 1,000,000;

x · 1 = 3.42 · 1,000,000;

x = 3,420,000 cm = 34.2 km.

Answer: the distance between points A and B on the ground is 34.2 km.

Example 3

The map scale is 1: 1,000,000. The distance between points on the ground is 38.4 km. What is the distance between these points on the map?

Solution.

The ratio of the unknown distance x between points A and B on the map to the distance in centimeters between the same points A and B on the ground is equal to the scale of the map.

38.4 km = 3,840,000 cm;

x: 3,840,000 = 1: 1,000,000;

x = 3,840,000 · 1: 1,000,000 = 3.84.

Answer: the distance between points A and B on the map is 3.84 cm.

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The frames of maps of derived scales are constructed by dividing the base sheet along the lines of parallels and meridians into several equal parts, i.e. The layout of sheets is always based on a geographic coordinate grid. We consider the following scales of maps and plans to be standard:
Layout scheme and nomenclature of topographic maps of derivative scales for the main coordinate system SK-42 in the Russian Federation:

Scale	Base sheet	divided into	Designation	Frame size
1: 1 000 000			N-37	4 x 6 degrees
1: 500 000	1: 1 000 000	4 sheets (A, B, C, D)	N-37-B	2 x 3 degrees
1: 200 000	1: 1 000 000	36 sheets (I-XXXVI)	N-37-XXIII	40" x 60"
1: 100 000	1: 1 000 000	144 sheets (1-144)	N-37-89	20" x 30"
1: 50 000	1: 100 000	4 sheets (A, B, C, D)	N-37-44-B	10" x 15"
1: 25 000	1: 100 000	16 sheets (a,b,c,d)	N-37-114-GB	5" x 7" 30"
1: 10 000	1: 100 000	64 sheets (1,2,3,4)	N-37-78-Bv-3	2" 30" x 3" 45"

Topographic maps for which the base layout is a scale of 1: 1,000,000 are considered to be medium-scale, and those for which the base is a scale of 1: 100,000 are considered large-scale. Sheets of topographic maps of large scales from 1: 50,000 and 1: 10,000 are formed by sequentially dividing the sheet of the previous scale into 4 parts with the corresponding additions of letters to the nomenclature.
Topographic maps at a scale of 1: 200,000 and smaller are open to us, for a scale of 1: 100,000 the order of use is defined - for official use, all larger scales of topographic maps are closed.

	This figure shows the division of a sheet of scale 1: 1,000,000 On 4 sheets of scale 1: 500,000 (A, B, C, D), For 36 sheets of scale 1:200,000 (indicated by Roman numerals), and For 144 sheets, scale 1:100,000 (indicated by Arabic numerals).
		This figure shows the division of a sheet of scale 1: 100,000: For 4 sheets of scale 1: 50,000 (A, B, C, D are added); Scale sheet division 1: 50,000 on 4 sheets scale 1: 25,000 (a, b, c, d are added); Scale sheet division 1: 25,000 on 4 sheets scale 1: 10,000 (add 1, 2, 3, 4); Three-digit numbers from 1 to 256 show division into sheets of scale 1: 5,000, but maps of this scale are very rare in practice.

The domestic nomenclature of topographic maps larger than 1: 100,000 in practice often causes errors and confusion (Vb - Bv, ...) and, according to the authors, is not very successful - it is very difficult to estimate from just the nomenclature number which sheet will be next. To make it easier to navigate, we provide a reference table for dividing into sheets of scale 1: 10,000.
Although the frames of all topographic maps have boundaries along a geographic grid, on the topographic map sheets themselves, starting from a scale of 1: 200,000 and for all maps larger, it is no longer geographical, but a rectangular, so-called kilometer grid with a step of 4000 m for a scale of 1: 200,000 and up to 1000 m for a scale of 1: 10,000, which is a display of the rectangular Gauss-Kruger coordinate system.
On the standard SK-42 topographic maps there is complete information about the coordinates of the sheet both in the geographic coordinate system and in the rectangular Gauss-Kruger coordinate system. The fragment of the topographic map below shows a corner with information about its coordinates and explains how to understand it correctly. This is a sheet of topographic map at a scale of 1: 200,000 with the nomenclature number N-38-XXII, made in the SK-42 coordinate system.

Topographic map angle 1: 200,000 and coordinate information:

at the very corner of the sheet, the geographical coordinates of this corner are written, 46° 00" east longitude and 54° 00" north latitude; At the top frame, the numbers 48, 52, 56, 60 are the kilometer grid coordinates, and together with the small number 85 next to 60 they show the exact Y coordinate value of this vertical line in the rectangular Gauss-Kruger coordinate system, equal to 8,560,000 m; that is, this map is from zone 8, and the coordinate of the line is 60 km east of the middle meridian of the zone; In the right frame, the numbers 76, 80, 84 are also kilometer grid coordinates, and together with the small number 59 next to 80 they show the exact X coordinate value of this horizontal line in the rectangular Gauss-Kruger coordinate system, equal to 5,980,000 m; is the distance to that line from the equator.

When practical problems of creating cartographic foundations for mapping regions are solved, it turns out that even in the Central European part of the Russian Federation, only rare regions lie entirely within the same Gauss-Kruger projection zone. To solve this problem, it is possible to expand the standard 6-degree zone, but with the caveat that area distortions will increase in the expansion zone. To ensure the possibility of combining adjacent map sheets from different zones, kilometer grid marks of the adjacent zone can be applied to the outer sheets, as shown in the figure. When using maps in GIS, this information seems to be of little use.

INTRODUCTION

The topographic map is reduced a generalized image of the area showing elements using a system of symbols.
In accordance with the requirements, topographic maps are highly geometric accuracy and geographical relevance. This is ensured by them scale, geodetic basis, cartographic projections and a system of symbols.
The geometric properties of a cartographic image: the size and shape of areas occupied by geographical objects, the distances between individual points, the directions from one to another - are determined by its mathematical basis. Mathematical basis cards includes as components scale, geodetic basis, and map projection.
What a map scale is, what types of scales there are, how to construct a graphic scale and how to use scales will be discussed in the lecture.

6.1. TYPES OF SCALES OF TOPOGRAPHIC MAPS

When drawing up maps and plans, horizontal projections of segments are depicted on paper in a reduced form. The degree of such reduction is characterized by scale.

Map scale (plan) - the ratio of the length of a line on a map (plan) to the length of the horizontal location of the corresponding terrain line

m = l K : d M

The scale of the image of small areas throughout the topographic map is practically constant. At small angles of inclination of the physical surface (on a plain), the length of the horizontal projection of the line differs very little from the length of the inclined line. In these cases, the length scale can be considered the ratio of the length of a line on the map to the length of the corresponding line on the ground.

The scale is indicated on maps in different versions

6.1.1. Numerical scale

Numerical scale expressed as a fraction with numerator equal to 1(aliquot fraction).

Or

Denominator M numerical scale shows the degree of reduction in the lengths of lines on a map (plan) in relation to the lengths of the corresponding lines on the ground. Comparing numerical scales with each other, the larger one is the one with the smaller denominator.
Using the numerical scale of the map (plan), you can determine the horizontal location dm lines on the ground

Example.
Map scale 1:50,000. Length of segment on the map lК= 4.0 cm. Determine the horizontal location of the line on the ground.

Solution.
By multiplying the size of the segment on the map in centimeters by the denominator of the numerical scale, we obtain the horizontal distance in centimeters.
d= 4.0 cm × 50,000 = 200,000 cm, or 2,000 m, or 2 km.

note that the numerical scale is an abstract quantity that does not have specific units of measurement. If the numerator of a fraction is expressed in centimeters, then the denominator will have the same units of measurement, i.e. centimeters.

For example, a scale of 1:25,000 means that 1 centimeter of map corresponds to 25,000 centimeters of terrain, or 1 inch of map corresponds to 25,000 inches of terrain.

To meet the needs of the economy, science and defense of the country, maps of various scales are needed. For state topographic maps, forest management tablets, forestry and afforestation plans, standard scales have been determined - scale series(Table 6.1, 6.2).

Scale series of topographic maps

Table 6.1.

Numerical scale	Card name	1cm card corresponds on the ground distance	1 cm2 card corresponds on the area area
	Five thousandth		0.25 hectare
	Ten-thousandth
	Twenty-five thousandth		6.25 hectares
	Fifty thousandth
	One hundred thousandth
	Two hundred thousandth
	Five hundred thousandth
	Millionth

Previously, this series included scales 1: 300,000 and 1: 2,000.

6.1.2. Named scale

Named scale called a verbal expression of a numerical scale. Under the numerical scale on the topographic map there is an inscription explaining how many meters or kilometers on the ground correspond to one centimeter of the map.

For example, on the map under a numerical scale of 1:50,000 it is written: “there are 500 meters in 1 centimeter.” The number 500 in this example is named scale value .
Using a named map scale, you can determine the horizontal distance dm lines on the ground. To do this, you need to multiply the value of the segment, measured on the map in centimeters, by the value of the named scale.

Example. The named scale of the map is “2 kilometers in 1 centimeter”. Length of a segment on the map lК= 6.3 cm. Determine the horizontal location of the line on the ground.
Solution. By multiplying the value of the segment measured on the map in centimeters by the value of the named scale, we obtain the horizontal distance in kilometers on the ground.
d= 6.3 cm × 2 = 12.6 km.

6.1.3. Graphic scales

To avoid mathematical calculations and speed up work on the map, use graphic scales . There are two such scales: linear And transverse .

Linear scale

To construct a linear scale, select an initial segment convenient for a given scale. This original segment ( A) are called basis of scale (Fig. 6.1).

Rice. 6.1. Linear scale. Measured segment on the ground
will CD = ED + CE = 1000 m + 200 m = 1200 m.

The base is laid on a straight line the required number of times, the leftmost base is divided into parts (segment b), to be smallest linear scale divisions . The distance on the ground that corresponds to the smallest division of the linear scale is called linear scale accuracy .

How to use a linear scale:

• place the right leg of the compass on one of the divisions to the right of zero, and the left leg on the left base;
• the length of the line consists of two counts: the count of whole bases and the count of divisions of the left base (Fig. 6.1).
• If a segment on the map is longer than the constructed linear scale, then it is measured in parts.

Transverse scale

For more accurate measurements use transverse scale (Fig. 6.2, b).

Figure 6.2. Transverse scale. Measured distance
PK = TK + PS + ST = 1 00 +10 + 7 = 117 m.

To construct it, several scale bases are laid out on a straight line segment ( a). Usually the length of the base is 2 cm or 1 cm. At the resulting points, perpendiculars to the line are installed AB and draw ten parallel lines through them at equal intervals. The leftmost base above and below is divided into 10 equal segments and connected by oblique lines. The zero point of the lower base is connected to the first point WITH top base and so on. Get a series of parallel inclined lines, which are called transversals.
The smallest division of the transverse scale is equal to the segment C 1 D 1 , (Fig. 6. 2, A). The adjacent parallel segment differs by this length when moving up the transversal 0С and along a vertical line 0D.
A transverse scale with a base of 2 cm is called normal . If the base of the transverse scale is divided into ten parts, then it is called hundredths . On the hundredth scale, the price of the smallest division is equal to one hundredth of the base.
The transverse scale is engraved on metal rulers, which are called scale rulers.

How to use a transverse scale:

• use a measuring compass to record the length of the line on the map;
• place the right leg of the compass on a whole division of the base, and the left leg on any transversal, while both legs of the compass should be located on a line parallel to the line AB;
• the length of the line consists of three counts: the count of integer bases, plus the count of divisions of the left base, plus the count of divisions up the transversal.

The accuracy of measuring the length of a line using a transverse scale is estimated at half the value of its smallest division.

6.2. VARIETIES OF GRAPHIC SCALES

6.2.1. Transitional scale

Sometimes in practice you have to use a map or aerial photograph, the scale of which is not standard. For example, 1:17,500, i.e. 1 cm on the map corresponds to 175 m on the ground. If you construct a linear scale with a base of 2 cm, then the smallest division of the linear scale will be 35 m. Digitization of such a scale causes difficulties in practical work.
To simplify the determination of distances on a topographic map, proceed as follows. The base of the linear scale is not taken as 2 cm, but is calculated so that it corresponds to a round number of meters - 100, 200, etc.

Example. It is required to calculate the length of the base corresponding to 400 m for a map of scale 1:17,500 (175 meters in one centimeter).
To determine what dimensions a 400 m long segment will have on a 1:17,500 scale map, we draw up the proportions:
on the ground on the plan
175 m 1 cm
400 m X cm
X cm = 400 m × 1 cm / 175 m = 2.29 cm.

Having solved the proportion, we conclude: the base of the transition scale in centimeters is equal to the value of the segment on the ground in meters divided by the value of the named scale in meters. The length of the base in our case
A= 400 / 175 = 2.29 cm.

If we now construct a transverse scale with the length of the base A= 2.29 cm, then one division of the left base will correspond to 40 m (Fig. 6.3).

Rice. 6.3. Transitional linear scale.
Measured distance AC = BC + AB = 800 +160 = 960 m.

For more accurate measurements, a transverse transition scale is built on maps and plans.

6.2.2. Steps scale

This scale is used to determine distances measured in steps during visual surveying. The principle of constructing and using the step scale is similar to the transition scale. The base of the step scale is calculated so that it corresponds to the round number of steps (pairs, triplets) - 10, 50, 100, 500.
To calculate the base value of the step scale, it is necessary to determine the shooting scale and calculate the average step length Shsr.
The average step length (pairs of steps) is calculated from the known distance traveled in the forward and reverse directions. By dividing the known distance by the number of steps taken, the average length of one step is obtained. When the earth's surface is tilted, the number of steps taken in the forward and reverse directions will be different. When moving in the direction of increasing relief, the step will be shorter, and in the opposite direction - longer.

Example. A known distance of 100 m is measured in steps. 137 steps were taken in the forward direction, and 139 steps in the reverse direction. Calculate the average length of one step.
Solution. Total distance covered: Σ m = 100 m + 100 m = 200 m. The sum of steps is: Σ w = 137 w + 139 w = 276 w. The average length of one step is:

Shsr= 200 / 276 = 0.72 m.

It is convenient to work with a linear scale, when the scale line is marked every 1 - 3 cm, and the divisions are signed with a round number (10, 20, 50, 100). Obviously, the value of one step of 0.72 m on any scale will have extremely small values. For a scale of 1:2,000, the segment on the plan will be 0.72 / 2,000 = 0.00036 m or 0.036 cm. Ten steps, on the appropriate scale, will be expressed as a segment of 0.36 cm. The most convenient basis for these conditions, in the opinion of author, the value will be 50 steps: 0.036 × 50 = 1.8 cm.
For those who count steps in pairs, a convenient base would be 20 pairs of steps (40 steps) 0.036 × 40 = 1.44 cm.
The length of the base of the step scale can also be calculated from proportions or by the formula
A = (Shsr × KS) / M
Where: Shsr - average value of one step in centimeters,
KS - number of steps at the base of the scale ,
M - scale denominator.

The length of the base for 50 steps on a scale of 1:2000 with the length of one step equal to 72 cm will be:
A= 72 × 50 / 2000 = 1.8 cm.
To construct the step scale for the example above, you need to divide the horizontal line into segments equal to 1.8 cm, and divide the left base into 5 or 10 equal parts.

Rice. 6.4. Step scale.
Measured distance AC = BC + AB = 100 + 20 = 120 sh.

6.3. SCALE ACCURACY

Scale accuracy (maximum scale accuracy) is a horizontal line segment corresponding to 0.1 mm on the plan. The value of 0.1 mm for determining scale accuracy is adopted due to the fact that this is the minimum segment that a person can distinguish with the naked eye.
For example, for a scale of 1:10,000 the scale accuracy will be 1 m. On this scale, 1 cm on the plan corresponds to 10,000 cm (100 m) on the ground, 1 mm - 1,000 cm (10 m), 0.1 mm - 100 cm (1m). From the above example it follows that If the denominator of the numerical scale is divided by 10,000, we obtain the maximum accuracy of the scale in meters.
For example, for a numerical scale of 1:5,000, the maximum scale accuracy will be 5,000 / 10,000 = 0.5 m.

Scale accuracy allows you to solve two important problems:

• determining the minimum sizes of objects and terrain that are depicted on a given scale, and the sizes of objects that cannot be depicted on a given scale;
• establishing the scale at which the map should be created so that it depicts objects and terrain features with predetermined minimum dimensions.

In practice, it is accepted that the length of a segment on a plan or map can be estimated with an accuracy of 0.2 mm. The horizontal distance on the ground, corresponding at a given scale to 0.2 mm (0.02 cm) on the plan, is called graphic scale accuracy . Graphic accuracy in determining distances on a plan or map can only be achieved when using a transverse scale.
It should be borne in mind that when measuring the relative position of contours on a map, the accuracy is determined not by the graphical accuracy, but by the accuracy of the map itself, where errors can average 0.5 mm due to the influence of errors other than graphic ones.
If we take into account the error of the map itself and the measurement error on the map, we can conclude that the graphical accuracy of determining distances on the map is 5 - 7 times worse than the maximum scale accuracy, i.e. it is 0.5 - 0.7 mm on the map scale.

6.4. DETERMINING AN UNKNOWN MAP SCALE

In cases where for some reason there is no scale on the map (for example, it was cut off when gluing), it can be determined in one of the following ways.

• By grid . It is necessary to measure the distance on the map between the grid lines and determine how many kilometers these lines are drawn through; This will determine the scale of the map.

For example, the coordinate lines are designated by the numbers 28, 30, 32, etc. (along the western frame) and 06, 08, 10 (along the southern frame). It is clear that the lines are drawn through 2 km. The distance on the map between adjacent lines is 2 cm. It follows that 2 cm on the map corresponds to 2 km on the ground, and 1 cm on the map corresponds to 1 km on the ground (named scale). This means that the scale of the map will be 1:100,000 (1 centimeter equals 1 kilometer).

• According to the nomenclature of the map sheet. The notation system (nomenclature) of map sheets for each scale is quite definite, therefore, knowing the notation system, it is not difficult to find out the scale of the map.

A map sheet at a scale of 1:1,000,000 (millionths) is designated by one of the letters of the Latin alphabet and one of the numbers from 1 to 60. The designation system for maps of larger scales is based on the nomenclature of sheets of a millionth map and can be represented by the following diagram:

1:1 000 000 - N-37
1:500,000 - N-37-B
1:200,000 - N-37-X
1:100,000 - N-37-117
1:50 000 - N-37-117-A
1:25 000 - N-37-117-A-g

Depending on the location of the map sheet, the letters and numbers that make up its nomenclature will be different, but the order and number of letters and numbers in the nomenclature of a map sheet of a given scale will always be the same.
Thus, if the map has the nomenclature M-35-96, then, by comparing it with the diagram shown, we can immediately say that the scale of this map will be 1:100,000.
For more information on card nomenclature, see Chapter 8.

• By distances between local objects. If there are two objects on the map, the distance between which on the ground is known or can be measured, then to determine the scale you need to divide the number of meters between these objects on the ground by the number of centimeters between the images of these objects on the map. As a result, we get the number of meters in 1 cm of this map (named scale).

For example, it is known that the distance from the settlement. Kuvechino to the lake Glubokoe 5 km. Having measured this distance on the map, we got 4.8 cm. Then
5000 m / 4.8 cm = 1042 m in one centimeter.
Maps at a scale of 1:104,200 are not published, so we round up. After rounding, we will have: 1 cm of the map corresponds to 1,000 m of terrain, i.e., the map scale is 1:100,000.
If there is a road with kilometer posts on the map, then it is most convenient to determine the scale by the distance between them.

• According to the dimensions of the arc length of one minute of the meridian . The frames of topographic maps along meridians and parallels are divided in minutes of arc of the meridian and parallel.

One minute of meridian arc (along the eastern or western frame) corresponds to a distance of 1852 m (nautical mile) on the ground. Knowing this, you can determine the scale of the map in the same way as by the known distance between two terrain objects.
For example, the minute segment along the meridian on the map is 1.8 cm. Therefore, in 1 cm on the map there will be 1852: 1.8 = 1,030 m. By rounding, we get the map scale of 1:100,000.
Our calculations obtained approximate scale values. This happened due to the proximity of the distances taken and the inaccuracy of their measurement on the map.

6.5. TECHNIQUES FOR MEASURING AND POSTPUTING DISTANCES ON A MAP

To measure distances on a map, use a millimeter or scale ruler, a compass-meter, and to measure curved lines, a curvimeter.

6.5.1. Measuring distances with a millimeter ruler

Using a millimeter ruler, measure the distance between given points on the map with an accuracy of 0.1 cm. Multiply the resulting number of centimeters by the value of the named scale. For flat terrain, the result will correspond to the distance on the ground in meters or kilometers.
Example. On a map of scale 1: 50,000 (in 1 cm - 500 m) the distance between two points is 3.4 cm. Determine the distance between these points.
Solution. Named scale: 1 cm 500 m. The distance on the ground between points will be 3.4 × 500 = 1700 m.
At angles of inclination of the earth's surface of more than 10º, it is necessary to introduce an appropriate correction (see below).

6.5.2. Measuring distances with a measuring compass

When measuring a distance in a straight line, the compass needles are placed at the end points, then, without changing the compass opening, the distance is measured using a linear or transverse scale. In the case when the opening of the compass exceeds the length of the linear or transverse scale, the whole number of kilometers is determined by the squares of the coordinate grid, and the remainder is determined in the usual order according to the scale.

Rice. 6.5. Measuring distances with a measuring compass on a linear scale.

To get the length broken line sequentially measure the length of each of its links, and then sum up their values. Such lines are also measured by increasing the compass solution.
Example. To measure the length of a broken line ABCD(Fig. 6.6, A), the legs of the compass are first placed at the points A And IN. Then, rotating the compass around the point IN. move the hind leg from the point A exactly IN", lying on the continuation of the straight line Sun.
Front leg from point IN transferred to point WITH. The result is a compass solution B"C=AB+Sun. By similarly moving the back leg of the compass from the point IN" exactly WITH", and the front one WITH V D. get a compass solution
C"D = B"C + CD, the length of which is determined using a transverse or linear scale.

Rice. 6.6. Line length measurement: a - broken line ABCD; b - curve A 1 B 1 C 1;
B"C" - auxiliary points

Long curved segments measured along chords by steps of a compass (see Fig. 6.6, b). The pitch of the compass, equal to an integer number of hundreds or tens of meters, is set using a transverse or linear scale. When rearranging the legs of the compass along the measured line in the directions shown in Fig. 6.6, b use arrows to count steps. The total length of the line A 1 C 1 is the sum of the segment A 1 B 1, equal to the step size multiplied by the number of steps, and the remainder B 1 C 1 measured on a transverse or linear scale.

6.5.3. Measuring distances with a curvimeter

Curve segments are measured with a mechanical (Fig. 6.7) or electronic (Fig. 6.8) curvimeter.

Rice. 6.7. Mechanical curvimeter

First, by rotating the wheel by hand, set the arrow to the zero division, then roll the wheel along the measured line. The reading on the dial opposite the end of the hand (in centimeters) is multiplied by the map scale and the distance on the ground is obtained. A digital curvimeter (Fig. 6.7.) is a high-precision, easy-to-use device. The curvimeter includes architectural and engineering functions and has an easy-to-read display. This device can process metric and Anglo-American (feet, inches, etc.) values, allowing you to work with any maps and drawings. You can enter your most frequently used measurement type and the instrument will automatically convert to scale measurements.

Rice. 6.8. Curvimeter digital (electronic)

To increase the accuracy and reliability of the results, it is recommended to carry out all measurements twice - in the forward and reverse directions. In case of minor differences in the measured data, the arithmetic mean of the measured values ​​is taken as the final result.
The accuracy of measuring distances using these methods using a linear scale is 0.5 - 1.0 mm on the map scale. The same, but using a transverse scale is 0.2 - 0.3 mm per 10 cm of line length.

6.5.4. Conversion of horizontal distance to slant range

It should be remembered that as a result of measuring distances on maps, the lengths of horizontal projections of lines (d) are obtained, and not the lengths of lines on the earth's surface (S) (Fig. 6.9).

Rice. 6.9. Slant range ( S) and horizontal distance ( d)

The actual distance on an inclined surface can be calculated using the formula:

where d is the length of the horizontal projection of line S;
v is the angle of inclination of the earth's surface.

The length of a line on a topographic surface can be determined using a table (Table 6.3) of the relative values ​​of corrections to the length of the horizontal distance (in %).

Table 6.3

Tilt angle

Rules for using the table

1. The first line of the table (0 tens) shows the relative values ​​of corrections at tilt angles from 0° to 9°, the second - from 10° to 19°, the third - from 20° to 29°, the fourth - from 30° up to 39°.
2. To determine the absolute value of the correction, it is necessary:
a) in the table based on the angle of inclination, find the relative value of the correction (if the angle of inclination of the topographic surface is not given by an integer number of degrees, then the relative value of the correction must be found by interpolating between the table values);
b) calculate the absolute value of the correction to the length of the horizontal distance (i.e., multiply this length by the relative value of the correction and divide the resulting product by 100).
3. To determine the length of a line on a topographic surface, the calculated absolute value of the correction must be added to the length of the horizontal alignment.

Example. The topographic map shows the horizontal length to be 1735 m, and the angle of inclination of the topographic surface to be 7°15′. In the table, the relative values ​​of the corrections are given for whole degrees. Therefore, for 7°15" it is necessary to determine the nearest larger and nearest smaller values ​​that are multiples of one degree - 8º and 7º:
for 8° the relative value of the correction is 0.98%;
for 7° 0.75%;
difference in table values ​​of 1º (60′) 0.23%;
the difference between a given angle of inclination of the earth's surface 7°15" and the nearest smaller tabulated value of 7º is 15".
We make up the proportions and find the relative value of the correction for 15":

For 60′ the correction is 0.23%;
For 15′ the correction is x%
x% = = 0.0575 ≈ 0.06%

Relative correction value for inclination angle 7°15"
0,75%+0,06% = 0,81%
Then you need to determine the absolute value of the correction:
= 14.05 m approximately 14 m.
The length of the inclined line on the topographic surface will be:
1735 m + 14 m = 1749 m.

At small angles of inclination (less than 4° - 5°), the difference in the length of the inclined line and its horizontal projection is very small and may not be taken into account.

6.6. MEASUREMENT OF AREA BY MAPS

Determining the areas of plots using topographic maps is based on the geometric relationship between the area of ​​a figure and its linear elements. The scale of the areas is equal to the square of the linear scale.
If the sides of a rectangle on a map are reduced by n times, then the area of ​​this figure will decrease by n 2 times.
For a map of scale 1:10,000 (1 cm 100 m), the scale of the areas will be equal to (1: 10,000) 2 or 1 cm 2 will be 100 m × 100 m = 10,000 m 2 or 1 hectare, and on a map of scale 1 : 1,000,000 per 1 cm 2 - 100 km 2.

To measure areas on maps, graphical, analytical and instrumental methods are used. The use of one or another measurement method is determined by the shape of the area being measured, the specified accuracy of the measurement results, the required speed of obtaining data and the availability of the necessary instruments.

6.6.1. Measuring the area of ​​a plot with straight boundaries

When measuring the area of ​​a plot with straight boundaries, the plot is divided into simple geometric shapes, the area of ​​each of them is measured geometrically and, by summing the areas of individual plots calculated taking into account the map scale, the total area of ​​the object is obtained.

6.6.2. Measuring the area of ​​a plot with a curved contour

An object with a curved contour is divided into geometric shapes, having previously straightened the boundaries in such a way that the sum of the cut off sections and the sum of the excesses mutually compensate each other (Fig. 6.10). The measurement results will be, to some extent, approximate.

Rice. 6.10. Straightening the curved boundaries of the site and
breaking down its area into simple geometric shapes

6.6.3. Measuring the area of ​​a site with a complex configuration

Measuring plot areas, having a complex irregular configuration, are often performed using palettes and planimeters, which gives the most accurate results. Grid palette It is a transparent plate with a grid of squares (Fig. 6.11).

Rice. 6.11. Square mesh palette

The palette is placed on the contour being measured and the number of cells and their parts found inside the contour is counted from it. The proportions of incomplete squares are estimated by eye, therefore, to increase the accuracy of measurements, palettes with small squares (with a side of 2 - 5 mm) are used. Before working on this map, determine the area of ​​one cell.
The area of ​​the plot is calculated using the formula:

P = a 2 n,

Where: A - side of the square, expressed in map scale;
n- the number of squares falling within the contour of the measured area

To increase accuracy, the area is determined several times with arbitrary rearrangement of the palette used to any position, including rotation relative to its original position. The arithmetic mean of the measurement results is taken as the final area value.

In addition to mesh palettes, dot and parallel palettes are used, which are transparent plates with engraved dots or lines. The points are placed in one of the corners of the cells of the grid palette with a known division value, then the grid lines are removed (Fig. 6.12).

Rice. 6.12. Spot palette

The weight of each point is equal to the cost of dividing the palette. The area of ​​the measured area is determined by counting the number of points inside the contour and multiplying this number by the weight of the point.
Equally spaced parallel lines are engraved on the parallel palette (Fig. 6.13). The area being measured, when the palette is applied to it, will be divided into a number of trapezoids with the same height h. The parallel line segments inside the contour (midway between the lines) are the midlines of the trapezoid. To determine the area of ​​a plot using this palette, it is necessary to multiply the sum of all measured center lines by the distance between parallel lines of the palette h(taking into account scale).

P = h∑l

Figure 6.13. A palette consisting of a system
parallel lines

Measurement areas of significant plots is carried out using cards using planimeter.

Rice. 6.14. Polar planimeter

A planimeter is used to determine areas mechanically. The polar planimeter is widely used (Fig. 6.14). It consists of two levers - pole and bypass. Determining the contour area with a planimeter comes down to the following steps. Having secured the pole and positioned the needle of the bypass lever at the starting point of the contour, a count is taken. Then the bypass pin is carefully guided along the contour to the starting point and a second reading is taken. The difference in readings will give the area of ​​the contour in divisions of the planimeter. Knowing the absolute value of the planimeter division, the contour area is determined.
The development of technology contributes to the creation of new devices that increase labor productivity when calculating areas, in particular the use of modern devices, including electronic planimeters.

Rice. 6.15. Electronic planimeter

6.6.4. Calculating the area of ​​a polygon from the coordinates of its vertices
(analytical method)

This method allows you to determine the area of ​​a plot of any configuration, i.e. with any number of vertices whose coordinates (x,y) are known. In this case, the numbering of vertices should be done clockwise.
As can be seen from Fig. 6.16, the area S of the polygon 1-2-3-4 can be considered as the difference between the areas S" of the figure 1y-1-2-3-3y and S" of the figure 1y-1-4-3-3y
S = S" - S".

Rice. 6.16. To calculate the area of ​​a polygon from coordinates.

In turn, each of the areas S" and S" is the sum of the areas of trapezoids, the parallel sides of which are the abscissas of the corresponding vertices of the polygon, and the heights are the differences in the ordinates of the same vertices, i.e.

S " = square 1у-1-2-2у + square 2у-2-3-3у,
S" = pl. 1у-1-4-4у + pl. 4у-4-3-3у
or: 2S " = (x 1 + x 2) (y 2 - y 1) + (x 2 + x 3 ) (y 3 - y 2)
2 S " = (x 1 + x 4) (y 4 - y 1) + (x 4 + x 3) (y 3 - y 4).

Thus,
2S = (x 1 + x 2) (y 2 - y 1) + (x 2 + x 3 ) (y 3 - y 2) - (x 1 + x 4) (y 4 - y 1) - (x 4 + x 3) (y 3 - y 4). Opening the brackets, we get
2S = x 1 y 2 - x 1 y 4 + x 2 y 3 - x 2 y 1 + x 3 y 4 - x 3 y 2 + x 4 y 1 - x 4 y 3

From here
2S = x 1 (y 2 - y 4) + x 2 (y 3 - y 1)+ x 3 (y 4 - y 2) + x 4 (y 1 - y 3) (6.1)
2S = y 1 (x 4 - x 2) + y 2 (x 1 - x 3)+ y 3 (x 2 - x 4)+ y 4 (x 3 - x 1) (6.2)

Let us present expressions (6.1) and (6.2) in general form, denoting by i the serial number (i = 1, 2, ..., n) of the vertices of the polygon:
(6.3)
(6.4)
Therefore, the doubled area of ​​a polygon is equal to either the sum of the products of each abscissa and the difference between the ordinates of the subsequent and previous vertices of the polygon, or the sum of the products of each ordinate and the difference between the abscissas of the previous and subsequent vertices of the polygon.
Intermediate control of calculations is the satisfaction of the conditions:

0 or = 0
Coordinate values ​​and their differences are usually rounded to tenths of a meter, and products - to whole square meters.
Complex formulas for calculating the area of ​​a plot can be easily solved using Microsoft XL spreadsheets. An example for a polygon (polygon) of 5 points is given in tables 6.4, 6.5.
In Table 6.4 we enter the initial data and formulas.

Table 6.4.

				y i (x i-1 - x i+1)
	Double area in m2			SUM(D2:D6)
	Area in hectares

In Table 6.5 we see the results of the calculations.

Table 6.5.

				y i (x i-1 -x i+1)
	Double area in m2
	Area in hectares

6.7. EYE MEASUREMENTS ON THE MAP

In the practice of cartometric work, eye measurements are widely used, which give approximate results. However, the ability to visually determine distances, directions, areas, slope steepness and other characteristics of objects from a map helps to master the skills of correctly understanding a cartographic image. The accuracy of visual determinations increases with experience. Visual skills prevent gross miscalculations in measurements with instruments.
To determine the length of linear objects on a map, one should visually compare the size of these objects with segments of a kilometer grid or divisions of a linear scale.
To determine the areas of objects, squares of a kilometer grid are used as a kind of palette. Each grid square of maps of scales 1:10,000 - 1:50,000 on the ground corresponds to 1 km 2 (100 hectares), scale 1:100,000 - 4 km 2, 1:200,000 - 16 km 2.
The accuracy of quantitative determinations on the map, with the development of the eye, is 10-15% of the measured value.

Video

Scale problems

Tasks and questions for self-control

1. What elements does the mathematical basis of maps include?
2. Expand the concepts: “scale”, “horizontal distance”, “numerical scale”, “linear scale”, “scale accuracy”, “scale bases”.
3. What is a named map scale and how do I use it?
4. What is a transverse map scale, and what is its purpose?
5. What transverse map scale is considered normal?
6. What scales of topographic maps and forest management tablets are used in Ukraine?
7. What is a transition map scale?
8. How is the transition scale base calculated?
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