Topographic maps. Layout and nomenclature
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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A topographic map is a universal-purpose geographic map that depicts the terrain in detail. A topographic map contains information about geodetic reference points, relief, hydrography, vegetation, soils, economic and cultural objects, roads, communications, boundaries and other terrain objects. Content completeness and accuracy topographic maps allow you to solve technical problems.
The science of creating topographic maps is topography.
All geographical maps, depending on their scale, are conventionally divided into the following types:
- topographic plans - up to 1:5 000 inclusive;
- large-scale topographic maps - from 1:10,000 to 1:200,000 inclusive;
- medium-scale topographic maps - from 1:200,000 (not including) to 1:1,000,000 inclusive;
- small-scale topographic maps - less than (less than) 1:1,000,000.
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 should not be drawn up for areas larger than 400 km² (that is, areas of land larger than 20x20 km). The main difference between topographic maps (in a narrow, strict sense) is their large scale, namely the scale of 1:200,000 and larger (the first two points, more strictly the second point: from 1:10,000 to 1:200,000 inclusive).
Geographical objects and their outlines are depicted in most detail on large-scale (topographic) maps. When you zoom out on a map, details have to be excluded and generalized. Individual objects are replaced by their collective meanings. Selection and generalization become obvious when comparing different-scale images of a populated area, which on a scale of 1:10,000 is given in the form of individual buildings, on a scale of 1:50,000 - by blocks, and on a scale of 1:100,000 - by punches. Selection and synthesis of content when compiling geographical maps called cartographic generalization. It aims to preserve and highlight on the map the typical features of the depicted phenomena in accordance with the purpose of the map.
Secrecy
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 intended for official use (DSP), and a smaller scale of 1:100,000 are unclassified.
Those working with maps up to a scale of 1:50,000 are required, in addition to a permit (license) from the Federal Service for State Registration, Cadastre and Cartography or a certificate from a self-regulatory organization (SRO), to obtain permission from the FSB, since such maps constitute a state secret. For the loss of a map of a scale of 1:50,000 or larger, in accordance with Article 284 of the Criminal Code of the Russian Federation “Loss of documents containing state secrets,” a penalty of up to three years in prison is provided.
At the same time, after 1991, secret maps of the entire territory of the USSR, stored in the headquarters of military districts located outside of Russia, appeared on the public market. Since the leadership of, for example, Ukraine or Belarus does not need to maintain the secrecy of maps of foreign territories.
The problem of the existing secrecy on cards became acute in February 2005 in connection with the launch of the project Google Maps, allowing anyone to use color satellite images high resolution(up to several meters), although in Russia any satellite image with a resolution of more than 10 meters is considered secret and requires a declassification procedure to be ordered from the FSB.
In other countries this problem permitted by the fact that not area, but object secrecy is used. With object secrecy, the free distribution of large-scale topographic maps and photographs of strictly defined objects, for example, areas of military operations, military bases and training grounds, and military ship sites, is prohibited. For this purpose, a methodology has been developed for creating topographic maps and plans of any scale that are not classified and intended for public use.
Scales of topographic maps and plans
Map scale- this 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.
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 private 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.
Main scales 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, 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 each). 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
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
To convert a numerical scale to a named scale, you need to convert the number in the denominator and corresponding to the number of centimeters into kilometers (meters). For example, 1: 100,000 in 1 cm - 1 km.
To convert a named scale to a numerical scale, you need to convert the number of kilometers to centimeters. For example, in 1 cm - 50 km 1: 5,000,000.
Nomenclature of topographic plans and maps
Nomenclature is a system of layout and designation of topographic plans and maps.
The division of a multi-sheet map into separate sheets according to a certain system is called map layout, and the designation of a sheet of a multi-sheet map is called nomenclature. In cartographic practice, the following map layout systems are used:
- along the lines of the cartographic grid of meridians and parallels;
- along the lines of a rectangular coordinate grid;
- along auxiliary lines parallel to the middle meridian of the map and a line perpendicular to it, etc.
The most widespread in cartography is the layout of maps along the lines of meridians and parallels, since in this case the position of each sheet of the map on earth's surface precisely determined by the values of the geographic coordinates of the corners of the frame and the position of its lines. Such a system is universal, convenient for depicting any territory of the globe, except for the polar regions. It is used in Russia, the USA, France, Germany and many other countries of the world.
The basis for the nomenclature of maps in the territory Russian Federation The international layout of map sheets at a scale of 1:1 000000 is required. To obtain one sheet of a map of this scale Earth divided by meridians and parallels into columns and rows (belts).
Meridians are drawn every 6°. The columns are counted from 1 to 60 from 180° of the meridian from 1 to 60 from west to east, counterclockwise. The columns coincide with the zones of the rectangular layout, but their numbers differ by exactly 30. So for zone 12, the column number is 42.
Column numbers
Parallels are drawn every 4°. The belts from A to W are counted from the equator to the north and south.
Row numbers
The 1:1,000,000 map sheet contains 4 1:500,000 map sheets, designated by capital letters A, B, C, D; 36 sheets of map 1:200,000, designated from I to XXXVI; 144 sheets of 1:100,000 map, designated from 1 to 144.
The 1:100,000 map sheet contains 4 1:50,000 map sheets, which are designated by capital letters A, B, C, D.
The 1:50,000 map sheet is divided into 4 1:25,000 map sheets, which are designated by lowercase letters a, b, c, d.
Within a sheet of map 1:1,000,000, the arrangement of numbers and letters when designating sheets of maps 1:500,000 and larger is done from left to right in rows and in the direction to the south pole. The initial row is adjacent to the northern frame of the sheet.
The disadvantage of this layout system is the change in the linear dimensions of the northern and southern frames of the map sheets depending on the geographic latitude. As a result, as they move away from the equator, the sheets take on the appearance of increasingly narrow strips stretched along the meridians. Therefore, topographic maps of Russia at all scales from 60 to 76° northern and southern latitudes are published in double longitude sheets, and in the range from 76 to 84° - in quadruple sheets (on a scale of 1:200,000 - folded) in longitude sheets.
The nomenclature of map sheets of scales 1:500,000, 1:200,000 and 1:100,000 is composed of the nomenclature of a map sheet of 1:1,000,000, followed by the addition of designations of map sheets of the corresponding scales. The nomenclatures of double, triple or quadruple sheets contain the designations of all individual sheets presented in the table:
Nomenclatures of topographic map sheets for the northern hemisphere.
| 1:1 000 000 | N-37 | P-47.48 | T-45,46,47,48 |
|---|---|---|---|
| 1:500 000 | N-37-B | R-47-A,B | T-45-A,B,46-A,B |
| 1:200 000 | N-37-IV | P-47-I,II | T-47-I,II,III |
| 1:100 000 | N-37-12 | P-47-9.10 | T-47-133, 134,135,136 |
| 1:50 000 | N-37-12-A | P-47-9-A,B | T-47-133-A,B, 134-A.B |
| 1:25 000 | N-37-12-A-a | R-47-9-A-a,b | T-47-12-A-a, b, B-a, b |
On sheets of the southern hemisphere, a signature (YUP) is placed to the right of the nomenclature.
N37
On the sheets of topographic maps of the entire scale series, along with the nomenclature, their coded digital designations (ciphers) are placed, which are necessary for recording maps using automated means. Coding of nomenclature consists of replacing letters and Roman numerals with Arabic numerals. In this case, the letters are replaced by their serial numbers in the alphabet. The numbers of belts and columns of the 1:1,000,000 map are always indicated by two-digit numbers, for which a zero is added to the single-digit numbers in front. The numbers of the 1:200,000 map sheets within the 1:1,000,000 map sheet are also designated by two-digit numbers, and the numbers of the 1:100,000 map sheets are indicated by three-digit numbers (one or two zeros are assigned to the front of single-digit and two-digit numbers, respectively).
Knowing the nomenclature of maps and the system for its construction, you can determine the scale of the map and the geographic coordinates of the corners of the sheet frame, that is, determine which part of the earth's surface a given map sheet belongs to. And, conversely, knowing the scale of the map sheet and the geographic coordinates of the corners of its frame, you can establish the nomenclature of this sheet.
To select the necessary sheets of topographic maps for a specific area and quickly determine their nomenclature, there are special prefabricated tables:
Composite tables are schematic blank maps of a small scale, divided by vertical and horizontal lines into cells, each of which corresponds to a specific sheet of a map of the appropriate scale. The prefabricated tables indicate the scale, signatures of meridians and parallels, designations of columns and zones of the map layout 1:1,000,000, as well as the rank order of the sheet numbers of larger scale maps within the sheets of the millionth map. Composite tables are used when preparing applications for necessary cards, as well as for geographical recording of topographic maps in troops and warehouses, drawing up documents on the cartographic provision of territories. A stripe or area of troop operations (route of movement, area of exercises, etc.) is plotted on the composite table of maps, then the nomenclature of sheets covering the stripe (area) is determined. For example, in an application for map sheets 1:100,000 of the area shaded in the figure, it is written O-36-132, 144, 0-37-121, 133; N-36-12, 24; N"37-1, 2, 13, 14.
Enlarging or reducing an image on paper is characterized by scale. On a geographic map, the image of the area is represented by a reduction scale.
Numerical scale map is expressed by the ratio of 1 to a number showing how many times the real segment has been reduced.
Most geographic maps are made on a scale of 1:20,000,000 or 1:25,000,000. This scale means that 1 cm on the map corresponds to 20,000,000 cm = 200 km or 25,000,000 cm = 25 km on the ground, since in scale records, the dimensions of the map and terrain units must match.
If the map shows a scale of 1:20,000,000, then by measuring the distance between points in centimeters and multiplying it by 20,000,000, you will get the real distance between points in centimeters.
To simplify calculations, you can immediately convert the scale to kilometers or meters on the ground.
For example, the distance between city A and city B was 3.5 cm on the map, map scale 1:25,000,000.
Solution:
1) 25,000,000 cm = 250 km
2) 3.5 * 250 = 875 (km)
In addition to the numerical scale, the map can also show linear scale .
The first square on the left shows the scale (1 cm on the map is equal to 200 m on the ground). By attaching a ruler to the map, we immediately determine from it how many meters this segment will be on the ground.
Scale is the ratio of 2 linear dimensions, which is used when creating drawings and models and allows you to show large objects in a reduced form, and small ones in an enlarged form. In other words, this is the ratio of the length of a segment on the map to the true length on the ground. Different practical situations may require you to know how to find the scale.
When does it become necessary to define scale?
How to find scale
This mainly happens in the following situations:
- when using a card;
- when making a drawing;
- in the manufacture of models of various objects.
Types of scale
A numerical scale should be understood as a scale expressed as a fraction.
Its numerator is one, and its denominator is a number showing how many times the image is smaller than the real object.
A linear scale is a measuring stick that you can see on maps. This segment is divided into equal parts, labeled with the values of distances commensurate with them on real terrain. A linear scale is convenient because it provides the ability to measure and plot distances on plans and maps.
A named scale is a verbal description of what distance one centimeter actually corresponds to on a map.
For example, there are 100,000 centimeters in one kilometer. In this case, the numerical scale would look like this: 1:100000.
How to find the map scale?
Take, for example, a school atlas and look at any page of it.
At the bottom you can see a ruler that indicates what distance on real terrain corresponds to one centimeter on your map.
The scale in atlases is usually indicated in centimeters, which will need to be converted to kilometers.
For example, when you see the inscription 1:9,500,000, you will understand that 95 kilometers of real terrain corresponds to only 1 cm of the map.
If, for example, you know that the distance between your city and the neighboring one is 40 km, then you can simply measure the distance between them on the map with a ruler and determine the ratio.
So, if by measuring you get a distance of 2 cm, you get a scale of 2:40=2:4000000=1:2000000. As you can see, finding the scale is not difficult at all.
Other uses of scale
When making models of airplanes, tanks, ships, cars and other objects, certain scaling standards are used. For example, it could be a scale of 1:24, 1:48, 1:144.
In this case, the manufactured models must be smaller than their prototypes exactly by the specified number of times.
Scaling may be necessary, for example, when enlarging a picture. In this case, the image is divided into cells of a certain size, for example, 0.5 cm. A sheet of paper will also need to be drawn into cells, but already enlarged by the required number of times (for example, the length of their sides can be one and a half centimeters, if the drawing needs to be enlarged 3 times) .
By drawing the contours of the original drawing onto a lined sheet, it will be possible to obtain an image very close to the original.
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Map scale. The scale of topographic maps is the ratio of the length of a line on the map to the length of the horizontal projection of the corresponding terrain line. In flat areas, with small angles of inclination of the physical surface, the horizontal projections of the lines differ very little from the lengths of the lines themselves, and in these cases the ratio of the length of the line on the map to the length of the corresponding terrain line can be considered a scale, i.e.
the degree of reduction in the lengths of lines on the map relative to their length on the ground. The scale is indicated under the southern frame of the map sheet in the form of a ratio of numbers (numerical scale), as well as in the form of named and linear (graphic) scales.
Numerical scale(M) is expressed as a fraction, where the numerator is one, and the denominator is a number indicating the degree of reduction: M = 1/m. So, for example, on a map at a scale of 1:100,000, the lengths are reduced in comparison with their horizontal projections (or with reality) by 100,000 times.
Obviously, the larger the scale denominator, the greater the reduction in lengths, the smaller the image of objects on the map, i.e. the smaller the scale of the map.
Named scale- an explanation indicating the ratio of the lengths of lines on the map and on the ground.
With M = 1:100,000, 1 cm on the map corresponds to 1 km.
Linear scale used to determine the lengths of lines in nature from maps. This is a straight line, divided into equal segments corresponding to “round” decimal numbers of terrain distances (Fig. 5).
Rice. 5. Designation of scale on a topographic map: a - the base of the linear scale: b - the smallest division of the linear scale; scale accuracy 100 m.
Scale size - 1 km
The segments a laid off to the right of zero are called basis of scale. The distance on the ground corresponding to the base is called linear scale value. To increase the accuracy of determining distances, the leftmost segment of the linear scale is divided into smaller parts, called the smallest divisions of the linear scale.
The distance on the ground expressed by one such division is the accuracy of the linear scale. As can be seen in Figure 5, with a numerical map scale of 1:100,000 and a linear scale base of 1 cm, the scale value will be 1 km, and the scale accuracy (with the smallest division of 1 mm) will be 100 m.
The accuracy of measurements on maps and the accuracy of graphical constructions on paper are associated both with the technical capabilities of measurements and with the resolution of human vision. The accuracy of constructions on paper (graphic accuracy) is generally considered to be 0.2 mm.
The resolution of normal vision is close to 0.1 mm.
Ultimate accuracy map scale - a segment on the ground corresponding to 0.1 mm on the scale of a given map. With a map scale of 1:100,000, the maximum accuracy will be 10 m, with a scale of 1:10,000 it will be 1 m.
Obviously, the possibilities of depicting contours in their actual outlines on these maps will be very different.
The scale of topographic maps largely determines the selection and detail of the objects depicted on them.
With a decrease in scale, i.e. as its denominator increases, the detail of the image of terrain objects is lost.
To meet the diverse needs of sectors of the national economy, science and defense of the country, maps of different scales are needed. A number of standard scales based on the metric decimal system of measures have been developed for state topographic maps of the USSR (Table.
| Table 1. Scales of topographic maps of the USSR | |||
| Numerical scale | Card name | 1 cm on the map corresponds to a distance on the ground | 1 cm2 on the map corresponds to an area on the ground |
| 1:5 000 | Five thousandth | 50 m | 0.25 ha |
| 1:10 000 | Ten-thousandth | 100 m | 1 ha |
| 1:25 000 | Twenty-five thousandth | 250 m | 6.25 ha |
| 1:50 000 | Fifty thousandth | 500 m | 25 hectares |
| 1:100 000 | One hundred thousandth | 1 km | 1 km2 |
| 1:200 000 | Two hundred thousandth | 2 km | 4 km2 |
| 1:500 000 | Five hundred thousandth | 5 km | 25 km2 |
| 1:1 000 000 | Millionth | 10 km | 100 km2 |
In the complex of cards named in table.
1, there are actual topographic maps of scales 1:5000-1:200,000 and survey topographic maps of scales 1:500,000 and 1:1,000,000. The latter are inferior in accuracy and detail to the depiction of the area, but individual sheets cover significant territories, and these maps are used for general familiarization with the terrain and for orientation when moving at high speed.
Measuring distances and areas using maps.
When measuring distances on maps, it should be remembered that the result is the length of horizontal projections of lines, and not the length of lines on the earth's surface. However, at small angles of inclination, the difference in the length of the inclined line and its horizontal projection is very small and may not be taken into account. So, for example, at an inclination angle of 2°, the horizontal projection is shorter than the line itself by 0.0006, and at 5° - by 0.0004 of its length.
When measuring from distance maps in mountainous areas, the actual distance on an inclined surface can be calculated
according to the formula S = d·cos α, where d is the length of the horizontal projection of the line S, α is the angle of inclination.
Inclination angles can be measured from a topographic map using the method indicated in §11. Corrections to the lengths of inclined lines are also given in the tables.
Rice. 6. Position of the measuring compass when measuring distances on a map using a linear scale
To determine the length of a straight line segment between two points, a given segment is taken from the map into a compass-measuring solution, transferred to the linear scale of the map (as indicated in Figure 6) and the length of the line is obtained, expressed in land measures (meters or kilometers).
In a similar way, measure the lengths of broken lines by taking each segment separately into a compass solution and then summing their lengths. Measuring distances along curved lines (along roads, borders, rivers, etc.)
etc.) are more complex and less accurate. Very smooth curves are measured as broken lines, having first been divided into straight segments. Winding lines are measured with a small constant opening of a compass, rearranging it (“walking”) along all the bends of the line. Obviously, finely sinuous lines should be measured with a very small compass opening (2-4 mm).
Knowing what length the compass opening corresponds to on the ground, and counting the number of its installations along the entire line, determine its total length. For these measurements, a micrometer or spring compass is used, the opening of which is adjusted by a screw passed through the legs of the compass.
7. Curvimeter
It should be borne in mind that any measurements are inevitably accompanied by errors (errors). According to their origin, errors are divided into gross errors (arising due to the inattention of the person making the measurements), systematic errors (due to errors in measuring instruments, etc.), random errors that cannot be fully taken into account (their reasons are not clear).
Obviously, the true value of the measured quantity remains unknown due to the influence of measurement errors. Therefore, its most probable value is determined. This value is the arithmetic average of all individual measurements x - (a1+a2+ …+аn):n=∑a/n, where x is the most probable value of the measured value, a1, a2…an are the results of individual measurements; 2 is the sign of the sum, n is the number of dimensions.
The more measurements, the closer the probable value is to the true value of A. If we assume that the value of A is known, then the difference between this value and the measurement of a will give the true measurement error Δ = A-a.
The ratio of the measurement error of any quantity A to its value is called relative error -. This error is expressed as a proper fraction, where the denominator is the fraction of the error from the measured value, i.e. Δ/A = 1/(A:Δ).
So, for example, when measuring the lengths of curves with a curvimeter, a measurement error of the order of 1-2% occurs, i.e. it will be 1/100 - 1/50 of the length of the measured line. Thus, when measuring a line 10 cm long, a relative error of 1-2 mm is possible.
This value on different scales gives different errors in the lengths of the measured lines. So, on a map of scale 1:10,000, 2 mm corresponds to 20 m, and on a map of scale 1:1,000,000 it will be 200 m.
It follows that more accurate measurement results are obtained when using large-scale maps.
Definition of areas plots on topographic maps is based on the geometric relationship between the area of the 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 a factor of n, then the area of this figure will decrease by a factor of n2.
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 cm2 - (100 m)2, i.e. in 1 cm2 - 1 hectare, and on a map of scale 1:1,000,000 in 1 cm2 - 100 km2.
To measure areas on maps, graphical and instrumental methods are used. The use of one or another measurement method is dictated 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.
8. Straightening the curved boundaries of the site and dividing its area into simple ones geometric figures: dots indicate cut-off areas, hatching indicates attached areas
When measuring the area of a plot with straight boundaries, divide the plot into simple geometric shapes, measure the area of each of them using a geometric method and, by summing the areas of individual plots calculated taking into account the map scale, obtain the total area of the object.
Plan scale
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. 8). The measurement results will be somewhat approximate.
Rice. 9. Square grid palette placed on the measured figure. Area of the plot P=a2n, a - side of the square, expressed on the map scale; n - number of squares falling within the contour of the measured area
Measuring the areas of areas with complex irregular configurations is often done using palettes and planimeters, which gives the most accurate results.
The grid palette (Fig. 9) is a transparent plate (made of plastic, organic glass or tracing paper) with an engraved or drawn grid of squares. 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 in land measures, i.e.
the price of dividing the palette.
Rice. 10. Dot palette - a modified square palette. Р=a2n
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.
10). 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.
11. A palette consisting of a system of parallel lines. The area of the figure is equal to the sum of the lengths of the segments (middle dotted lines) cut off by the contour of the area, multiplied by the distance between the lines of the palette.
Equally spaced parallel lines are engraved on the parallel palette. The measured area will be divided into a number of trapezoids with the same height when the palette is applied to it (Fig. 11). The segments of parallel lines inside the contour in the middle between the lines are the midlines of the trapezoids. Having measured all the middle lines, multiply their sum by the length of the gap between the lines and obtain the area of the entire area (taking into account the areal scale).
The areas of significant areas are measured from maps using a planimeter.
The most common is the polar planimeter, which is not very difficult to operate. However, the theory of this device is quite complex and is discussed in geodesy manuals.
12. Polar planimeter
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How to find out the map scale
A topographic map is a projection of a real ground mathematical model onto a plane in reduced form. The amount of relief image decreases and is called the denominator of the scale. In other words, the scale of a map is the ratio of the distance between two objects measured along it to the distance between the same objects measured on the ground. Knowing the scale of the map, you can always calculate the actual size and distance between objects located on the earth's surface. instructions
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© CompleteRepair.Ru
Geography lesson in 6th grade on the topic “Scale. Types of scale"
By scale, maps are divided into three groups: small-scale (1:1,000,000, 1:500,000, 1:300,000, 1:200,000); medium-scale (1:100000, 1:50,000, 1:25,000); large-scale (1:10000,1:5000, 1:2000,1:1000,1:500).
Large-scale topographic maps are the most accurate and suitable for detailed design.
Small-scale maps are intended: for a general study of the area during the general design of the development of the national economy, for taking into account the resources of the earth's surface and water space, for the preliminary design of large engineering facilities, for the needs of the country's defense.
Medium-scale maps have more detail and higher accuracy; designed for detailed design in agriculture, design of roads, routes, power lines, for preliminary development of planning and rural development settlements, to determine mineral reserves.
Large-scale maps and plans are prepared for more accurate detailed design (drawing technical projects, irrigation, drainage and landscaping, development master plans cities, design of engineering networks and communications, etc.).
The more important the survey task, the larger the scale required, but this is associated with high costs, so large-scale surveys must have an engineering justification.
Sheets of maps are published in a unified system of layout and nomenclature and represent a horizontal projection of a spheroidal trapezoid - a certain area of the earth's surface.
The nomenclature of topographic maps is usually called the designation of its individual sheets (trapezoids). The nomenclature of trapezoids is based on a sheet of map at a scale of 1:1000000, called the international map.
Types of scales
The scale can be written in numbers or words, or depicted graphically.
- Numerical.
- Named.
- Graphic.
Numerical scale
The numerical scale is signed with numbers at the bottom of the plan or map.
For example, a scale of “1: 1000” means that all distances on the plan are reduced by 1000 times. 1 cm on the plan corresponds to 1000 cm on the ground, or, since 1000 cm = 10 m, 1 cm on the plan corresponds to 10 m on the ground.
Named scale
The named scale of a plan or map is denoted in words.
For example, it may be written “1 cm - 10 m”.
Linear scale
It is most convenient to use a scale depicted as a straight line segment divided into equal parts, usually centimeters (Fig. 15). This scale is called linear and is also shown at the bottom of the map or plan.
Please note that when drawing a linear scale, the zero is set 1 cm from the left end of the segment, and the first centimeter is divided into five parts (2 mm each).
Next to each centimeter there is a sign indicating what distance it corresponds to on the plan.
One centimeter is divided into parts, next to which it is written what distance on the map they correspond to. Using a measuring compass or ruler, measure the length of any segment on the plan and, applying this segment to a linear scale, determine its length on the ground.
Application and use of scale
Knowing the scale, you can determine the distances between geographical objects and measure the objects themselves.
If the distance from the road to the river on a plan with a scale of 1: 1000 (“1 cm - 10 m”) is 3 cm, then on the ground it is 30 m.
Material from the site http://wikiwhat.ru
Let’s assume that from one object to another there are 780 m. It is impossible to show this distance in full size on paper, so you will have to draw it to scale. For example, if all distances are depicted 10,000 times smaller than in reality, i.e.
e. 1 cm on paper will correspond to 10 thousand cm (or 100 m) on the ground. Then, to scale, the distance in our example from one object to another will be equal to 7 cm and 8 mm.
Pictures (photos, drawings)
On this page there is material on the following topics:
What does the numerical scale show?
Report geographical scope
Scale definition of koroikr
Scale 1: 10 abstract
Causes of the revolution in Europe 1848-184
Questions for this article:
What is scale?
What does the scale show?
What can you measure with a scale?
How big is the lake if on a film with a scale of 1: 2000 (“1 cm - 20 m”) its length is 5 cm?
What does scale 1:5000, 1:50000 mean?
Which one is larger? Which scale is more convenient for a land plot plan, and which for a large city plan?
Material from the site http://WikiWhat.ru
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.
Geometric properties 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 options
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 one with the smaller denominator is called larger.
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. Standard scales have been determined for state topographic maps, forest management tablets, forestry and afforestation plans - scale series(Table 6.1, 6.2).
Scale series of topographic maps
Table 6.1.
| Numerical scale |
Card name |
1 cm card corresponds |
1 cm2 card corresponds |
|---|---|---|---|
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 scaleNamed 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.
To avoid mathematical calculations and speed up work on the map, use graphic scales . There are two such scales: linear And transverse .
Linear scaleTo 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 0C 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 scaleSometimes 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 build 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 build transverse scale with base length 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 scaleThis 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 plot the scale of steps for the above example you need horizontal line divide 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
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 d)
) and horizontal distance (
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.
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:
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 problemsTasks and questions for self-control
- What elements does it include? mathematical basis kart?
- Expand the concepts: “scale”, “horizontal distance”, “numerical scale”, “linear scale”, “scale accuracy”, “scale bases”.
- What is a named map scale and how do I use it?
- What is a transverse map scale, and what is its purpose?
- What transverse map scale is considered normal?
- What scales of topographic maps and forest management tablets are used in Ukraine?
- What is a transition map scale?
- How is the transition scale base calculated? Previous
