Solving tasks for topographic plans. Scale and its application Distance measurement with a caliper

called the scale, which is expressed as a fraction, the numerator of which is equal to one, and the denominator shows how many times the horizontal distance of the terrain line is reduced when displaying the horizontal distance of the line on a plan or map.

Numerical scale- unnamed value. It is written as follows: 1: 1000, 1: 2000, 1: 5000, etc., and in such a record 1000, 2000 and 5000 are called the denominator of the scale M.

The numerical scale suggests that one unit of line length on the plan (map) contains exactly the same number of units of length on the ground. So, for example, one unit of line length on the 1: 5000 plan contains exactly 5000 of the same length units on the ground, namely: one centimeter of the line length on the 1: 5000 plan corresponds to 5000 centimeters on the ground (i.e. 50 meters on the ground ); one millimeter of the line length on the 1: 5000 plan contains 5000 millimeters on the ground (i.e., one millimeter of the line length on the 1: 5000 plan contains 500 centimeters or 5 meters on the ground), etc.

When working with a plan, in some cases use linear scale.

Linear scale

- graphic construction, (Fig. 1) which is an image of a certain numerical scale.
Fig. 1

Linear scale base is called a segment AB of a linear scale (the main part of the scale), which is usually equal to 2 cm. It is translated into the appropriate length on the ground and signed. The leftmost base of the scale is divided into 10 equal parts.

Smallest division of the base of the linear scale equals 1/10 of the scale base.

Example: for a linear scale (used when working on a topographic plan of a scale of 1: 2000), shown in Figure 1, the base of the AB scale is 2 cm (i.e. 40 meters on the ground), and the smallest division of the base is 2 mm, which is scale 1: 2000 corresponds to 4 m on the ground.

Section cd (Fig. 1), taken from a topographic plan of scale 1: 2000, consists of two scale bases and two smallest divisions of the base, which, as a result, corresponds to 2x40m + 2x2m = 88 m on the ground.

A more accurate graphical definition and construction of line lengths can be done using another graphical construction - a transverse scale (Fig. 2).

Transverse scale

- graphic construction for the most accurate measurement and plotting of distances on the topographic plan (map). The scale accuracy is called a horizontal segment on the ground, which corresponds to a value of 0.1 mm on the plan of a given scale. This characteristic depends on the resolution of the naked human eye, which (resolution) allows us to consider the minimum distance on the topographic plan of 0.1 mm. On the ground, this value will already be equal to 0.1 mm x M, where M is the denominator of the scale

The base AB of the normal transverse scale is equal, as in the linear scale, also 2 cm. The smallest division of the base is CD = 1/10 AB = 2mm. The smallest division of the transverse scale is cd = 1/10 CD = 1/100 AB = 0.2mm (which follows from the similarity of triangle BCD and triangle Bcd).

Thus, for a numerical scale of 1: 2000, the base of the transverse scale will correspond to 40 m, the smallest division of the base (1/10 of the base) is 4 m, and the smallest division of the 1/100 AB scale is 0.4 m.

Example: segment ab (Fig. 2), taken from a plan of 1: 2000 scale, corresponds to 137.6 m on the ground (3 bases of the transverse scale (3x40 = 120 m), 4 smallest divisions of the base (4x4 = 16 m) and 4 smallest scale divisions (0.4x4 = 1.6 m), i.e. 120 + 16 + 1.6 = 137.6 m).

Let us dwell on one of the most important characteristics of the concept of "scale".

Accuracy of scale is called a horizontal segment on the ground, which corresponds to a value of 0.1 mm on the plan of a given scale. This characteristic depends on the resolution of the naked human eye, which (resolution) allows us to consider the minimum distance on the topographic plan of 0.1 mm. On the ground, this value will already be equal to 0.1 mm x M, where M is the scale denominator.

Fig. 2

The transverse scale, in particular, allows you to measure the length of a line on a plan (map) of 1: 2000 scale precisely with the accuracy of this scale.

Example: 1 mm of plan 1: 2000 contains 2000 mm of terrain, and 0.1 mm, respectively, 0.1 x M (mm) = 0.1 x 2000 mm = 200 mm = 20 cm, i.e. 0.2 m.

Therefore, when measuring (plotting) the length of the line on the plan, its value should be rounded with scale accuracy. Example: when measuring (plotting) a line 58.37 m long (Fig. 3), its value at a scale of 1: 2000 (with an accuracy of 0.2 m) is rounded to 58.4 m, and at a scale of 1: 500 (accuracy scale of 0.05 m) - the length of the line is rounded up to 58.35 m.

The scale of the map is the ratio of the length of the line 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.

Consider the types of scales.

Numerical scale

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

Numerical scale - 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 that indicate the lengths of mutually corresponding segments on the map and in nature.

For example, 1 centimeter is 5 kilometers (1 cm is 5 km).

Linear scale

it auxiliary measuring ruler applied to maps to facilitate measuring distances.

Plan scale and map scale

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

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

However, in practical measurements on the map, its main scale is used.

Scale Expression Forms

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

The numerical scale is expressed as a fraction, in which:

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

In Russia, standard numerical scales are 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 also create topographic maps in 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 is the one with the larger denominator M, and conversely, the smaller the denominator M, the larger the scale of the plan or map.

So the 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 the specified 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 dated 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:

One centimeter is 50 m.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 the signed values ​​of the lengths of the terrain lines commensurate with them. The linear scale allows you to measure or plot distances on maps and plans without calculations.

Scale accuracy

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

Since the accuracy of the scale expresses the length of the horizontal distance 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 scale map 1:25 000 scale accuracy is 2.5 m; for card 1:100 000 - 10 m, etc.

Scales of topographic maps

numerical scale cards	title cards	1 cm on the map corresponds to on the grounddistance	1 cm 2 on the map corresponds to in the area of ​​the square
	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 their 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 - 10000 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 - 10m (0.01 km) on the ground

Example 1

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

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

Solution:

For an easier conversion of a numerical scale to a named scale, you need to calculate how many zeros the number in the denominator ends in.

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

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

Example for a scale of 1: 500,000

There are five zeros in the denominator after the digit. Closing them, we get for the named scale: 1 cm on the map 5 kilometers on the ground.

If after the digit in the denominator there are less than five zeros, then, covering 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 we close 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 overlay on maps to help you measure distances.

Example 2

Convert the named scale to numeric:

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

Solution:

For an easier translation of a named scale into a numerical one, you need to convert the distance on the terrain indicated in the named scale to centimeters.

If the distance on the ground is expressed in meters, then to get 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, therefore, for a numerical scale, we assign two zeros and get: 1:10 000 .

For a scale of 1 cm - 5 km, we assign 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.

Map types depending on scales

Maps, depending on the scale, 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

Topographic maps are such maps, the content of which makes it possible to solve various technical problems using them.

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

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

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

The maps take into account the "sphericity" of the earth, but the plans do not. Because of this, plans are not drawn up for areas over 400 km² (ie plots of land approximately 20 km × 20 km).

• Standard scales for topographic maps

In our country, the following scales of topographic maps are adopted:

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 range 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 secret, topographic maps of a scale of 1: 100,000 are DSP (for official use), and smaller ones are unclassified.

Currently, there is a methodology for creating topographic maps and plans of any scale that do not have a secrecy stamp and are intended for open use.

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

Once upon a time there was a Capricious King. Once he traveled around his kingdom and saw how great and beautiful his land is. 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 the 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 mountain ranges, large cities and large lakes and rivers were indicated.

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

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

The capricious King died without waiting for the end of the work. The heirs, one after another, ascended the throne and died in turn, and the map was all drawn up and drawn up. Each king hired new cartographers to map the kingdom, but each time he was dissatisfied with the fruits of labor, finding the map insufficiently detailed.

Finally, the cartographers drew an incredible map! She depicted the entire kingdom in great detail - and was exactly the same size as the kingdom itself. Now no one could find the difference between the map and the kingdom.

Where were the Capricious Kings going to keep their wonderful map? The casket for such a card is not enough. You will need a huge room like a hangar, and in it the map will lie in many layers. But is such a card needed? After all, a full-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, horticulture, horticulture, which are owned, it is useful to familiarize yourself with the procedure for drawing up the cut-offs.
  
• From January 1, 2018, the exact boundaries of the site must be recorded in the cadastral passport, since it will be simply impossible to buy, sell, mortgage or donate land without an accurate description of the boundaries. This is regulated by amendments to the Land Code. And a total revision of the borders at the initiative of municipalities began on June 1, 2015.
• On March 1, 2015, a new Federal Law "On Amendments to the Land Code of the Russian Federation and Certain Legislative Acts of the Russian Federation" (N 171-ФЗ dated June 23, 2014) came into force, in accordance with which, in particular, the procedure for the purchase of land plots was simplified from municipalities.You can get acquainted with the main provisions of the law.
• Regarding the registration of houses, baths, garages and other buildings on land owned by citizens, a new dacha amnesty will improve the situation.

INTRODUCTION

The topographic map is reduced a generalized image of the area, showing the elements using a system of conventional signs.
In accordance with the required requirements, topographic maps are distinguished by a high geometric precision and geographic relevance. This is ensured by their scale, geodetic base, cartographic projections and system of conventional signs.
The geometrical properties of a cartographic image: the size and shape of areas occupied by geographic objects, distances between individual points, directions from one to another - are determined by its mathematical basis. Mathematical basis maps include as components scale, geodetic base, and cartographic projection.
What is the scale of the map, what types of scales are, how to build a graphical scale and how to use the scales will be discussed in the lecture.

6.1. TYPES OF SCALE TOPOGRAPHIC MAPS

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

Map scale (plan) - the ratio of the length of the line on the map (plan) to the length of the horizontal distance 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 ratio of the length of the line on the map to the length of the corresponding line on the ground can be considered the scale of length.

The scale is indicated on the 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 of the lengths of lines on the map (plan) in relation to the lengths of the corresponding lines on the ground. Comparing numerical scales among themselves, the larger one is called the one with the smaller denominator.
Using the numerical scale of the map (plan), you can determine the horizontal distance dm ground lines

Example.
The scale of the map is 1:50 000. The length of the segment on the map lK= 4.0 cm. Determine the horizontal distance of the line on the ground.

Solution.
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 to the fact that the numerical scale is an abstract quantity that does not have specific units of measurement. If the numerator of the 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 the map corresponds to 25,000 centimeters of terrain, or 1 inch of the 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 plans, forestry and afforestation plans, standard scales have been determined - scale series(Tables 6.1, 6.2).

Scale series of topographic maps

Table 6.1.

Numerical scale	Card name	1cm card matches on terrain distance	1cm2 card matches in the area of ​​the square
	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 of 1: 300,000, and 1: 2,000.

6.1.2. Named scale

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

Example... The named scale of the map is "1 centimeter 2 kilometers". The length of the segment on the map lK= 6.3 cm. Determine the horizontal distance of the line on the ground.
Solution... Multiplying the size of the segment measured on the map in centimeters by the value of the named scale, we get 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 graphical scales ... There are two such scales: linear and transverse .

Linear scale

To build a linear scale, an initial segment is selected that is 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 extreme left base is divided into parts (segment b), to be smallest divisions on a linear scale ... The distance on the ground, which corresponds to the smallest division of the linear scale, is called linear scale accuracy .

How to use a linear scale:

• put 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: counting whole bases and counting divisions of the left base (Fig. 6.1).
• If a segment on the map is longer than the built linear scale, then it is measured in parts.

Transverse scale

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

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

To build it, on a straight line segment, several scale bases are laid ( a). Usually the length of the base is 2 cm or 1 cm.At the obtained points, set perpendiculars to the line AB and draw ten parallel lines through them at regular intervals. The extreme left base above and below is divided into 10 equal segments and connected with oblique lines. The zero point of the lower base is connected to the first point WITH top base and so on. A series of parallel oblique lines is obtained, which are called transversals.
The smallest division of the transverse scale is equal to the line 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 the 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 centesimal . On a hundredth scale, 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 the cross scale:

• using a caliper to record the length of the line on the map;
• put 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: counting whole bases, plus counting divisions of the left base, plus counting divisions up the transversal.

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

6.2. VARIETIES OF GRAPHIC ZOOMS

6.2.1. Transitional scale

Sometimes in practice it is necessary 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 the production of practical work.
To simplify the determination of distances on a topographic map, proceed as follows. The base of the linear scale is taken not 2 cm, but calculated so that it corresponds to the 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 with a scale of 1: 17,500 (175 meters in one centimeter).
To determine what dimensions a segment of 400 m in length will have on a map of 1:17 500 scale, we compose 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 decided the proportion, we conclude: the base of the transitional scale in centimeters is equal to the size 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.

Now if you build 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. Transient linear scale.
Measured distance AC = BC + AB = 800 +160 = 960 m.

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

6.2.2. Step scale

Use this scale to determine the distances measured in steps during eye shooting. The principle of constructing and using a step scale is similar to a transition scale. The base of the scale of steps is calculated so that it corresponds to the round number of steps (pairs, triples) - 10, 50, 100, 500.
To calculate the size of the base of the scale of steps, it is necessary to determine the scale of the survey and calculate the average step length Shsr.
The average stride length (pairs of strides) is calculated from the known distance traveled in forward and backward 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 higher relief, the stride will be shorter, and in the opposite direction, longer.

Example... The known distance of 100 m is measured in steps. We walked 137 steps forward and 139 steps backward. Calculate the average length of one stride.
Solution... Total 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 size of one step 0.72 m at 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, in the corresponding scale, will be expressed as a segment of 0.36 cm. The most convenient basis for these conditions, in the opinion author, there will be a value of 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) .036 x 40 = 1.44 cm.
The base length of the step scale can also be calculated from proportions or by the formula
a = (Shsr × KSh) / M
where: Shsr - average value of one step in centimeters,
KSh - number of steps at the base of the scale ,
M - denominator of scale.

The base length for 50 steps on a scale of 1: 2,000 with one step length equal to 72 cm will be:
a= 72 × 50/2000 = 1.8 cm.
To build the scale of steps for the above example, 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. Scale of steps.
Measured distance AC = BC + AB = 100 + 20 = 120 sh.

6.3. SCALE ACCURACY

Scale accuracy (maximum scale accuracy) is a segment of the horizontal distance of the line corresponding to 0.1 mm on the plan. The value of 0.1 mm for determining the accuracy of the scale is taken 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.In 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 given example it follows that if the denominator of the numerical scale is divided by 10,000, then we get the ultimate accuracy of the scale in meters.
For example, for a numerical scale of 1: 5,000, the limiting accuracy of the scale will be 5,000 / 10,000 = 0.5 m.

Scale accuracy allows two important tasks to be accomplished:

• determination of the minimum sizes of objects and terrain items that are depicted at a given scale, and the sizes of objects that cannot be depicted on a given scale;
• setting the scale in which the map should be created so that objects and terrain objects with predetermined minimum dimensions are depicted on it.

In practice, it is assumed 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 on a given scale of 0.2 mm (0.02 cm) on the plan, is called graphic scale accuracy . Graphic accuracy of determining distances on a plan or map can only be achieved using a transverse scale.
It should be borne in mind that when measuring the relative position of the contours on the map, the accuracy is determined not by the graphic accuracy, but by the accuracy of the map itself, where errors can be on 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 error of measurements on the map, then we can conclude that the graphic accuracy of determining the distances on the map is 5 - 7 worse than the limiting accuracy of the scale, that is, it is 0.5 - 0.7 mm at the map scale.

6.4. DETERMINING AN UNKNOWN MAP SCALE

In cases where, for some reason, the scale on the map is absent (for example, cut off when gluing), it can be determined in one of the following ways.

• On a coordinate grid ... It is necessary to measure the distance on the map between the grid lines and determine how many kilometers these lines are drawn; this will determine the scale of the map.

For example, coordinate lines are indicated by 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 after 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 (in 1 centimeter, 1 kilometer).

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

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

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 sheet of a map of a given scale will always be the same.
Thus, if the map has the M-35-96 nomenclature, then, comparing it with the given diagram, we can immediately say that the scale of this map will be 1: 100,000.
For more information on the nomenclature of cards, see Chapter 8.

• By the distance between local objects. If there are two objects on the map, the distance between which is known on the ground 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 off. After rounding, we will have: 1 cm of the map corresponds to 1,000 m of the terrain, i.e. the scale of the map is 1: 100,000.
If there is a road with kilometer pillars on the map, then the scale is most conveniently determined by the distance between them.

• By the dimensions of the arc length of one minute of the meridian ... The frames of topographic maps along the meridians and parallels have divisions in minutes of the meridian arc and parallel.

One minute of the 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, 1 cm on the map will be 1852: 1.8 = 1,030 m. After rounding, we get the scale of the map 1: 100,000.
In our calculations, approximate values ​​of the scales are obtained. This happened due to the proximity of the distances taken and the inaccuracy of their measurement on the map.

6.5. TECHNIQUE FOR MEASURING AND STAYING DISTANCES ON THE MAP

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

6.5.1. Measuring distances with a millimeter ruler

Using a millimeter ruler, measure the distance between the specified 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 terrain distance in meters or kilometers.
Example. On a map with a scale of 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: at 1 cm 500 m.The distance on the ground between the points will be 3.4 × 500 = 1700 m.
When the angles of inclination of the earth's surface are more than 10º, it is necessary to introduce an appropriate correction (see below).

6.5.2. Distance measurement with a caliper

When measuring the distance in a straight line, the needles of the compass are set at the end points, then, without changing the solution of the compass, the distance is measured along a linear or transverse scale. In the case when the compass solution 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 by the usual order in scale.

Rice. 6.5. Measurement of distances with a compass-gauge on a linear scale.

To get the length broken line the length of each of its links is measured sequentially, and then their values ​​are summed up. Such lines are also measured by extending the compass solution.
Example... To measure the length of a polyline ABCD(fig. 6.6, a), the legs of the compass are first set at the points A and V... Then, rotating the compass around the point V... move the hind leg out of the point A exactly V"lying on the continuation of the straight line Sun.
Front leg from point V transfer to point WITH... The result is a compass solution B "C=AB+Sun... Moving the back leg of the compass in the same way from the point V" exactly WITH", and the front from 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 Sections measured along the chords with the steps of a compass (see Fig. 6.6, b). The step 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 arrows, consider the 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

Curved segments are measured with a mechanical (Figure 6.7) or electronic (Figure 6.8) curvimeter.

Rice. 6.7. Mechanical curvimeter

First, turning the wheel by hand, set the arrow to zero division, then roll the wheel along the measured line. The countdown on the dial opposite the end of the arrow (in centimeters) is multiplied by the magnitude of the map scale and the distance on the ground is obtained. The 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 handle metric and Anglo-American (feet, inches, etc.) values, which allows you to work with any maps and drawings. The most commonly used type of measurement can be entered and the instrument will automatically translate scale measurements.

Rice. 6.8. Digital curvimeter (electronic)

To improve the accuracy and reliability of the results, it is recommended to carry out all measurements twice - in forward and backward directions. In case of slight differences in the measured data, the arithmetic mean of the measured values ​​is taken as the final result.
The accuracy of measuring distances by the indicated methods using a linear scale is 0.5 - 1.0 mm on a 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 the horizontal projections of the lines (d) are obtained, and not the lengths of the 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 the line S;
v is the angle of inclination of the earth's surface.

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

Table 6.3

Tilt angle

Rules for using the table

1. The first row of the table (0 tens) shows the relative values ​​of the corrections at tilt angles from 0 ° to 9 °, in the second - from 10 ° to 19 °, in the third - from 20 ° to 29 °, in the fourth - from 30 ° up to 39 °.
2. To determine the absolute value of the correction, you must:
a) in the table, by the angle of inclination, find the relative value of the correction (if the angle of inclination of the topographic surface is not an integer number of degrees, then it is necessary to find the relative value of the correction by interpolating between the tabular 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 the line on the topographic surface, the calculated absolute value of the correction must be added to the length of the horizontal distance.

Example. On the topographic map, the horizontal length of 1735 m is determined, the angle of inclination of the topographic surface is 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 higher and the nearest lower value multiples of one degree - 8º and 7º:
for 8 ° the relative value of the correction is 0.98%;
for 7 ° 0.75%;
the difference in tabular values ​​is 1º (60 ′) 0.23%;
the difference between the given angle of inclination of the earth's surface 7 ° 15 "and the nearest lower tabular value of 7 ° 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 tilt 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 sloped 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. AREA MEASUREMENT BY MAPS

Determination of the areas of sites 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 the rectangle on the map are reduced by n times, then the area of ​​this figure will be reduced by n 2 times.
For a map with a scale of 1:10 000 (in 1 cm 100 m) the scale of areas will be (1: 10 000) 2 or in 1 cm 2 it will be 100 m × 100 m = 10 000 m 2 or 1 hectare, and on a map with a scale of 1 : 1,000,000 in 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 due to the shape of the measured area, the specified accuracy of the measurement results, the required speed of data acquisition and the availability of the necessary instruments.

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

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

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

An object with a curvilinear 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 surpluses mutually compensate each other (Fig. 6.10). The measurement results will be approximate to some extent.

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

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

Measuring the area of ​​plots, having a complex misconfiguration, more often they are produced using pallets and planimeters, which gives the most accurate results. Mesh palette is a transparent plate with a grid of squares (Fig. 6.11).

Rice. 6.11. Square Grid Palette

The palette is applied to the measured contour and the number of cells and their parts inside the contour is counted using it. Fractions of incomplete squares are assessed by eye, therefore, to improve 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 by the formula:

P = a 2 n,

Where: a - the side of the square, expressed in terms of the scale of the map;
n- the number of squares that fall within the contour of the measured area

To improve accuracy, the area is determined several times with an arbitrary permutation of the used pallet 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 grid pallets, point and parallel pallets 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 division value of the palette. The area of ​​the area to be measured is determined by counting the number of points inside the contour and multiplying this number by the point weight.
Equally spaced parallel straight lines are engraved on a parallel palette (Fig. 6.13). The measured area, when the palette is applied to it, will be divided into a row of trapezoids with the same height h... The parallel line segments within the outline (midway between the lines) are the midline of the trapezoid. To determine the area of ​​the site using this palette, you need to multiply the sum of all measured midlines by the distance between the parallel lines of the palette h(subject to scale).

P = h∑l

Fig 6.13. Palette consisting of a system
parallel lines

Measurement areas of significant plots produced by cards using planimeter.

Rice. 6.14. Polar planimeter

The planimeter is used to determine areas mechanically. The polar planimeter is widespread (Fig. 6.14). It consists of two levers - pole and bypass. Determination of the contour area with a planimeter is reduced to the following steps. After fixing the pole and setting the needle of the bypass lever at the starting point of the contour, take a reading. Then the bypass spire 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 division price of the planimeter, the area of ​​the contour 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, among which are electronic planimeters.

Rice. 6.15. Electronic planimeter

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

This method allows you to determine the area of ​​the site of any configuration, i.e. with any number of vertices, the coordinates of which (x, y) are known. In this case, the vertices must be numbered 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 figures 1y-1-2-3-3y and S" of figures 1y-1-4-3-3y
S = S "- S".

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

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

S "= square 1y-1-2-2y + square 2y-2-3-3y,
S "= pl 1y-1-4-4u + pl. 4y-4-3-3y
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). Expanding 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 represent expressions (6.1) and (6.2) in general form, denoting by i the ordinal number (i = 1, 2, ..., n) of the vertices of the polygon:
(6.3)
(6.4)
Therefore, the doubled area of ​​the polygon is either the sum of the products of each abscissa and the difference between the ordinates of the next 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.
An intermediate control of the 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 ​​\ u200b \ u200bplot can be easily solved using MicrosoftXL spreadsheets. An example for a polygon (polygon) of 5 points is shown 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 m 2			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 m 2
	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 the distance, direction, area, steepness of the slope and other characteristics of objects from the map contributes to mastering the skills of a correct understanding of the cartographic image. The accuracy of eye measurements increases with experience. Eye-gazing skills prevent gross miscalculations in measurements with instruments.
To determine the length of linear objects on the map, you 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, the squares of the kilometer grid are used as a kind of palette. Each square of the grid of maps with scales of 1: 10,000 - 1: 50,000 on the ground corresponds to 1 km 2 (100 ha), a scale of 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

Scope Tasks

Assignments 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 the transverse scale of the map, for what purpose is it intended?
5. What is the normal transverse scale of the map?
6. What are the scales of topographic maps and forest management plans used in Ukraine?
7. What is the Transitional Map Scale?
8. How is the base of the transition scale calculated?
Previous

Scale(it. Maßstab, letters. "Measuring stick": Maß"measure", Stab"Stick") - in the general case, the ratio of two linear dimensions. In many areas of practical application, scale refers to the ratio of the size of the image to the size of the displayed object.

The concept is most common in geodesy, cartography and design - the ratio of the size of the image of an object to its actual size. A person is not able to depict large objects, for example, a house, in full size, therefore, when depicting a large object in a drawing, drawing or model, the size of the object is reduced several times: two, five, ten, one hundred, a thousand, and so on. The number showing how many times the depicted object is reduced is the scale. The scale is also used when depicting the microcosm. A person cannot depict a living cell, which he examines through a microscope, in full size and therefore increases the size of its image several thousand times. The number showing how many times the increase or decrease of a real phenomenon is made when it is depicted, is defined as a scale.

Scale in surveying, mapping and design

Scale shows how many times each line plotted on a map or drawing is less or more than its actual size. There are three types of scale: numerical, named, graphic.

The scales on maps and plans can be represented numerically or graphically.

Numerical scale written in the form of a fraction, in the numerator of which there is one, and in the denominator - the degree of reduction of the projection. For example, a scale of 1: 5,000 shows that 1 cm on the plan corresponds to 5,000 cm (50 m) on the ground.

The larger is the scale with the smaller denominator. For example, a scale of 1: 1,000 is larger than a scale of 1: 25,000. In other words, if large scale the object is displayed larger (larger), with more small scale- the same object is rendered smaller (smaller).

Named scale shows what distance on the ground corresponds to 1 cm on the plan. It is recorded, for example: "In 1 centimeter 100 kilometers", or "1 cm = 100 km".

Graphic scales are subdivided into linear and transverse.

• Linear scale is a graphical scale in the form of a scale bar divided into equal parts.
• Transverse scale is a graphical scale in the form of a nomogram, the construction of which is based on the proportionality of the segments of parallel straight lines intersecting the sides of the corner. The transverse scale is used for more accurate measurements of line lengths on plans. The transverse scale is used as follows: measure the length on the bottom line of the transverse scale so that one end (right) is on the whole division OM, and the left goes beyond 0. If the left leg falls between tenth divisions of the left segment (from 0), then raise both legs of the meter up until the left leg hits the intersection of some transvensal and some horizontal line. In this case, the right leg of the meter should be on the same horizontal line. The smallest CP = 0.2 mm, and the accuracy is 0.1.

Scale accuracy- this is a segment of the horizontal space of the line, corresponding to 0.1 mm on the plan. The value of 0.1 mm for determining the accuracy of the scale is taken 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 equal to 1 m.In 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 (1 m).

The scale of images in drawings should be selected from the following range:

When designing master plans for large objects, it is allowed to use a scale of 1: 2,000; 1: 5,000; 1:10 000; 1:20 000; 1:25 000; 1:50 000.
In necessary cases, it is allowed to use the magnification scales (100n): 1, where n is an integer.

Scale in photography

Main article: Linear magnification

When photographing, the scale is understood as the ratio of the linear size of the image obtained on a photographic film or photosensitive matrix to the linear size of the projection of the corresponding part of the scene onto a plane perpendicular to the direction to the camera.

Some photographers measure scale as the ratio of the size of an object to the size of its image on paper, screen, or other media. The correct method for determining the scale depends on the context in which the image is used.

Scale is important when calculating depth of field. Photographers have access to a very wide range of scales - from almost infinitely small (for example, when photographing celestial bodies) to very large (without using special optics, it is possible to obtain scales of the order of 10: 1).

Macro photography is traditionally defined as shooting at a scale of 1: 1 or larger. However, with the widespread use of compact digital cameras, this term also began to refer to shooting close to the lens (as a rule, closer than 50 cm) small objects. This is due to the necessary change in the operating mode of the autofocus system in such conditions, however, from the point of view of the classical definition of macro photography, this interpretation is incorrect.

Scale in Modeling

Main article: Scale (modeling)

For each type of large-scale (bench) modeling, large-scale series have been determined, consisting of several scales of varying degrees of reduction, and for different types of modeling (aircraft modeling, ship modeling, railway, automobile, military equipment), their own, historically established, large-scale series are defined, which usually do not intersect ...

The scale in modeling is calculated by the formula:

Where: L is the original parameter, M is the required scale, X is the desired value

For example:

At a scale of 1/72, and the original setting is 7500 mm, the solution will look like;

7500 mm / 72 = 104.1 mm.

The resulting value is 104.1 mm, which is the desired value at a scale of 1/72.

Time scale

In programming

In time-sharing operating systems, it is extremely important to provide individual tasks with so-called "real-time", in which the processing of external events is provided without additional delays and gaps. For this, the term "real time scale" is also used, but this is a terminological convention that has nothing to do with the original meaning of the word "scale".

In film technology

Main article: Accelerated filming # Time scale Main article: Time lapse filming # Time scale

The time scale is a quantitative measure of the slowdown or acceleration of movement, equal to the ratio of the projected frame rate to the shooting frame rate. So, if the projected frame rate is 24 frames per second, and filming was performed at 72 frames per second, the time scale is 1: 3. A 2: 1 time scale means twice as fast as the normal process flow on the screen.

In mathematics

Scale is the ratio of two linear dimensions. In many areas of practical application, scale refers to the ratio of the size of the image to the size of the displayed object. In mathematics, scale is defined as the ratio of the distance on the map to the corresponding distance in the real world. A scale of 1: 100,000 means that 1 cm on the map corresponds to 100,000 cm = 1,000 m = 1 km on the ground.

/ WHAT IS THE SCALE

Scale. Scale views

Geography. 7th grade

What is scale?

The scale shows how many times the distance on the map is less than the corresponding distance on the ground.

A scale of 1: 10,000 (read one ten-thousandth) shows that each centimeter on the map corresponds to 10,000 centimeters on the ground.

What does scale mean

Scale views

What types of scale are shown here? Which one is missing?

Write in 1 cm -

Since there are 100 centimeters in 1 meter, you need to remove two zeros

Since there are 1000 meters in 1 kilometer, you need to remove three more zeros (if possible)

Write the remaining number after the dash, indicate meters or kilometers

How to convert a numeric scale to a named scale

			in 1 cm - 5 m
			in 1 cm - 200 m
			1 cm - 30 km

Converting a scale from numeric to named

Check the answers

h1 cm - 5 m

h1 cm - 15 m

h1 cm - 500 m

h1 cm - 2 km

h1 cm - 30 km

h1 cm - 600 km

h1 cm - 15 km

Exercises. Convert scale from numerical to named

How to calculate 1:50 scale?

The scale is used to place an area on the drawing, which is actually many times larger. At a scale of 1:50, all dimensions are taken 50 times smaller than they actually are. For example, the drawing is drawn at a scale of 1:50. On it, the size of 50 meters is taken as 1 meter. If you want to depict a bench 5 meters long, then in the figure its length will be 10 cm. Such a small scale is used in construction drawings for the graphic representation of a small area (landscape design). Conclusion: when executing a drawing with a scale of 1:50, all original dimensions must be divided by 50.

Mirra-mi

A scale of 1 to 50 means that all objects and lines in the drawing are reduced 50 times than they actually are. That is, 1 cm in the drawing is 50 cm in reality. Therefore, while reading such a drawing, each centimeter must be multiplied by 50:

1 cm is 50 cm,

2 cm is 100 cm,

10 cm is 500 cm, etc.

A scale of 1:50 means that the object (drawing, map, graph, drawing, object, sketch, etc.) that we see is reduced in comparison with its original dimensions by fifty times. Where the length is shown, for example, one centimeter in the original means fifty centimeters.

Zolotynka

To understand what the 1:50 scale is, consider an example: suppose we have a model car produced at a scale of 1 in 50. This means that the real car is 50 times larger than our model.

The same applies to maps: when we depict on a scale some terrain on a piece of paper or a computer screen, we reduce the distances by 50 times, but we will definitely preserve all the features of the terrain and all the proportions. The scale clearly shows the relationship between distances on the map and distances on the ground. This makes the map convenient for us, as we receive visual information with which we can easily calculate ground distances.

Those. in order to create a model at a scale of 1 to 50 (anything - object, terrain), you need to divide the real size by 50.

Azamatik

To do this, let's use an example.

A scale of 1 to 50 means, for example, that 50 kilometers is taken as 1 kilometer; 50 meters is taken as 1 meter; 50 centimeters as 1 centimeter, etc.

Let's take a real football field, which is 100 meters long and 50 meters wide.

To depict this field on a sheet of paper at a scale of 1 to 50, divide both the width and the length by 50 (50 m).

Therefore, this football field on a scale of 1:50 will be 2 meters long and 1 meter wide.

Moreljuba

The scale is a very necessary and important thing. It is very important when creating terrain drawings and maps. If we are talking about a scale of 1:50, then this means that all real objects when transferring them to our drawing should be 50 times reduced in size. In other words, the sizes of objects should be divided by 50. For example, if we need to draw an object 100 centimeters long on a drawing, we reduce it to 2 centimeters (100/50).

Quite simply, if this is some kind of drawing, then this means that all the details, for example, the model of the ship, are reduced by 50 times and to represent the true size of the ship from which this drawing was made, you will need to increase the model by 50 times, that is, multiply the size all parts by 50.

Raziyusha

If you need to make rooms, some object on a scale of 1:50, then you need to do it like this: divide each length by 50 cm, draw the result on paper. Let's say a wall with a length of 6 m in the drawing will be 12 cm long.How is this calculated:

6 m = 600 cm,

600: 50 = 12 cm.

Pollock tail

It turns out that all the objects in the picture are reduced by fifty times. In order to calculate the scale of the object, you need to measure the picture with a regular ruler after 1 cm. Multiply by 50. Actually, this is the real scale of the object.

The question is on the verge of fantasy. A scale of one to fifty is the ratio of one scale unit containing 50 real scale units. For example, 1 cm of the established scale contains 50 cm of the real one.

What is scale?

Daria remizova

Scale
(German Maßstab, from Maß - measure, size and Stab - stick), the ratio of the length of the segments in a drawing, plan, aerial photograph or map to the lengths of the corresponding segments in nature. The numerical Scale thus defined is an abstract number, greater than 1 in cases of drawings of small parts of machines and devices, as well as many micro-objects, and less than 1 in other cases, when the denominator of the fraction (with the numerator equal to 1) shows the degree of reduction in the size of the image of objects relative to their actual sizes. The scale of plans and topographic maps is a constant value; The scale of geographical maps is a variable value. For practice, a linear scale is important, that is, a straight line divided into equal segments with captions indicating the lengths of the corresponding segments in nature. For a more accurate drawing and measurement of lines on the plans, a so-called transverse scale is built. A transverse scale is a linear scale, parallel to which a series of equidistant horizontal lines are drawn, intersected by perpendiculars (verticals) and oblique lines (transversals). The principle of building and using a transverse scale. is clear from the figure given for a numerical scale of 1: 5000. The section of the transverse scale, marked with dots in the figure, corresponds on the ground with a line 200 + 60 + 6 = 266 m. , sometimes without any inscriptions. This makes it easy to use it in the case of any numerical scale used in practice.
A scale of 1: 200 means that 1 unit of measurement in a figure or drawing corresponds to 200 units of measurement in space. For example: topographic map - the atlas of the Tver region has a scale of 1: 200000. This means that 1 centimeter on the map is equal to 2 kilometers on the ground.

Dmitry mosendz

A scale of 1: 200 means that 1 unit of measurement in a figure or drawing corresponds to 200 units of measurement in space. For example: topographic map - the atlas of the Tver region has a scale of 1: 200000. This means that 1 centimeter on the map is equal to 2 kilometers on the ground.

Betuganov Astemir

Project Manager:

Shopagova Alla Sergeevna

Institution:

MCOU "School No. 27", Nalchik

In the presented research paper in mathematics on the topic "Scale and its application" I will try to find out at what scale it will be convenient to place objects on A4 sheet. Working on a research project about scale will help solidify my knowledge of mathematics.

In my research project in mathematics "Scale and its application" I will need to refine and correlate the mathematical calculations with the data obtained.

In the course of my research in mathematics about scale and its applications, I hope that the scales I set will allow me to arrange objects on an A4 album sheet.

Also, in the practical part of my work, I will consider and mathematically solve interesting problems in terms of distance and scale.

Introduction
Main part
1. Determination of the scale.
2. Solving interesting problems on a scale.
conclusions
Applications.

Introduction

In the 6th grade mathematics lessons, we went through this interesting topic, from which we learned how, using a scale, you can find the distance on the ground, knowing the length of the segment on the map, corresponding to this distance on the ground, and vice versa.

When drawing images of objects on paper, we most often have to change their actual sizes: large objects have to be depicted in a reduced form, and small ones - to enlarge.

Areas of the earth's surface are depicted on paper in a reduced form. An example of such an image is any map, plan. And small details are shown in the drawings in an enlarged form.

But a drawing, map or plan should give an idea of ​​the real dimensions of the objects. Therefore, a special entry is made on the drawings and maps, showing the ratio of the length of the segment on the map or drawing to its real length.

The topic of my research project in mathematics is “ Scale and its application».

Objective of the project: find out at what scale it will be convenient to place objects on A4 sheet.

Project objectives:

1. to consolidate school knowledge in mathematics;
2. clarify whether the mathematical calculations are comparable with the data obtained.

Hypothesis: patterns are most efficiently drawn 1:10, apartment layout 1: 100; house passport 1: 1000; city ​​map 1: 10000; area map 1: 100000.

Expected Result: the scales I have set will allow you to arrange objects on the album sheet.

Equipment:
ruler, pencil, compasses, calculator, map.
sheet A 4, ruler, pencil.

Scale determination

Scale- this is a fraction, where the numerator is one, and the denominator is the number that shows how many times the distance on the terrain plan is reduced than on the terrain.

For example: 1: 1000 (one thousandth) means that all distances on the ground are reduced by a factor of one thousand. The larger the number in the denominator of the fraction, the greater the decrease and the greater the coverage of the territory.

• numerical, expressed in numbers 1: 1000;
• named, expressed in words, that is, we translate cm into m: in 1cm 10m, 10m is the magnitude of the scale;
• linear knowing the magnitude of the scale, we can determine the distances.

Let's look at the map. The scale is indicated at the top (1: 500,000). They say that the map was made on a scale of one five hundred thousandth. This means that 1 cm on the map corresponds to 500,000 cm on the ground. This means that a 1 cm segment on the map corresponds to a 5 km segment on the terrain.

And if I take a 3 cm segment on the map, then on the ground it will be a 15 km segment.

I downloaded a map of the Kabardino-Balkarian Republic from the Internet. A map of the republic with a scale of 1: 10000, that is, 1 cm 100 meters, and a scale of the surroundings 1: 100000 in 1 cm 1 kilometer. I immediately found my native village on it.

So, the scale (it. Maßstab, letters. " measuring stick»: Maß « measure», Stab « stick») - in general, the ratio of two linear dimensions.

In many areas of practical application scale is the ratio of the image size to the size of the displayed object .

The concept of scale is most common in geodesy, cartography and design - the ratio of the actual size of an object to the size of its image.

A person is not able to depict large objects, for example, a house, in full size, and therefore when depicting a large object in a drawing, drawing, layout, and so on, a person reduces the size of the object several times: two, five, ten, one hundred, one thousand, etc. so on again. The number showing how many times the depicted object is reduced, there is a scale.

The scale is also used when depicting the microcosm. A person cannot depict a living cell, which he examines through a microscope, in full size and therefore increases the size of its image several times.

The number showing how many times the increase or decrease of the real phenomenon is made when it is depicted, is defined as a scale.

Some photographers measure scale as the ratio of the size of an object to the size of its image on paper, screen, or other media.

The correct method for determining the scale depends on the context in which the image is used.

conclusions

Compared their assumptions, put forward in my hypothesis with inscriptions on patterns, maps and technical plans of a house and apartment. It turned out that in some places I was mistaken 10 and even 100 times.

• patterns are most efficiently drawn 1:10;
• apartment layout 1: 100;
• house passport 1: 1000;
• city ​​map 1: 10000;
• area map 1: 100000.

In fact, the plan of an apartment is usually taken on a scale of 1: 200; the scale of the maps turned out to be exactly the same as in the original, but they are located as much on 6 album sheets!

So once again, I am convinced that before making a guess, you need to recalculate several times.

Thus, we formed the concept of scale, map, drawing, worked out the solution of problems for calculating the length of a segment on the ground and on the map.

Solving problems at scale

Objective 1. The distance between the two cities is 400 km. Find the length of the line segment connecting these cities on a map drawn at a scale of 1: 5,000,000.

Solution:
400km = 400000m = 40000000cm
40,000,000: 5,000,000 = 40: 5 = 8 (cm)

Objective 2. The straight-line distance from Moscow to St. Petersburg is approximately 635 km from center to center. The length of the route along the highway is 700 km.
How many times should this distance be reduced so that it can be depicted on a slide as a segment 14 cm long?

Solution:
700km = 700000m = 70000000cm
70,000,000cm: 14cm = 5,000,000 (times)

Objective 3. Using a physical map of Russia, determine the real distance between Moscow and St. Petersburg.
M1: 20,000,000 if the distance on the map is 3 cm.