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What is the scale in 1 cm 10 meters? Numerical, linear and transverse scales

Scale(German) Massstab, lit. "measuring stick": Maß"measure", Stab"stick") - in the general case, the ratio of two linear dimensions. In many areas of practical application, scale is the ratio of the size of the image to the size of the depicted object.

The concept is most common in geodesy, cartography and design - the ratio of the size of the image of an object to its natural size. A person is not able to depict large objects, for example a house, in life-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. Scale is also used when depicting the microworld. A person cannot depict a living cell, which he examines through a microscope, in natural size and therefore increases the size of its image several thousand times. The number showing how many times the real phenomenon is increased or decreased when depicting it is defined as scale.

Scale in geodesy, cartography and design

Scale shows how many times each line drawn on a map or drawing is smaller or larger than its actual dimensions. There are three types of scale: numerical, named, graphic.

Scales on maps and plans can be presented numerically or graphically.

Numerical scale written as a fraction, the numerator of which is one, and the denominator is 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 scale is the one whose denominator is smaller. For example, a scale of 1:1,000 is larger than a scale of 1:25,000. In other words, with more on a large scale the object is depicted larger (larger), with more small scale- the same object is depicted smaller (smaller).

Named scale shows what distance on the ground corresponds to 1 cm on the plan. It is written, for example: “There are 100 kilometers in 1 centimeter”, or “1 cm = 100 km”.

Graphic scales are divided into linear and transverse.

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

Scale accuracy- this 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 (1 m).

The scales of images in the drawings must 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.
If necessary, it is allowed to use magnification scales (100n):1, where n is an integer.

Scale in photography

Main article: Linear increase

When taking photographs, scale is understood as the ratio of the linear size of the image obtained on photographic film or a light-sensitive matrix to the linear size of the projection of the corresponding part of the scene onto a plane perpendicular to the direction of 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 technique for determining scale depends on the context in which the image is being 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 the use of special optics it is possible to obtain scales of the order of 10:1).

Macro photography traditionally refers to 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 small objects located close to the lens (usually closer than 50 cm). 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, such an interpretation is incorrect.

Scale in modeling

Main article: Scale (modeling)

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

Scale in modeling is calculated using the formula:

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

For example:

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

7500 mm / 72 = 104.1 mm.

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

Time scale

In programming

IN operating systems With time sharing, it is extremely important to provide individual tasks with the so-called “real-time mode”, in which the processing of external events is ensured without additional delays and omissions. For this purpose, 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: Time-lapse photography#Time scale Main article: Slow Motion#Time Zoom

Time scale is a quantitative measure of slowing down or accelerating movement, equal to the ratio of the projection frame rate to the shooting frame rate. So, if the projection frame rate is 24 frames per second, and the film was shot at 72 frames per second, the time scale is 1:3. A 2:1 time scale means the process on the screen is twice as fast as normal.

In mathematics

Scale is the ratio of two linear dimensions. In many areas of practical application, scale is the ratio of the size of the image to the size of the depicted object. In mathematics, scale is defined as the ratio of a distance on a map to the corresponding distance on the real terrain. 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 SCALE

Scale. Types of scale

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 (one ten-thousandth is read) shows that each centimeter on the map corresponds to 10,000 centimeters on the ground.

What does scale mean?

Types of scale

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

Record 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 numerical scale to a named scale


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

Converting scale from numerical to named

Check the answers

h1 cm – 5 m

h1 cm – 15 m

h1 cm – 500 m

in 1 cm – 2 km

in 1 cm – 30 km

h1 cm – 600 km

in 1 cm – 15 km

Exercises. Convert scale from numeric to named

How to calculate 1:50 scale?

Scale is used to place on a drawing an area that is actually many times larger. At a scale of 1:50, all dimensions are taken 50 times smaller than in reality. For example, the drawing is drawn at a scale of 1:50. On it, a size of 50 meters is taken as 1 meter. If you want to depict a shop 5 meters long, then in the drawing its length will be 10 cm. Such a small scale is used in construction drawings to graphically depict a small area (landscape design). Conclusion: when making 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 in the drawing all objects and lines are reduced 50 times than they actually are. That is, 1 cm in the drawing is 50 cm in reality. Therefore, when 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 fifty times compared to its original dimensions. Where the length is shown, for example, one centimeter in the original means fifty centimeters.

Zolotynka

To understand what 1:50 scale is, let's look at an example: Let's say we have a car model produced at 1:50 scale. This means that the real car is 50 times larger than our model.

The same applies to maps: when we depict a certain area to scale on a piece of paper or a computer screen, we reduce the distances by 50 times, but we make sure to preserve all the features of the area and all the proportions. The scale clearly demonstrates the relationship between distances on the map and distances on the ground. This makes the map useful for us, as we get visual information with which we can easily calculate ground distances.

Those. in order to create a model on a scale of 1 to 50 (anything - an object, a location) you need actual size divide 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 is the same as 1 centimeter, etc.

Let's take a real football field, whose length is 100 meters and its width is 50 meters.

To depict this field on a sheet of paper on a scale of 1 to 50, we divide both the width and 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

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 transferred to our drawing, must be reduced in size by 50 times. In other words, the sizes of objects must be divided by 50. For example, if you need to draw an object 100 centimeters long, we reduce it to 2 centimeters (100/50).

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

Raziyusha

If you need to make rooms or 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 6 m long 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 objects in the picture are reduced by fifty times. In order to calculate the scale of an 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 borders on 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 lengths of segments in a drawing, plan, aerial photograph or map to the lengths of the corresponding segments in nature. The numerical scale defined in this way is an abstract number greater than 1 in cases of drawings of small parts of machines and instruments, as well as many micro-objects, and less than 1 in other cases when the denominator of the fraction (with a numerator equal to 1) shows the degree of reduction in the size of the image of objects relative to their real ones sizes. The scale of plans and topographic maps is a constant value; The scale of geographical maps is a variable value. Important for practice linear scale, that is, a straight line divided into equal segments with captions indicating the lengths of the corresponding segments in kind. For 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 number of equally spaced horizontal lines are drawn, intersected by perpendiculars (verticals) and oblique lines (transversals). The principle of constructing 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 in the figure with dots, corresponds on the ground to the line 200 + 60 + 6 = 266 m. A transverse scale is also called a metal ruler on which an image of such a drawing is carved with very thin lines , sometimes without any inscriptions. This makes it easy to use in the case of any numerical scale used in practice.
A scale of 1:200 means that 1 unit of measurement in a picture or drawing corresponds to 200 units of measurement in space. For example: a topographic map - 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 picture or drawing corresponds to 200 units of measurement in space. For example: a topographic map - 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.

Theme "Scale"

Materials for preparing for the lesson

T.V. KONSTANTINOV
Ph.D. ped. sciences, senior lecturer
E.A. KUZNETSOVA
Kaluga State Pedagogical University
them. K.E. Tsiolkovsky

Means of education

A terrain plan (preferably your own area), a physical map of the hemispheres, a physical map of Russia, measuring instruments (measuring tape, range finder).

Terms and concepts

Scale ( from German - measure and Stab - stick) - 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.

Geographical sciences and professions of geographers

Geodesy (Greek - land division) - a science that studies the shape and size of the Earth, methods of measuring distances, angles and heights on the earth's surface.
Topography(Greek - place and - write) - a section of geodesy devoted to measurements on the ground to create maps and plans.
Cartography- the science of maps, their creation and use. Cartography also studies globes, plans and other images of the earth's surface, in addition, maps and globes of the starry sky and other planets.

Geographer's Toolkit

A measuring compass is a tool for transferring dimensions to drawings. When working with geographical maps used to determine distances between points and individual sections of the map.
Curvimeter - a mechanical portable device designed to measure the lengths of winding lines using maps. It consists of a round box with a dial and pointer, and a small wheel at the bottom. The divisions on the dial scale can indicate the path traveled by the wheel on the map (in cm), or immediately show the distance on the ground, depending on the scale of the map.
Rangefinders - devices various types, used to determine distances without directly measuring them with a measuring tape or tape measure.
Measuring tape - the main instrument used to measure distances before the invention of rangefinders. It is a steel strip, usually 20 m long, secured to the ground with long (about 0.5 m) steel pins.

Geographic nomenclature

Local names: settlement where students live, streets, shops, educational institutions, nearby bodies of water, various local landforms, etc.

Independent work of students

Determining distances on maps using a scale

Purpose of the work: developing skills in working with different types of scales; developing the ability to determine distances on maps using scale.
Equipment: geography atlas for 6th grade, curvimeter or thread about 20 cm long, workbook.

Exercise 1. Convert the numerical scale of the map to a named one:

a) 1: 200,000
b) 1: 10,000,000
c) 1: 25,000

Rule for students. To more easily convert a numerical scale into a named one, you need to count how many zeros the number in the denominator ends with. For example, on a scale of 1:500,000, there are five zeros in the denominator after the number 5.
If after the number in the denominator there is five and more zeros, then by covering (with a 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 scale 1: 500,000. The denominator after the number is five zeros, closing them, we get for a named scale: 1 cm on the map is 5 kilometers on the ground.
If there are less than five zeros after the number in the denominator, then by closing two zeros, we get the number of meters on the ground corresponding to 1 centimeter on the map. If, for example, in the denominator of a scale of 1: 10,000 we close two zeros, we get: 1 cm - 100 m.
Answer: a) 1 cm - 2 km; b) 1 cm - 100 km; c) 1 cm - 250 m.

Task 2. Convert the named scale to a numerical one:

a) 1 cm - 500 m

b) 1 cm - 10 km

c) 1 cm - 250 km

Rule for students. To more easily convert a named scale to a numerical one, you need to convert the distance on the ground indicated in the named scale into centimeters. If the distance on the ground is expressed in meters, in order to obtain the denominator of the numerical scale, you need to assign two zeros, if in kilometers, then five zeros.
For example, for a named scale of 1 cm - 100 m, the distance on the ground is expressed in meters, so for the numerical scale we assign two zeros and get: 1: 10,000. For a scale of 1 cm - 5 km we assign five zeros to the five and get: 1 : 500,000.
Answers: a) 1: 50,000; b) 1: 1,000,000; c) 1: 25,000,000.

Task 3. Determine the distance between points by physical map Russia in the 6th grade atlas:

a) Moscow and Murmansk
b) Mount Narodnaya (Ural Mountains) and Mount Belukha (Altai Mountains)
c) Cape Dezhnev (Chukchi Peninsula) and Cape Lopatka (Kamchatka Peninsula)

Rule for students. When determining the distance on a map between points, you should:
1. Using a ruler, measure the distance in centimeters between points. For example, the distance between the cities of Moscow and Astrakhan on the map is 6.5 cm.
2. Find out by the named scale how many kilometers (meters) on the ground correspond to 1 cm on the map.
(On the physical map of Russia in the 6th grade geographic atlas, 1 cm on the map corresponds to 200 km on the ground.)
3. Multiply the distance between points measured with a ruler by the number of kilometers (meters) on the ground for a given scale.

6.5 x 200 = 1300 km.

Answers: a) 1460 km; b) 2240 km; c) 2500 km* * .

Task 4. Measure the length of rivers using the physical map of Russia in the 6th grade atlas:

a) Oka;
b) the Ural River;
c) Kama.

Measurements of winding lines on a map (in this case, rivers) are carried out using a curvimeter or a thread.
How to measure the length of a river using a thread (rule for students).
1. The thread must be moistened, otherwise it will be difficult to lay it on the paper.
2. Attach the thread to the curved line (to the river - from source to mouth) so that it follows all the bends of the river.
3. Mark the source and mouth points on the thread (with your fingers or tweezers) (you can carefully cut the thread with scissors at these points).
4. Straighten the thread, attach the noticed (or cut) section of the thread to the ruler and measure how many centimeters it contains. The measurement result is multiplied by the number of kilometers on the ground for a given scale. (You can attach a string to the linear scale on the map and immediately read the length of the river.)
Answers: a) approximately 920 km; b) approximately 1300 km; c) approximately 1200 km.
Note. The accuracy of measurements of curved sections is low, so the students’ answers may differ somewhat from the answers of their friends. Surely, the results of measuring with a thread on a small-scale map will VERY diverge from the river lengths indicated in textbooks and reference books. The real length of the Oka is 1500 km, the Urals - 2400 km, the Kama - 1800 km. It is imperative to tell students these numbers so that the “clumsy” numbers of independent measurement do not stick in their memory (and they have a great chance of sticking precisely because they were obtained independently). It is also necessary to explain where this discrepancy comes from: a small-scale map cannot reflect many medium and small turns, bends of the river, they are all “straightened”. This explanation will come in handy in the topic “Scale”: it will make it easier to understand the differences between maps of different scales.

Figures and facts

Scales of topographic maps

Numerical scale	Name cards	1 cm on the map corresponds to on the ground distance	1 cm 2 on the map corresponds on the ground area
1: 5 000 1: 10 000 1: 25 000 1: 50 000 1: 100 000 1: 200 000 1: 500 000 lll 1: 1 000 000	Five thousandth Ten-thousandth Twenty-five thousandth Fifty thousandth One hundred thousandth Two hundred thousandth Five hundred thousandth, or half a millionth Millionth	50 m 100 m 250 m 500 m 1 km 2 km 5 km lll 10 km	0.25 ha 1 ha 6.25 ha 25 hectares 1 km 2 4 km 2 25 km 2 ll 100 km 2

Cards have other names. Let us determine what scales the following names refer to: hundred-meter, half-kilometer, kilometer, two-kilometer, five-kilometer, ten-kilometer.
What type of scale are the names given in the table based on? And those given in the previous paragraph?

(reading for students)

A story about a 1:1 scale map

Once upon a time there lived a Capricious King. One day he traveled around his kingdom and saw how large and beautiful his land was. He saw winding rivers, huge lakes, high mountains and wonderful cities. He became proud of his possessions and wanted the whole world to know about them. And so, the Capricious King ordered cartographers to create a map of the kingdom. The cartographers worked for a whole year and finally presented the King with a wonderful map on which all the mountain ranges were marked, big cities and large lakes and rivers.
However, the Capricious King was not satisfied. He wanted to see on the map not only the outlines of mountain ranges, but also an image of each mountain peak. Not only large cities, but also small ones and villages. He wanted to see small rivers flowing into rivers.
The cartographers set to work again, worked for many years and drew another map, twice the size of the previous one. But now the King wanted the map to show passes between mountain peaks, small lakes in the forests, streams, and peasant houses on the outskirts of villages. Cartographers drew more and more maps.
The Capricious King died before the work was completed. The heirs, one after another, ascended the throne and died in turn, and the map was drawn up and drawn up. Each king hired new cartographers to map the kingdom, but each time he was dissatisfied with the fruits of the labor, finding the map insufficiently detailed.
Finally, the cartographers drew the Incredible Map. The map depicted the entire kingdom in great detail - and was exactly the same size as the kingdom itself. Now no one could tell the difference between the map and the kingdom.
Where were the Capricious Kings going to keep their wonderful map? The casket is not enough for such a map. You will need a huge room like a hangar, and in it the map will lie in many layers. But is such a card necessary? After all, a life-size map can be successfully replaced by the terrain itself.

Dependence of map detail on scale

If you have ever flown on an airplane, then you probably remember how at the beginning of the flight, when the plane just takes off from the ground, the outlines of the airport, houses, and squares float beneath it. But the higher he rises into the air, the fewer details are visible through the porthole, but the wider the space that opens to the eye becomes wider. The detail of the maps also changes when the scale is reduced.
On large-scale maps, where no more than 500 m of earthly space fits in 1 cm of area, small area depicted in great detail.
Small-scale maps, where up to several thousand kilometers fit into 1 cm, show huge areas of the Earth, but with little detail. Both cards are needed, depending on their purpose.
If you are wondering which countries you will fly over when traveling from Moscow to Melbourne, you need to open a small-scale map, and when going to the forest to pick mushrooms or go hiking with friends, you need to take a large-scale map with you so as not to get lost.

Homework for those interested

Determine the scale of maps of your area

Find maps that show the area where you live. If you don’t have such maps at home, ask your acquaintances and friends, a geography teacher, a librarian or a bookstore seller for help.
Write down the scales of maps depicting your area. Which scale is larger, which is smaller?
Compare maps of different scales and find out which scale maps show a larger territory, and which ones show a smaller one.
Determine which scale maps show the area in more detail, and which scales show less detail.
Draw a conclusion about how the area of the depicted territory and its detail depend on the scale of the map.

Find your location on the map

Using a map of your region (region, republic...), determine the distance from your settlement to the regional (regional, republican) center, if you do not live in it, or to some other settlement, if you are in the center of the region ( regions, republics).

On old maps a named scale could show what distance on the ground corresponds to one inch or other archaic linear measure on a map.
Here and below, calculations were made using the atlas “Geography. Beginner course. 6th grade.": Atlas. - M.: Bustard; Publishing house DIK, 1999. - 32 p. Of course, at this stage of training the teacher has not yet addressed the issues of distance distortion associated with map projection.

Scale is the degree to which lines are reduced when they are transferred to a plan or map.

A numerical scale is a proper fraction, the numerator of which is one, and the denominator is a number (M) showing the degree of reduction of the lines.

For example, a numerical scale or 1:2000 shows that all lines on the ground are reduced by M = 2000 times, or 1 cm on a plan or map corresponds to 2000 cm in reality, or one centimeter contains 20 m.

A linear scale is a graph that helps determine the distances between points on a map or plan.

Constructing a linear scale involves drawing a straight line on paper, dividing it into equal segments of 2 or 1 cm, and dividing the first segment into smaller divisions, for example, 2 or 1 mm (Fig. 52).

Rice. 52. Linear scale

In Fig. 52 it can be seen that one centimeter on a map of scale 1:10000 is 100 m on the ground. Two centimeters will contain 200 m. A two-centimeter segment is divided into 20 parts, therefore, 1 mm on the map will correspond to 10 m on the ground. The delayed distance on a linear scale is 590 m.

A transverse scale is a graph by which distances are determined on a plan or map with an accepted accuracy of 0.2 mm. Such a graph is shown in Fig. 53.

Fig.53. Normal transverse scale

On this graph the segment ab is the smallest division of the transverse scale. The transverse scale base A is 2 cm and can be divided into m equal parts. The height H of this scale is 2.5 cm and generally includes n equal parts.

A segment, and a segment.

From the relation we get .

For normal transverse scale m = n=10, then

ab= 0.2 mm.

Transverse scale accuracy t– this is the distance on the ground corresponding to the accuracy of graphic constructions of 0.2 mm:

where M is the denominator of the numerical scale.

For example, the accuracy of a transverse scale of 1:25000 will be

or t = 5 m.

Example1. Determine the length of the measured distance se at scales 1:5000 and 1:25000.

On a scale of 1:5000, 2 cm is 100 m in reality, and on a scale of 1:25000 - 500 m. Since the base of the scale is divided into 10 equal parts, then one tenth of it (segment CD) corresponds to a distance of 10 m on a scale of 1:5000, and on a scale of 1:25000 – 50 m. The height of the scale H is divided into 10 equal parts, so in the segment ab contains 1 m when using a scale of 1:5000 and 5 m when using a scale of 1:25000.

In order to measure the distances between points on the map, it is necessary to touch the points with the compass needles and apply the resulting compass solution to the transverse scale so that one needle is at the intersection of the inclined and horizontal scale lines (point s), and the other - on the horizontal and vertical lines (point e). Measured segment se consists of three parts so, or And re. These parts correspond to distances on the ground on a scale of 1:5000 40 + 6 + 4 = 446 m, and on a scale of 1:25000 – 200 + 30 + 2000 = 2230 m.

Example 2. Determine on a map of scale 1:25000 the distance between the point in square 6507 “Mark 214.3” and the point in square 6508 “Mark 197.1” (see Fig. 2).

As a result of measurements on this map, and not in its schematic representation, the result was obtained: 1480 m.

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 distinguished by high 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).

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

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

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

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

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

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

Scale series of topographic maps

Table 6.1.

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

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

6.1.2. Named scale

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

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

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

6.1.3. Graphic scales

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

Linear scale

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

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

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

How to use a linear scale:

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

Transverse scale

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

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

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

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

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

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

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

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

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

6.2.2. Steps scale

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

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

Shsr= 200 / 276 = 0.72 m.

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

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

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

6.3. SCALE ACCURACY

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

Scale accuracy allows you to solve two important problems:

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

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

6.4. DETERMINING AN UNKNOWN MAP SCALE

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

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

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

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

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

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

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

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

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

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

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

6.5. TECHNIQUES FOR MEASURING AND POSTPUTING DISTANCES ON A MAP

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

6.5.1. Measuring distances with a millimeter ruler

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

6.5.2. Measuring distances with a measuring compass

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

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

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

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

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

6.5.3. Measuring distances with a curvimeter

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

Rice. 6.7. Mechanical curvimeter

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

Rice. 6.8. Curvimeter digital (electronic)

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

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

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

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

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

Line length per topographic surface can be determined using the table (Table 6.3) of the relative values of amendments to the length of the horizontal distance (in %).

Table 6.3

Tilt angle

Rules for using the table

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

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

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

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

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

6.6. MEASUREMENT OF AREA BY MAPS

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

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

6.6.1. Measuring the area of a plot with straight boundaries

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

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

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

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

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

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

Rice. 6.11. Square mesh palette

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

P = a 2 n,

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

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

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

Rice. 6.12. Spot palette

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

P = h∑l

Figure 6.13. A palette consisting of a system
parallel lines

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

Rice. 6.14. Polar planimeter

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

Rice. 6.15. Electronic planimeter

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

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

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

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

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

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

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

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

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

Table 6.4.

y i (x i-1 - x i+1)

Double area in m2

SUM(D2:D6)

Area in hectares

In Table 6.5 we see the results of the calculations.

Table 6.5.

				y i (x i-1 -x i+1)






	Double area in m2
	Area in hectares

6.7. EYE MEASUREMENTS ON THE MAP

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

Video

Scale problems

Tasks and questions for self-control

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?

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

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

Solution:

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

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

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

Example for scale 1: 500,000

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

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

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

in 1 cm - 100 m.

Answers:

1 cm - 2 km;

1 cm - 100 km;

in 1 cm - 250 m.

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

Convert the named scale to a numerical one:

in 1 cm - 500 m

1 cm - 10 km

1 cm - 250 km

Solution:

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

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

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

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

Answers:

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

topographic plans - 1:400 - 1:5 000;

large-scale topographic maps - 1:10,000 - 1:100,000;

medium-scale topographic maps - from 1:200,000 - 1:1,000,000;

small-scale topographic maps - less than 1:1,000,000.

Scale maps:

1:10,000 (1cm =100m)

1:25,000 (1cm = 100m)

1:50,000 (1cm = 500m)

1:100,000 (1cm =1000m)

are called large-scale.

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

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

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

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

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

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

15.03.2022

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