On any geographical map, you can see approximately the following inscription: "Scale 1: 100,000." Traditionally, the first number is 1, and the second can vary. If there is no inscription, then there is certainly a tiny ruler divided into equal segments, or a nomogram. These signs indicate the ratio of the size of an object on a map or plan to its actual size.

You will need

• Roulette or compasses
• Ruler

Instruction

1. If you have a plan on which the different objects, and you need to know at what scale this plan done - start with measurements. Select an object, one that is nearby. Measure it on the plan and write down the results.

2. Measure the actual object. Use a tape measure for this. In order to avoid mistakes, make a peg and hook a tape measure loop on it. Drive a peg into the ground so that the zero mark of the tape measure is on the tier starting point the length or width of the object.

3. Determine the scale. It is more convenient for everyone to write it down in numbers. Write down the size of the object on the plan, after that - the one that turned out when measured on the territory. Let's say you have a barn 5 meters long on the plan occupies 2.5 cm. Convert meters to centimeters. That is, it turns out that you have 500 cm in 2.5 cm. Calculate how many centimeters of territory are contained in 1 cm on the plan. For this more divide by less. It turns out 2.5:500 = 1:200, that is, 1 cm on the plan corresponds to 2 m on the territory.

4. In order to determine the scale more correctly, take several measurements. Let's say measure the barn on the site and the distance from the gate to the pond. The plans are different, and the dimensions of one or another object can be applied unsatisfactorily correctly. If there are discrepancies, make another frosted. The image of the object, the one that does not correspond to the other two, correct on the plan.

Scale is a numerical designation of parameters related to real objects that are unthinkable to depict in natural size. The figure applies their layouts.

Instruction

1. The scale is recorded in several ways, say, numerically - 1: 1000000. The size ratio can also be indicated in this form: 1 cm 10 km is a named scale. Line method display is shown as a ticked line.

2. When considering scale in relation to cartography, the appearance of a particular map will depend on the ratios used. The larger it is, the more detailed the area will be depicted. The detail is also influenced by the nature of the territory, which is sparsely inhabited, say, easier to depict. Maps are large, medium and small scale. Large-scale maps are when 1 cm is from 100 to 2000 meters, medium-scale maps are 1 cm to 10 km, small-scale maps are 1 cm more than 10 km.

3. Scale matters in photography as well. With the help of lenses, photographers change the size from hefty small to hefty large. The methodology of the metamorphosis of scale depends on the specifics of the surveys. If this small objects, say, insects, the scale increases, if huge, it decreases.

4. The representation is also used in many sciences. In mathematics it is the ratio of numbers, in programming it is the scale of time, in astronomy it is the scale of the universe. The meaning of the word is also used in the construction industry.

5. Firms are distinguished by the scale of their activities. There are, let's say territorial organizations, but there is also a federal tier. Different in scale and people. True, not with physical point vision, there psychological representation figure scale. This means human qualities set goals and results of activities.

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Note!
The size of a reduced object is relative to its natural size. The distance between objects can be changed by several centimeters, meters, kilometers. The scale of reality changes a lot, but all parameters must remain proportional. If proportions are not observed, it will be unthinkable to analyze the distances and sizes of objects.

With the need to present the real dimensions of the object depicted in the drawing, a person is faced more closely at school. In a drawing lesson, it may be necessary to draw a detail on a scale of 1:2 or 1:4, in a geography lesson - to calculate the exact distance between two cities. In order to cope with the task, you need to know how the scale is translated.

You will need

• - geographic map;
• – detail drawing;
• - calculator;
• - drawing accessories.

Instruction

1. If you need to draw details on a 1:1 scale, this means that 1 cm of the surface will correspond to 1 cm in the drawing. Measure the surface you want to depict and draw it on paper at natural size.

2. Other scales are also used in drawing. 1:2 means that the detail in the drawing should be half as large as in reality. If the scale is 1; 4, this means that 1 cm in the drawing is equal to 4 cm of the part. It also happens the other way around. Not at all small object it is allowed to draw, say, on a scale of 4:1, 10:1, etc. If you see a similar designation in front of you, it means that the object in the picture is four or ten times larger than it really is.

3. In geography, scale conversion is also required. make out geographical map. In one of the lower corners, you will see either a ruler with numbers, or primitive numbers - say, 1:50,000. The numbers, finally, are larger than in the drawing, but the rule for translating them is exactly the same, that is, in the above example, per 1 cm of the map brought to 50,000 cm earth's surface, i.e. 500 m. This is a relatively large scale map. Looking at the atlas of the world, you will see much more impressive figures.

4. Quite often it is necessary to translate the scale not of a linear measure, but of a square one, that is, to determine how much square centimeters. To do this, measure the area you need by any comfortable method. Say, with palette support. In order to find out the real area of ​​the territory, you need to convert the linear scale into a square one, that is, build the number of centimeters contained in 1 cm of the map into a square. Multiply the resulting number by the area of ​​the plot shown on the map. This way you will know how much square meters occupies the territory you care about.

5. Occasionally there is a need to translate the scale of a three-dimensional object. For example, at a labor lesson, a teacher can give the task to make a part depicted in a technical drawing on a certain scale. You need to find out how much material this will require. The translation thesis will be the same. First, find out how many real centimeters this or that line in the drawing corresponds to. Determine the volume of the part from the drawing. This is a simple mathematical problem, the method of solving it depends on the shape of a particular part. The number that indicates the scale, cube, and then multiply by the volume of the part, calculated according to the drawing.

Useful advice
You can try to draw a simple plan on your own, setting yourself a certain scale. Let's say a 1:10 scale for a room plan would absolutely fit. Measure the length of the walls and large items, define them mutual arrangement and draw a plan exactly according to the data received.

Note!
The scale is greater than smaller denominator the fraction with which it is written. 1:100 is larger than 1:2,000. It is more comfortable to measure an object with an assistant. If there is no assistant, and there was no peg at hand, firmly press the tape measure against the wall of the object. It is more comfortable to measure everyone on the ground - say, on the bottom of the wall.

The scale is the ratio of the line (size) in the drawing to the line (size) in nature.

The scale in the drawing is indicated as a fraction, which shows the multiplicity of increase or decrease in natural dimensions when depicted in the drawing. Such a scale is called numerical.

All construction drawings are made on a reduced scale, since the elements shown in the drawing are much larger than a sheet of drawing paper. Plans and sections of buildings are reduced by 100 or 200 times, depending on the size of the building. General plan done on a scale of 1:500 or 1:1000, i.e., 500 or 1000 times smaller than the actual size of the site.

To find out the real size of any object or line shown in a drawing drawn on a scale of 1:100, you need to measure this line in centimeters and multiply by 100. For example, the measured line in the drawing is 2 cm. Knowing from the scale that the dimensions of the drawing reduced by 100 times, multiply 2 cm by 100 m and get 200 cm or 2 m. This means that the dimensions shown in the drawing by a line 2 cm long are equal to 2 m in nature.

Using a numerical scale creates some inconvenience, since every time you measure a line in a drawing, you have to make small calculations. It is much more convenient to use a linear scale, which allows you to determine the actual dimensions of the object from the drawing without calculation.

The linear scale is constructed as follows: a numerical scale of 1:100 is given, at which 1 cm in the drawing is equal to 1 m in kind. Several segments equal to 1 cm are laid on a horizontal line. From each division point, perpendiculars are restored to a straight line.

Above the first perpendicular (division) they put (counting from left to right) the number 1, above the second - 0. To the right of zero, all divisions are numbered with ordinal numbers, starting from one, and after the last division they put the letter "m" (meter), showing that one scale division equal to 1 cm corresponds in kind to 1 m.

The length of each division (in our example 1 cm) is called the base of the scale. The first division (from 1 to 0) is divided by 10 equal parts. Each division in the drawing will be equal to 1 mm, and in kind - a value 100 times larger, i.e. 100 mm, or 10 cm.

Using a linear scale is very easy. It is necessary to measure the line on the drawing with a compass and combine the ends of the compass with the scale. If the dimensions of the line in the drawing exactly match the main divisions of the scale, for example, from zero to the right to the third division, then the dimensions of this line in kind will be 3 m.

If the length of the measured line does not coincide with the main divisions of the scale, for example, more than three, but less than four main divisions, then by placing the leg of the compass in the third division, they look at what small division (to the left of zero) the other leg of the compass will align with. Suppose it is compatible with the fourth small division. Knowing that each small division is equal to 0.1 m, or 10 cm, the length of the measured line is 3.4 m, or 340 cm.

If it turns out that the length of the measured line does not exactly match the small divisions of the scale, for example, more than four, but less than five small divisions, the size of the line has to be determined only approximately, depending on the position of the leg of the compass in relation to adjacent small divisions, but this will be not exactly.

Thus, it can be seen that the constructed scale can measure lines with an accuracy of 10 cm.

"Handbook of the Assistant Sanitary Doctor
and assistant epidemiologist,
ed. corresponding member of the USSR Academy of Medical Sciences
prof. N.N. Litvinova

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 requirements, topographic maps are highly geometric accuracy and geographic fit. This is provided by their scale, geodetic base, cartographic projections and a system of symbols.
Geometric Properties cartographic image: the size and shape of the areas occupied geographic features, distances between individual points, directions from one to another - are determined by its mathematical basis. Mathematical basis cards include as constituent parts scale, a geodesic base, and a map projection.
What is the scale of the map, what types of scales are there, how to build a graphical scale and how to use the scales will be considered in the lecture.

6.1. TYPES OF SCALE OF TOPOGRAPHIC MAP

When compiling maps and plans, horizontal projections of segments are depicted on paper in a reduced form. The degree of such a decrease 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 location of the corresponding terrain line

m = l K : d M

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

The scale is indicated on the maps in different options

6.1.1. Numerical scale

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

Or

Denominator M the numerical scale shows the degree of reduction in the lengths of the lines on the map (plan) in relation to the lengths of the corresponding lines on the ground. Comparing numerical scales, the largest is the one whose denominator is smaller.
Using the numerical scale of the map (plan), you can determine the horizontal distance dm lines on the ground

Example.
Map scale 1:50 000. The length of the segment on the map lk\u003d 4.0 cm. Determine the horizontal location of the line on the ground.

Solution.
Multiplying the value of the segment on the map in centimeters by the denominator of the numerical scale, we get the horizontal distance in centimeters.
d\u003d 4.0 cm × 50,000 \u003d 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 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 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 government topographic maps, forest management plans, forestry plans and forest plantations, standard scales are defined - scale range(Tables 6.1, 6.2).

Scale series of topographic maps

Table 6.1.

Numerical scale	Map name	1 cm card corresponds on the ground distance	1 cm2 card corresponds on the territory of the square
	five thousandth		0.25 hectare
	ten thousandth
	twenty-five thousandth		6.25 hectares
	fifty thousandth
	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 called verbal expression numerical scale. Under the numerical scale on the topographic map there is an inscription explaining how many meters or kilometers on the ground corresponds to one centimeter of the map.

For example, on the map under a numerical scale of 1:50,000 it is written: "in 1 centimeter 500 meters." Numeral 500 in this example there is named scale value .
Using a named map scale, you can determine the horizontal distance dm lines on the ground. To do this, it is necessary 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". The length of the segment on the map lk\u003d 6.3 cm. Determine the horizontal location of the line on the ground.
Solution. 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 build a linear scale, choose an initial segment that is convenient for a given scale. This original segment ( a) are called scale base (Fig. 6.1).

Rice. 6.1. Linear scale. Measured segment on the ground
will be 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 the smallest divisions of the linear scale . 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:

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

Cross scale

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

Fig 6.2. Cross 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. Perpendiculars to the line are set at the points obtained. AB and pass through them ten parallel lines through equal intervals. The leftmost base from above and below is divided into 10 equal segments and connected by oblique lines. zero point the lower base is connected to the first point FROM 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 by 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 hundreds . On a 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.

How to use the transverse scale:

• fix the length of the line on the map with a measuring compass;
• put the right leg of the compass on an integer 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: a count of integer bases, plus a count of divisions of the left base, plus a count of 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. VARIETY OF GRAPHIC SCALE

6.2.1. transitional scale

Sometimes in practice it is necessary to use a map or an 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 a linear scale is not taken to be 2 cm, but 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 at a scale of 1:17,500 (175 meters in one centimeter).
To determine what dimensions a segment of 400 m long 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 transitional 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 a base length a\u003d 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 \u003d BC + AB \u003d 800 +160 \u003d 960 m.

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

6.2.2. Step scale

Use this scale to determine the distances measured in steps during eye survey. The principle of constructing and using the scale of steps is similar to the transitional scale. The base of the scale of steps is calculated so that it corresponds to the round number of steps (pairs, triplets) - 10, 50, 100, 500.
To calculate the value of the base of the scale of steps, it is necessary to determine the survey scale and calculate the average step length Shsr.
The average step length (pair of steps) is calculated from known distance traversed in a straight line 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 reverse side- longer.

Example. A known distance of 100 m is measured in steps. There are 137 steps in the forward direction and 139 steps in the reverse direction. Calculate the average length of one step.
Solution. Total covered: Σ m = 100 m + 100 m = 200 m. The sum of the steps is: Σ w = 137 w + 139 w = 276 w. Average length 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 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 \u003d 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, according to the 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) 0.036 × 40 = 1.44 cm.
The length of the base of the steps 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 - scale denominator.

The length of the base for 50 steps on a scale of 1:2,000 with a step length of 72 cm will be:
a= 72 × 50 / 2000 = 1.8 cm.
To build the scale of steps for the above example, it is necessary 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 \u003d BC + AB \u003d 100 + 20 \u003d 120 sh.

6.3. SCALE ACCURACY

Scale Accuracy (maximum scale accuracy) is a segment of the horizontal line, corresponding to 0.1 mm on the plan. The value of 0.1 mm for determining the accuracy of the scale 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. 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 above example, it follows that if the denominator of the numerical scale is divided by 10,000, then we get the maximum scale accuracy 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 tasks:

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

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. Horizontal distance on the ground, corresponding to a given scale of 0.2 mm (0.02 cm) on the plan, is called graphic accuracy of scale . Graphical 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 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 graphical ones.
If we take into account the error of the map itself and the measurement error on the map, then we can conclude that the graphical accuracy of determining distances on the map is 5–7 worse than the maximum scale accuracy, i.e., it is 0.5–0.7 mm on the map scale.

6.4. DETERMINATION OF UNKNOWN MAP SCALE

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

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

For example, the coordinate lines are indicated 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. Distance on the map between adjacent lines equals 2 cm. It follows that 2 cm on the map correspond to 2 km on the ground, and 1 cm on the map - 1 km on the ground (named scale). This means that the scale of the map will be 1:100,000 (1 kilometer in 1 centimeter).

• 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 easy to find out the scale of the map.

A map sheet at a scale of 1:1,000,000 (millionth) is indicated by one of the letters 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 map sheet of a given scale will always be the same.
Thus, if a map has the M-35-96 nomenclature, then by comparing it with the above diagram, we can immediately say that the scale of this map will be 1:100,000.
See Chapter 8 for details on card nomenclature.

• 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 n.p. Kuvechino to the lake. Deep 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 on a scale of 1:104 200 are not published, so we make rounding. 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 length of the arc of one minute of the meridian . Frames of topographic maps along the meridians and parallels have divisions in minutes of the meridian and parallel arcs.

One minute of the meridian arc (along the eastern or western frame) corresponds to a distance of 1852 m on the ground ( nautical mile). Knowing this, it is possible to 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 a map scale of 1:100,000.
In our calculations, approximate values ​​of the scales were obtained. This happened due to the approximation of the distances taken and the inaccuracy of their measurement on the map.

6.5. TECHNIQUE FOR MEASURING AND PUTTING DISTANCES ON A MAP

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

6.5.1. Measuring distances with a millimeter ruler

Use a millimeter ruler to 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: in 1 cm 500 m. The distance on the ground between the points will be 3.4 × 500 = 1700 m.
At angles of inclination of the earth's surface more than 10º, it is necessary to introduce an appropriate correction (see below).

6.5.2. Measuring distances with a compass

When measuring 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 read off on a linear or transverse scale. In the case when the opening of the compass exceeds the length of the linear or transverse scale, the integer number of kilometers is determined by the squares of the coordinate grid, and the remainder - by the usual scale order.

Rice. 6.5. Measuring distances with a compass-meter on a linear scale.

To get the length broken line sequentially measure the length of each of its links, and then summarize their values. Such lines are also measured by increasing the compass solution.
Example. To measure the length of a polyline ABCD(Fig. 6.6, a), the legs of the compass are first placed at points BUT and AT. Then, rotating the compass around the point AT. move the back leg from the point BUT exactly AT" lying on the continuation of the line sun.
Front leg from point AT transferred to a point FROM. The result is a solution of the compass B "C"=AB+sun. Moving the back leg of the compass in the same way from the point AT" exactly FROM", and the front of FROM in D. get a solution of the compass
C "D \u003d 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 curves measured along the chords with compass steps (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, count the steps. The total length of the line A 1 C 1 is made up of the segment A 1 B 1, equal to step 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 (Fig. 6.7) or electronic (Fig. 6.8) curvimeter.

Rice. 6.7. Curvimeter mechanical

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

Rice. 6.8. Curvimeter digital (electronic)

To improve the accuracy and reliability of the results, it is recommended that all measurements be carried out twice - in the forward and reverse directions. In case of slight differences in the measured data for final result average is taken arithmetic value measured values.
The accuracy of measuring distances by these 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. Converting 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 spacing ( 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 - the angle of inclination of the earth's surface.

Line length for topographic surface can be determined using the table (Table 6.3) of the relative values ​​of the corrections to the length of the horizontal span (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 the corrections at inclination 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 absolute value amendments, it is necessary:
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 given by an integer number of degrees, then the relative value of the correction must be found by interpolation between the tabular values);
b) calculate the absolute value of the correction to the length of the horizontal span (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 distance.

Example. On the topographic map, the length of the horizontal laying is 1735 m, 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 larger and nearest smaller multiples of one degree - 8º and 7º:
for 8° relative correction value 0.98%;
for 7° 0.75%;
difference in tabular values ​​in 1º (60') 0.23%;
difference between given angle the slope of the earth's surface 7 ° 15 "and the nearest smaller tabular value of 7º is 15".
We make proportions and find the relative amount of the correction for 15 ":

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

Relative value corrections 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 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 MAP

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

To measure areas on maps, graphic, analytical and instrumental methods are used. The use of one or another method of measurement is due to the shape of the measured area, given accuracy 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 plot with rectilinear boundaries, the plot is divided into simple geometric figures, measure the area of ​​\u200b\u200beach of them in a geometric way and, summing up the areas individual sections, calculated taking into account the scale of the map, get total area object.

6.6.2. Measuring the area of ​​a plot 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 excesses mutually compensate each other (Fig. 6.10). The measurement results will be approximate to some extent.

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

6.6.3. Measurement of the area of ​​a plot with a complex configuration

Measurement of plot areas, having a complex irregular configuration, more often produced using pallets and planimeters, which gives the most accurate results. grid palette is a transparent plate with a grid of squares (Fig. 6.11).

Rice. 6.11. Square Mesh Palette

The palette is placed on the measured contour and the number of cells and their parts inside the contour is counted. The proportions of incomplete squares are estimated 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 \u003d a 2 n,

Where: a - the side of the square, expressed on 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 palette used in any position, including rotation relative to its original position. The arithmetic mean of the measurement results is taken as the final value of the area.

In addition to grid palettes, dot and parallel palettes are used, which are transparent plates with engraved dots or lines. 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. dot palette

Weight of each point equal to the price 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.
Equidistant parallel lines are engraved on the parallel palette (Fig. 6.13). The measured area, when applied to it with a palette, will be divided into a series of trapezoids with the same height h. Segments of parallel lines inside the contour (in the middle between the lines) are the middle lines of the trapezoid. To determine the area of ​​​​a plot using this palette, it is necessary to multiply the sum of all measured middle lines by the distance between the parallel lines of the palette h(taking into account the scale).

P = h∑l

Figure 6.13. Palette consisting of a system
parallel lines

Measurement areas of significant plots made on cards with the help of planimeter.

Rice. 6.14. polar planimeter

The 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. After fixing the pole and setting the needle of the bypass lever at the starting point of the circuit, a reading is taken. 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 value of the division of the planimeter, determine the area of ​​the contour.
The development of technology contributes to the creation of new devices that increase labor productivity in calculating areas, in particular, the use of modern appliances 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 ​​​​a site of any configuration, i.e. with any number of vertices whose coordinates (x, y) are known. In this case, the numbering of the vertices should be done in a clockwise direction.
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 the calculation of the area of ​​a polygon by coordinates.

In turn, each of the areas S "and S" is the sum of the areas of the trapezium, parallel sides 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 "\u003d pl. 1u-1-2-2u + pl. 2u-2-3-3u,
S" \u003d pl 1y-1-4-4y + pl. 4y-4-3-3y
or: 2S " \u003d (x 1 + x 2) (y 2 - y 1) + (x 2 + x 3) (y 3 - y 2)
2S " \u003d (x 1 + x 4) (y 4 - y 1) + (x 4 + x 3) (y 3 - y 4).

In this way,
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 \u003d 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 \u003d 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 view, denoting by i serial number(i = 1, 2, ..., n) polygon vertices:
(6.3)
(6.4)
Therefore, twice the area of ​​the polygon is equal to 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 of the abscissas of the previous and subsequent vertices of the polygon.
intermediate control computation is to satisfy 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 by calculating the area of ​​the plot can be easily solved using spreadsheets MicrosoftXL. 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 square 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 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 the distance, direction, area, steepness of the slope and other characteristics of objects on the map contributes to mastering the skills correct understanding cartographic image. The accuracy of eye measurements increases with experience. Eye skills prevent gross miscalculations in instrument measurements.
To determine the length of linear objects on the 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 square of the grid of maps of scales 1:10,000 - 1:50,000 on the ground corresponds to 1 km 2 (100 ha), scale 1:100,000 - 4 km 2, 1:200,000 - 16 km 2.
Accuracy quantitative determinations on the map, with the development of the eye, is 10-15% of the measured value.

Video

Scaling tasks

Tasks and questions for self-control

1. What elements does it include mathematical basis kart?
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 you use it?
4. What is the transverse scale of the map, for what purpose is it intended?
5. What transverse map scale is considered normal?
6. What scales of topographic maps and forest management tablets are used in Ukraine?
7. What is a transitional map scale?
8. How is the base of the transitional scale calculated?
Previous

To depict the surface of the Earth on maps, cartographers had to decide math problem. It was necessary to reduce the image and determine which objects could be shown on a geographical map with a particular reduction.

Why is scale needed?

On old maps and plans, the real area is shown in a reduced form. But different areas are reduced in different ways. Therefore, according to old maps you can determine the outlines of objects, but not their sizes. To measure the length of a river or the distance between cities, you need to reduce the image of the area and all objects in certain number once. To do this, you need to use a scale.

Scale is the ratio of two numbers, such as 1:100 or 1:1000. The ratio shows how many times one number is greater than another. A scale of 1:100 means that the image is one hundred times smaller than the depicted object, and a scale of 1:1000 means that a thousand times. How less number, showing a decrease, the larger the scale, and vice versa. Scale 1:100 is larger than scale 1:1000 and smaller than scale 1:50.

The scale on the plan, map, shows how many times the length of each line is reduced compared to its actual length on the ground. Using the scale, you can measure the distances between individual geographical objects and determine the size of the objects themselves.

How is scale recorded?

The scale on plans and maps is usually depicted in three types: numerical, named, linear.

Numerical scale written as a ratio of numbers: 1:100, 1:500, 1:100,000. On this scale, the first number is the distance in the image, and the second number is the real distance on the ground in the same units of measurement. At a scale of 1:100,000, a distance of 1 centimeter on the map corresponds to 100,000 centimeters on the ground. 100,000 centimeters is 1000 meters or 1 kilometer. The scale, expressed in the form of the words "1 kilometer in 1 centimeter", is called named scale.

Linear scale- a line divided into centimeter segments. The segments to the right of zero show what distance on the ground corresponds to 1 centimeter on a plan or map. The segment to the left of zero is divided into five smaller parts for greater measurement accuracy. By measuring the distance between objects using a measuring compass, you can apply it to a linear scale and get distances on the ground. Using a linear scale, determine the length of curved lines ( coastline seas, rivers or roads).

Image scale and details

Depending on the scale, the degree of detail of the image changes. The larger the scale, the more detailed the parts of the Earth with all geographical objects are depicted. But on large-scale images (1:200,000 and larger), only a small area of ​​the earth's surface fits. On small-scale maps (smaller than 1:1000,000), where 1 centimeter corresponds to several thousand kilometers on the ground, even the entire surface of the Earth can be shown. However, the amount of detail and terrain detail here is low.

Often in training and practical purposes have to create plans and maps varying degrees details and therefore scale.