How to calculate area on a map. Measuring areas according to plan and map

Instructions

Go to the Google search engine and click on the word “Maps”, which is located at the top of the search engine. On the right side you will see a map, and on the left there are two buttons: “Routes” and “My Places”. Click on "Routes". Two windows “A” and “B” will appear under it, that is, the starting and ending reference points. Let’s say you are in Ufa, and you need to find out how long the road to Perm will take. In this case, enter “Ufa” in box “A”, and “Perm” in box “B”. Click on the button again under the “Routes” windows. The route will appear on the map, and under the “A” and “B” windows, how many kilometers are from one city to another, as well as how much time it takes to get there by car. If you are interested in walking walk, click on the button with the image of a pedestrian, which is located above windows “A” and “B”. The service will rebuild the route and automatically calculate distance and expected travel time.

In the event that it is necessary distance from point “A” to “B”, located in the same locality, you should proceed according to the above scheme. The only difference is that you need to add a street and, possibly, a house number separated by a comma to the name of the area. (For example, “A”: Moscow, Tverskaya 5 and “B”: Moscow, Tsvetnoy Boulevard, 3).

There are situations when you are interested distance between objects “directly”: through fields, forests and rivers. In this case, click on the cog icon in the top corner of the page. In the expanded menu that appears, select “Laboratory” Google Maps» and enable the distance measurement tool, save the changes. A ruler has appeared in the lower left corner of the map, click on it. Mark the starting point and then the end point. A red line will appear between these points on the map, and the distance will be shown in the panel on the left side.

Useful advice

You can choose one of two units of measurement: kilometers or miles;
- by clicking on several points on the map, you can determine the distance between many points;
- if you log into the service using your profile, Google maps will remember your settings in the Google Maps Lab.

Sources:

measure distance on a map

When going on a summer tourist trip on foot, by car or kayak, it is advisable to know in advance the distance that will need to be covered. To measure length paths, you can’t do without a map. But it’s easy to determine from the map direct distance between two objects. But what about, for example, measuring the length of a winding water route?

You will need

Area map, compass, strip of paper, curvimeter

Instructions

Technique one: using a compass. Set a compass angle suitable for measuring length, otherwise known as its pitch. The pitch will depend on how tortuous the line to be measured is. Typically, the pitch of the compass should not exceed one centimeter.

Place one leg of the compass at the starting point of the measured path length, and place the second needle in the direction of movement. Consistently turn the compass around each of the needles (it will resemble steps along the route). The length of the proposed path will be equal to the number of such “steps” multiplied by the steps of the compass, taking into account the scale of the map. The remainder, smaller than the pitch of the compass, can be measured linearly, that is, along a straight line.

The second method involves having a regular strip of paper. Place the strip of paper on its edge and align it with the route line. Where the line bends, bend the strip of paper accordingly. After that all that remains is to measure length the resulting segment of the path along the strip, of course, again taking into account the scale of the map. This method is only suitable for measuring the length of small sections of the path.

Topic 7. MEASUREMENT OF DISTANCES AND AREA BY TOPOGRAPHIC MAPS

7.1. 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.

7.1.1. Measuring distances with a millimeter ruler

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

7.1.2. Measuring distances with a measuring compass

When measuring a distance in a straight line, the needles of the compass are placed at the end points, then, without changing the opening of the compass, 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 grid, and the remainder - in the usual order of scale.

Rice. 7.1. 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. 7.2, 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 to the point 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 moving the back leg of the compass from the point in the same way IN" to the point 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. 7.2. Line length measurement: a - broken line ABCD; b - curve A1B1C1;
B"C" - auxiliary points

Long curved segments measured along chords using compass steps (see Fig. 7.2, 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. 7.2, 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.

7.1.3. Measuring distances with a curvimeter

Curve segments are measured with a mechanical (Fig. 7.3) or electronic (Fig. 7.4) curvimeter.

Rice. 7.3. Mechanical curvimeter

First, by rotating the wheel by hand, set the arrow to the zero division, then roll the wheel along the line being measured. 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. 7.4.) 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. 7.4. 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.

7.1.4. 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. 7.5).

Rice. 7.5. Slant range ( S) and horizontal distance ( d)

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

Where d- length of the horizontal projection of the line S;
α - angle of inclination of the earth's surface.

The length of a line on a topographic surface can be determined using a table ( table 7.1) relative values of amendments to the length of the horizontal installation (in%) .

Table 7.1

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 absolute value amendments, 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. On topographic map horizontal length determined 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 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" 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.

7.2. 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 the 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 in 1 cm 2 – 100 km 2.
To measure areas on maps, graphical, analytical and instrumental methods are used. The use of one or another measurement method is 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.

7.2.1. Measuring the area of a plot with straight boundaries

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

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

Object with curvilinear contour are 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. 7.6). The measurement results will be, to some extent, approximate.

Rice. 7.6. Straightening the curved boundaries of the site and
breaking down its area into simple geometric shapes

7.2.3. 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. 9.9).

Rice. 7.7. 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.

Rice. 7.8. 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. 7.9). The area being measured, when the palette is applied to it, will be divided into a number of trapezoids with the same height h. Segments parallel lines inside the contour (midway between the lines) are the midlines of the trapezoids. 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 the parallel lines of the palette h(taking into account scale).

P = h∑ l

Figure 7.9. A palette consisting of a system
parallel lines

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

Rice. 7.10. Polar planimeter

A planimeter is used to determine areas mechanically. The polar planimeter is widely used (Fig. 7.10). 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. 7.11. Electronic planimeter

7.2.4. 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. 7.12, area S polygon 1-2-3-4 can be considered as the difference in area S" figures 1у-1-2-3-3у And S" figures 1y-1-4-3-3у
S = S" - S".

Rice. 7.12. To calculate the area of a polygon from coordinates.

In turn, each of the areas S" And S" represents 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" = pl. 1у-1-2-2у + pl. 2у-2-3-3у,
S" = pl. 1у-1-4-4у + pl. 4у-4-3-3у
or:

2S " = (x 1+ x 2)(at 2 – at 1) + (x 2+ x 3 ) (at 3 - y 2)
2 S" = (x 1+ x 4)(at 4 – at 1) + (x 4+ x 3)(at 3 - at 4).
Thus,
2S = (x 1+ x 2)(at 2 – at 1) + (x 2+ x 3 ) (at 3 - y 2) – (x 1+ x 4)(at 4 – at 1) - (x 4+ x 3)(at 3 - at 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 at 1 - x 4 y 3

From here
2S = x 1 (y 2 - at 4) + x 2 (y 3 - y 1)+ x 3 (y 4 - at 2 )+x 4 (at 1 - at 3 ) (7.1)
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) (7.2)

Let us represent expressions (7.1) and (7.2) in general view, denoting by i serial number ( i = 1, 2, ..., p) polygon vertices:
2S = (7.3)
2S = (7.4)

Hence, the doubled area of a polygon is equal to either the sum of the products of each abscissa by the difference between the ordinates of the subsequent and previous vertices of the polygon, or the sum of the products of each ordinate by the difference of 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 plot area can be easily solved using spreadsheets MicrosoftXL . An example for a polygon (polygon) of 5 points is given in tables 7.2, 7.3.
In Table 7.2 we enter the initial data and formulas.

Table 7.2.


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





	Double area in m2			SUM(D2:D6)
	Area in hectares

In Table 7.3 we see the calculation results.

Table 7.3.

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






	Double area in m2
	Area in hectares

7.3. 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. Accuracy eye definitions increases with experience. Visual skills prevent gross miscalculations in measurements with instruments.
To determine lengths of linear objects Using the map, you should visually compare the size of these objects with segments of a kilometer grid or divisions of a linear scale.
To determine area of objects The squares of the 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.

Questions and tasks for self-control

Explain how to measure a straight line on a map.

Explain the procedure for measuring a polyline map.

Explain how to measure a curved line on a map using a measuring compass.

Explain how to measure a curved line on a map using a curvimeter.

How can you determine the length of a linear object using a topographic map?

What area on the ground corresponds to one square of the coordinate grid of a map at a scale of 1:25,000?

To determine on a map the distance between terrain points (objects, objects), using a numerical scale, you need to measure on the map the distance between these points in centimeters and multiply the resulting number by the scale value (Fig. 20).

Rice. 20. Measuring distances on a map with a measuring compass

on a linear scale

For example, on a map at a scale of 1:50,000 (scale value 500 m), the distance between two landmarks is 4.2 cm.

Therefore, the required distance between these landmarks on the ground will be equal to 4.2 500 = 2100 m.

A small distance between two points in a straight line is easier to determine using a linear scale (see Fig. 20). To do this, a measuring compass is sufficient, the solution of which equal to the distance between given points on the map, apply it to a linear scale and take a reading in meters or kilometers. In Fig. 20 the measured distance is 1250 m.

Large distances between points along straight lines are usually measured using a long ruler or measuring compass. In the first case, a numerical scale is used to determine the distance on the map using a ruler. In the second case, the opening (“step”) of the measuring compass is set so that it corresponds to an integer number of kilometers, and an integer number of “steps” are plotted on the segment measured on the map. The distance that does not fit into the whole number of “steps” of the measuring compass is determined using a linear scale and added to the resulting number of kilometers.

In this way, distances are measured along winding lines. In this case, the “step” of the measuring compass should be 0.5 or 1 cm, depending on the length and degree of tortuosity of the line being measured (Fig. 21).

Rice. 21. Measuring distances along curved lines

To determine the length of a route on a map, a special device called a curvimeter is used. It is convenient for measuring curved and long lines. The device has a wheel, which is connected by a gear system to an arrow. When measuring distance with a curvimeter, you need to set its needle to the zero division, and then roll the wheel along the route so that the scale readings increase. The resulting reading in centimeters is multiplied by the scale value and the distance on the ground is obtained.

The accuracy of determining distances on a map depends on the scale of the map, the nature of the measured lines (straight, winding), the chosen method of measuring the terrain and other factors.

The most accurate way to determine the distance on the map is in a straight line. When measuring distances using a measuring compass or a ruler with millimeter divisions, the average measurement error on flat areas of the terrain usually does not exceed 0.5–1 mm on the map scale, which is 12.5–25 m for a map of scale 1: 25,000 , scale 1: 50,000 – 25–50 m, scale 1: 100,000 – 50–100 m. In mountainous areas with steep slopes, errors will be greater. This is explained by the fact that when surveying a terrain, it is not the length of the lines on the Earth’s surface that is plotted on the map, but the length of the projections of these lines onto the plane.

With a slope steepness of 20° and a distance on the ground of 2120 m, its projection onto the plane (distance on the map) is 2000 m, i.e. 120 m less. It is calculated that with an inclination angle (steepness of the slope) of 20°, the resulting distance measurement result on the map should be increased by 6% (add 6 m per 100 m), with an inclination angle of 30° - by 15%, and with an angle of 40° - by 23 %.

When determining the length of a route on a map, it should be taken into account that road distances measured on the map using a compass or curvimeter are shorter than the actual distances. This is explained not only by the presence of ups and downs on the roads, but also by some generalization of road convolutions on maps. Therefore, the result of measuring the length of the route obtained from the map should, taking into account the nature of the terrain and the scale of the map, be multiplied by the coefficient indicated in the table. 3.

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METHODOLOGICAL INSTRUCTIONS FOR LABORATORY WORK

FOR THE COURSE “GEODESY Part 1”

7. MEASUREMENT OF AREA ACCORDING TO PLAN OR MAP

To solve the series engineering problems It is required to determine the areas of various areas of the area from a plan or map. The determination of areas can be done graphically. analytical and mechanical methods.

7.1. Graphical method for determining area

The graphical method is used to determine small areas (up to 10-15 cm 2) from a plan or map and is used in two versions: a) with a breakdown of the intended area into geometric shapes; b) using palettes.

In the first option, the area of the site is divided into the simplest geometric figures: triangles, rectangles, trapezoids (Fig. 19, a), the corresponding elements of these figures are measured (base lengths and heights) and the areas of these figures are calculated using geometric formulas. The area of the entire plot is determined as the sum of the areas of individual figures. The division of the area into figures should be done in such a way that the figures can be large sizes, and their sides coincided as closely as possible with the contour of the site.

To control, the area of the site is divided into other geometric shapes and the area is re-determined. The relative discrepancy in the results of double determinations of the total area of the site should not exceed 1: 200.

For small areas (2-3 cm 2) with clearly defined curved boundaries, it is advisable to determine the area using using a square palette(Fig. I9, b). The palette can be made on tracing paper by drawing it with a grid of squares with sides of 2-5 mm. Knowing the length of the side and the scale of the plan, you can calculate the area of the square of the palette I KB.

To determine the area of the site, the tent is randomly placed on the plan and the number of complete squares is counted N 1 , located inside the contour of the site. Then evaluate each incomplete square by eye (in tenths) and find the total number N 2 for all incomplete squares on the boundaries of the contour. Then the total area of the measured area S= s KB *(N 1 + N 2 ). For control, the tent is deployed approximately 45 A and the area is re-determined. The relative error in determining the area with a square palette is 1: 50 - 1: 100. When determining areas, several larger areas (up to 10 cm2) can be used linear palette(Fig. 19, c), which can be made on tracing paper by drawing a series of parallel lines at equal intervals (2-5 mm). The palette is applied to this area in such a way that extreme points area (points m and n in Fig. 19, c) are located in the middle between the parallel lines of the palette. Then measure the length of the lines using compasses and a scale ruler. l 1 , l 2 ….., l n , which are the middle lines of the trapezoid into which the area of a given area is divided using a palette. Then the area of the plot S= a(l 1 + l 2 +……+ l n ), Where a- linear palette step, i.e. distance between parallel lines. For control, the palette is drawn at 60-90° relative to the original position and the area of the area is re-determined. The relative error in determining the area by a linear tent depends on its pitch and is 1: 50 - 1: 100
7.2. Analytical method for determining area If you collect enough points along the contour of the area of the measured area to approximate this area with the required accuracy by a polygon formed by these points (Fig. 19, a), and then measure the coordinates on the map X And at all points, then the area of the site can be determined analytically. For a polygon about the number of vertices n when they are digitized clockwise, the area will be determined by the formulas For control, calculations are performed using both formulas. The accuracy of the analytical method depends on the density of the set of points along the contour of the measured area. With a significant number of points, it is advisable to carry out calculations using computers or microcalculators = 7.3. Mechanical method determining the area using a planimeter A planimeter is a mechanical device for measuring area. In engineering and geodetic practice, using a planimeter, the areas of fairly large areas are measured from plans or maps. Of the many designs of planimeters greatest distribution received polar planimeters. The polar planimeter (Fig. 20) consists of two levers - pole 1 and bypass 4. At the bottom of the weight 2, attached to one end of the pole lever, there is a needle - the planimeter pole. At the second end of the pole lever there is a pin with a spherical head, which is inserted into a special socket in carriage 5 of the bypass lever. At the end of the bypass lever there is a lens 3, on which there is a circle with a bypass point in the center. Carriage 5 has a counting mechanism, consisting of a counter of 6 whole revolutions of the counting wheel and the counting wheel itself 7. For readings on the counting wheel there is a special device - vernier 8. When tracing the contour of a section of the bypass lens 3, the rim of the counting wheel and roller 9 rolls or slides along the paper , forming, together with the contour point, three reference points of the planimeter. In modern planimeters, a carriage with a counting mechanism can move along the bypass lever, thereby changing its length, and be fixed in a new position. The circumference of the counting wheel is divided into 100 parts, every tenth stroke is digitized. The planimeter count consists of four digits: the first digit is the smaller digit of the revolution counter closest to the pointer (thousands divisions of the planimeter), the second and third digits are the hundreds and tens divisions on the counting wheel, preceding the zero stroke of the vernier; the fourth digit is the number of the vernier stroke, which coincides with the nearest stroke of the counting wheel (division unit). Before measuring the area of an area, the planimeter is installed on the map so that its pole is located outside the area being measured, and the pole and bypass arms form approximately a right angle. In this case, the place where the pole is secured is chosen so that during the detour of the entire figure, the angle between the bypass and pole levers is no less than 30° and no more than 150°. Having aligned the contour point of the planimeter with a certain starting point of the contour of the section, the initial reading is taken using the counting mechanism no and smoothly trace the entire contour clockwise. Returning to the starting point, take the final count n. Count difference ( n -no) expresses the area of a figure in planimeter divisions. Then the area of the measured area Where µ is the cost of dividing the planimeter, i.e. area corresponding to one planimeter division. To control and improve the accuracy of measurement results, the area of the site is measured at two positions of the planimeter pole relative to the counting mechanism: “pole left” and “pole right”. Before measuring areas, it is necessary to determine the division priceplanimeter µ. To do this, choose a figure whose area is ½ O known in advance (for example, one or more grid squares). In order to obtain higher accuracy this figure trace along the contour 4 times: 2 times in the “pole right” position and 2 times in the “pole left” position. For each round, the initial and final readings are taken and their difference is calculated (n i- n oi) . The discrepancies between the difference values for “pole right” and “pole left” should not exceed 2 divisions for a figure area of up to 200 division, 3 divisions - with a figure area from 200 to 2000 divisions and 4 divisions - with a figure area over 2000 divisions of the planimeter. If the discrepancies do not exceed acceptable values, then the average is calculated.difference of counts (n- no) Wedand calculate the price of dividing the planimeter using the formula / (n - n o ) Wed The division value is calculated with an accuracy of 3-4 significant figures. The table (p. 39) shows an example of recording the measurement results of the planimeter division price and determining the area of the site on the map. The accuracy of determining areas with a polar planimeter depends on the size of the measured areas. The smaller the area of the site, the greater the relative error in its determination. It is recommended to use a planimeter to measure the areas of plots on the plan (map) of at least 10-12 cm 2. At favorable conditions measurements, the relative error in determining areas using a planimeter is approximately 1: 400. 8. DESCRIPTION OF THE CARD When carrying out engineering and geodetic surveys, drawing up technical documentation requires the performer to have a good knowledge of conventional signs and basic patterns of placement of natural objects (for example, mutual consistency of relief, hydrography, vegetation, settlements, road network, etc.). Often there is a need to describe certain areas of the map. To describe a map area, it is recommended to use the following scheme. I. Name (nomenclature) of the card. 2. Output: 2.1. Where, when and by whom was the map compiled and published? 2.2. What cartographic materials is it made from? 3.1. Map scale. 3.2. Longitude and latitude of map frames. 3.3. Kilometer grid, frequency of its lines and their digitization. 3.4. Location on the map of the described area. 3.5. Geodetic basis on the described map (types of reference marks, their number). 4. Physiographic elements: hydrography (seas, rivers, lakes, canals, irrigation and drainage systems); relief, its character, dominant heights and lowest places, their marks; vegetation cover. 5. Socio-economic elements: settlements, transport routes, communications, industry, agriculture and forestry, cultural elements. As an example, the following description of one of the sections of the map at a scale of 1: 25,000 is given. I. Map U-34-37-V-v (Dreams). 2. Output: 2.1. The map was prepared for publication in 1981 by the GUGK and printed in 1982. Photographed by A.P. Ivanov. 2.2. The map was compiled based on materials from an aerial phototopographic survey of 1980. 3. Mathematical elements of the map: 3.1. Map scale 1: 25,000. 3.2. The map sheet is limited in longitude by the meridians 18 o 00' 00'' (in the west) and І8°07'"З0'' (in the east) and in latitude - by parallels 54 o 40' 00'' (in the south) and 54°45 '00'' (in the north). 3.3. The map shows a kilometer grid of rectangular coordinates (every 1 km). The grid squares on the map have side dimensions of 40 mm (on the map scale, 1 cm corresponds to 250 m on the ground). The map sheet contains 9 horizontal lines of the kilometer grid (from x = 6065 km in the south to x = 6073 km in the north) and 8 vertical lines grid (from y = 4307 km in the west to y = 4314 km in the east). 3.4. The described map area occupies four squares of the kilometer grid (from x 1 = 6068 km to x 2 = 6070 km and from y 1 = 4312 km to y 2 = 4314 km) east of the central map area. Determining the area of a plot using a planimeter

Pole position		Number			Counts				Difference r=n- n 0			Average r cp		Relative error (rpp- rpl)/ r cp		Division price µ= s o/ r cp	Contour area S= µ * r cp
					n 0		n
1. Determination of the price of planimeter division (S o = 4 km 2 = 400 ha)
PP			2	0112 0243		6414 6549				6302 6306			6304	1:3152 0.06344 ha/division.
PL			2	0357 0481		6662 6788				6305 6307			6306
2. Determination of the area of the site
PP PL	2				0068 0106			0912 0952			846				1:472 0.06344 ha/division. 59.95 hectares

3.5. On the described section of the map there is one point of the geodetic network, installed on Mount Mikhalinskaya. 4. Physiographic elements. In the northeastern corner of the described area flows the Sot River, over 250 m wide. The direction of its flow is from northwest to southeast, the flow speed is 0.1 m/s. A permanent river bank signal sign has been installed on the western bank of the river. The banks of the river are swampy and covered with meadow vegetation. In addition, there are isolated bushes on the eastern bank of the river. In the area described, two streams flow into the Sot River, flowing along the bottom of ravines leading to the river. In addition to the indicated ravines, another ravine leads to the crayfish and in the southwestern part of the site there are two ravines covered with continuous vegetation. The terrain is hilly, with elevation differences of over 100 m. The dominant heights are Mount Bolshaya Mikhalinskaya with a peak elevation of 213.8 m in the western part of the site and Mount Mikhalinskaya with a peak elevation of 212.8 m in the southern part of the site. From these heights the relief rises towards the river (with a water mark of about 108.2 m). In the northern section the coast is steep (with a cliff height of up to 10 m). There is also a slight decrease in the relief from the indicated heights to the southwest. In the southern part of the site there is the Northern forest, occupying about 0.25 km 2 and located in the saddle between the indicated heights and to the east of the saddle. The predominant tree species in forest - pine, the average height of trees is about 20 m, the average thickness of trees is 0.20 m, the distance between trees is 6 m. In the southern part of the site, an area of open forest and cut down forest adjoins the Northern forest. On the western slope of Mount Mikhalinskaya there is a separate standing tree, having the value of a landmark. 5. Socio-economic elements. There are no settlements in the described area, but immediately beyond its borders in the southwest there is the settlement of Mikhalino, numbering 33 houses. The area of the plot includes partly the gardens of this settlement. There are three dirt (country) roads on the site. One of them runs from west to southwest of the site, the other runs from southwest to north and turns into a field road at the very edge of the site. At the point of this transition, the road branches and a third dirt road runs from north to southeast local) road. From this third road in the southeast another floor road branches off in a southerly direction. There are no other socio-economic elements in this area of the map.
9. PREPARATION OF THE REPORT The report on laboratory work on the topographic map consists of an explanatory note and graphic documents. The explanatory note contains a write-off of the laboratory work performed and an explanation of the results obtained. The explanatory note is drawn up on separate sheets of writing paper (standard format 210 x 297 mm). Each laboratory work must have the name and information about the card on which it was performed, and the date the work was completed. The explanatory note must have front page, on which it is necessary to indicate the name of the faculty, group, the name of the student who completed the work, the name of the teacher who issued the assignment and checked the work, and the date the work was completed. Graphic documents are a copy and a topographic profile. These documents are included in the explanatory note. A copy of the map is drawn in ink on tracing paper, and copies the border design of the map (design and degree frames, signatures), and the kilometer grid. Copies of those parts of the map that are necessary to illustrate the solution of a particular problem are also made onto a copy of the map on tracing paper, for example, when designing a line of a given slope, when determining the boundaries of a drainage area, when describing a section of the map. The topographic profile is drawn in ink on graph paper, and the profile line must be shown on a copy of the map and the horizontal lines immediately adjacent (1 cm in each direction) to the profile line must be copied on it. Other graphic diagrams and drawings illustrating the solution of topographic map problems may be included in the text explanatory note. All drawings must be made carefully, without blots, in compliance with dimensions, symbols and fonts. The pages of the explanatory note must be numbered, and the note itself must have a table of contents. The count is submitted to the teacher for verification, after which it is defended by the student in class.

Map scale. The scale of topographic maps is the ratio of the length of a line on the map to the length of the horizontal projection of the corresponding terrain line. In flat areas, with small angles of inclination of the physical surface, the horizontal projections of the lines differ very little from the lengths of the lines themselves, and in these cases the ratio of the length of the line on the map to the length of the corresponding terrain line can be considered a scale, i.e. the degree of reduction in the lengths of lines on the map relative to their length on the ground. The scale is indicated under the southern frame of the map sheet in the form of a ratio of numbers (numerical scale), as well as in the form of named and linear (graphic) scales.

Numerical scale(M) is expressed as a fraction, where the numerator is one, and the denominator is a number indicating the degree of reduction: M = 1/m. So, for example, on a map at a scale of 1:100,000, the lengths are reduced in comparison with their horizontal projections (or with reality) by 100,000 times. Obviously, the larger the scale denominator, the greater the reduction in lengths, the smaller the image of objects on the map, i.e. the smaller the scale of the map.

Named scale- an explanation indicating the ratio of the lengths of lines on the map and on the ground. With M = 1:100,000, 1 cm on the map corresponds to 1 km.

Linear scale used to determine the lengths of lines in nature from maps. This is a straight line divided into equal segments corresponding to “round” decimal numbers terrain distances (Fig. 5).

Rice. 5. Designation of scale on a topographic map: a - the base of the linear scale: b - the smallest division of the linear scale; scale accuracy 100 m. Scale size - 1 km

The segments a laid off to the right of zero are called basis of scale. The distance on the ground corresponding to the base is called linear scale value. To increase the accuracy of determining distances, the leftmost segment of the linear scale is divided into smaller parts, called the smallest divisions of the linear scale. The distance on the ground expressed by one such division is the accuracy of the linear scale. As can be seen in Figure 5, with a numerical map scale of 1:100,000 and a linear scale base of 1 cm, the scale value will be 1 km, and the scale accuracy (with the smallest division of 1 mm) will be 100 m. Accuracy of measurements from maps and accuracy of graphical constructions on paper connected as with technical capabilities measurements and resolution human vision. The accuracy of constructions on paper (graphic accuracy) is generally considered to be 0.2 mm. The resolution of normal vision is close to 0.1 mm.

Ultimate accuracy map scale - a segment on the ground corresponding to 0.1 mm on the scale of a given map. At a map scale of 1:100,000, the maximum accuracy will be 10 m; at a scale of 1:10,000 it will be 1 m. Obviously, the possibilities of depicting contours in their actual outlines on these maps will be very different.

The scale of topographic maps largely determines the selection and detail of the objects depicted on them. With a decrease in scale, i.e. as its denominator increases, the detail of the image of terrain objects is lost.

To meet the diverse needs of industries national economy, science and national defense require maps of different scales. A number of standard scales based on the metric decimal system of measures have been developed for state topographic maps of the USSR (Table 1).

Table 1. Scales of topographic maps of the USSR
Numerical scale	Card name	1 cm on the map corresponds to a distance on the ground	1 cm 2 on the map corresponds to the area on the ground
1:5 000	Five thousandth	50 m	0.25 ha
1:10 000	Ten-thousandth	100 m	1 ha
1:25 000	Twenty-five thousandth	250 m	6.25 ha
1:50 000	Fifty thousandth	500 m	25 hectares
1:100 000	One hundred thousandth	1 km	1 km 2
1:200 000	Two hundred thousandth	2 km	4 km 2
1:500 000	Five hundred thousandth	5 km	25 km 2
1:1 000 000	Millionth	10 km	100 km 2

In the complex of cards named in table. 1, there are actual topographic maps of scales 1:5000-1:200,000 and survey topographic maps of scales 1:500,000 and 1:1,000,000. The latter are inferior in accuracy and detail to the depiction of the area, but individual sheets cover significant territories, and these maps are used for general familiarization with the area and for orientation when moving at high speed.

Measuring distances and areas using maps. When measuring distances on maps, it should be remembered that the result is the length of horizontal projections of lines, and not the length of lines on the earth's surface. However, at small angles of inclination, the difference in the length of the inclined line and its horizontal projection is very small and may not be taken into account. So, for example, at an inclination angle of 2°, the horizontal projection is shorter than the line itself by 0.0006, and at 5° - by 0.0004 of its length.

When measuring from distance maps in mountainous areas, the actual distance on an inclined surface can be calculated

according to the formula S = d·cos α, where d is the length of the horizontal projection of the line S, α is the angle of inclination. Inclination angles can be measured from a topographic map using the method indicated in §11. Corrections to the lengths of inclined lines are also given in the tables.

Rice. 6. Position of the measuring compass when measuring distances on a map using a linear scale

To determine the length of a straight line segment between two points, a given segment is taken from the map into a compass-measuring solution, transferred to the linear scale of the map (as indicated in Figure 6) and the length of the line is obtained, expressed in land measures (meters or kilometers). In a similar way, measure the lengths of broken lines by taking each segment separately into a compass solution and then summing their lengths. Measuring distances along curved lines (along roads, borders, rivers, etc.) is more complex and less accurate. Very smooth curves are measured as broken lines, having first been divided into straight segments. Winding lines are measured with a small constant opening of a compass, rearranging it (“walking”) along all the bends of the line. Obviously, finely sinuous lines should be measured with a very small compass opening (2-4 mm). Knowing what length the compass opening corresponds to on the ground, and counting the number of its installations along the entire line, determine its total length. For these measurements, a micrometer or spring compass is used, the opening of which is adjusted by a screw passed through the legs of the compass.

Rice. 7. Curvimeter

It should be borne in mind that any measurements are inevitably accompanied by errors (errors). According to their origin, errors are divided into gross errors (arising due to the inattention of the person making the measurements), systematic errors (due to errors in measuring instruments, etc.), random errors that cannot be fully taken into account (their reasons are not clear). Obviously, the true value of the measured quantity remains unknown due to the influence of measurement errors. Therefore, its most probable value is determined. This value is the arithmetic average of all individual measurements x - (a 1 +a 2 + …+a n):n=∑a/n, where x is the most probable value of the measured value, a 1, a 2 … a n are the results of individual measurements ; 2 is the sign of the sum, n is the number of dimensions. The more measurements, the closer the probable value is to the true value of A. If we assume that the value of A is known, then the difference between this value and the measurement of a will give the true measurement error Δ=A-a. The ratio of the measurement error of any quantity A to its value is called relative error-. This error is expressed as a proper fraction, where the denominator is the fraction of the error from the measured value, i.e. Δ/A = 1/(A:Δ).

So, for example, when measuring the lengths of curves with a curvimeter, a measurement error of the order of 1-2% occurs, i.e. it will be 1/100 - 1/50 of the length of the measured line. Thus, when measuring a line 10 cm long, a relative error of 1-2 mm is possible. This value on different scales gives different errors in the lengths of the measured lines. So, on a map of scale 1:10,000, 2 mm corresponds to 20 m, and on a map of scale 1:1,000,000 it will be 200 m. It follows that more accurate measurement results are obtained when using large-scale maps.

Definition of areas plots on topographic maps is based on the geometric relationship between the area of the figure and its linear elements. The scale of the areas is equal to the square of the linear scale. If the sides of a rectangle on a map are reduced by a factor of n, then the area of this figure will decrease by a factor of n2. For a map of scale 1:10,000 (1 cm - 100 m), the scale of the areas will be equal to (1:10,000)2 or 1 cm 2 - (100 m) 2, i.e. in 1 cm 2 - 1 hectare, and on a map of scale 1:1,000,000 in 1 cm 2 - 100 km 2.

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

Rice. 8. Straightening the curved boundaries of the site and dividing its area into simple geometric shapes: dots indicate cut-off areas, hatching indicates attached areas

When measuring the area of a plot with straight boundaries, divide the plot into simple geometric shapes, measure the area of each of them geometrically and, by summing the areas of individual plots, calculated taking into account the map scale, obtain the total area of the object. An object with a curved contour is divided into geometric shapes, having previously straightened the boundaries in such a way that the sum of the cut off sections and the sum of the excesses mutually compensate each other (Fig. 8). The measurement results will be somewhat approximate.

Rice. 9. Square grid palette placed on the measured figure. Area of the plot P=a 2 n, a is the side of the square, expressed on a map scale; n - number of squares falling within the contour of the measured area

Measuring the areas of areas with complex irregular configurations is often done using palettes and planimeters, which gives the most accurate results. The grid palette (Fig. 9) is a transparent plate (made of plastic, organic glass or tracing paper) with an engraved or drawn grid of squares. The palette is placed on the contour being measured and the number of cells and their parts found inside the contour is counted from it. The proportions of incomplete squares are estimated by eye, therefore, to increase the accuracy of measurements, palettes with small squares (with a side of 2-5 mm) are used. Before working on this map, determine the area of one cell in land measures, i.e. the price of dividing the palette.

Rice. 10. Dot palette - a modified square palette. Р=a 2 n

In addition to mesh palettes, dot and parallel palettes are used, which are transparent plates with engraved dots or lines. The points are placed in one of the corners of the cells of the grid palette with a known division value, then the grid lines are removed (Fig. 10). The weight of each point is equal to the cost of dividing the palette. The area of the measured area is determined by counting the number of points inside the contour and multiplying this number by the weight of the point.

Rice. 11. A palette consisting of a system of parallel lines. The area of the figure is equal to the sum of the lengths of the segments (middle dotted lines) cut off by the contour of the area, multiplied by the distance between the lines of the palette. P = р∑l

Equally spaced parallel lines are engraved on the parallel palette. The measured area will be divided into a number of trapezoids with the same height when the palette is applied to it (Fig. 11). The segments of parallel lines inside the contour in the middle between the lines are the midlines of the trapezoids. Having measured all the middle lines, multiply their sum by the length of the gap between the lines and get the area of the entire area (taking into account the areal scale).

The areas of significant areas are measured from maps using a planimeter. The most common is the polar planimeter, which is not very difficult to operate. However, the theory of this device is quite complex and is discussed in geodesy manuals.