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Scale. Determination of distances. Solving tasks according to topographic plans Measuring the area of a plot with a curved contour

The scale is the ratio of the length of a line in a drawing, plan or map to the actual length of the corresponding line. It shows how many times the distance on the map is reduced relative to the real distance on the ground. If, for example, the scale geographic map 1: 1,000,000, which means that 1 cm on the map corresponds to 1,000,000 cm on the ground, or 10 km.

Distinguish between numerical, linear and named scales. .

Numerical scale is depicted as a fraction, in which the numerator is equal to one, and the denominator is a number that shows how many times the lines on the map (plan) are reduced relative to the lines on the ground. For example, a scale of 1: 100,000 shows that all linear dimensions on the map are reduced by a factor of 100,000. Obviously, the larger the denominator of the scale, the smaller the scale, and the smaller the denominator, the larger. The numerical scale is a fraction, so the numerator and denominator are given in the same measurements (centimeters).

Linear scale is a straight line divided into equal segments. These segments correspond to a certain distance on the depicted area; divisions are indicated by numbers. The measure of length along which the divisions are plotted on the scale bar is called the base of the scale. In our country, the scale base is taken to be 1 cm. The number of meters or kilometers corresponding to the scale base is called the scale value. When building a linear scale, the figure 0 , from which the divisions are counted, are usually placed not at the very end of the scale line, but stepping back one division (base) to the right; on the first segment to the left of 0, the smallest divisions of a linear scale are applied - millimeters. The distance on the ground corresponding to one smallest division of the linear scale corresponds to the scale accuracy, and 0.1 mm corresponds to the extreme scale accuracy. The linear scale in comparison with the numerical one has the advantage that it makes it possible to determine the actual distance on the plan and map without additional calculations.

Named scale - scale expressed in words, for example, 1 cm 32 km.

Measuring distances on the map and plan.

Measuring distances using a scale... You need to draw a straight line (if you want to know the distance in a straight line) between two points and use a ruler to measure this distance in centimeters, and then multiply the resulting number by the magnitude of the scale. For example, on a scale map 1: 100 000 (in 1 cm 1 km) the distance is 5 cm, i.e., on the ground, this distance is 1 * 5 = 5 (km)... You can also measure the distance on the map using a compass-measuring device. In this case, it is convenient to use a linear scale.

Measuring distances using a degree network. To calculate distances on a map or globe, you can use the following values: arc length 1 ° meridian and 1 ° equator is approximately 111 km. For meridians this is always true, and the length of an arc of 1 ° along the parallels decreases towards the poles. At the equator, it can also be taken equal to 111 km. And at the poles - 0 (since the pole is a point). Therefore, it is necessary to know the number of kilometers corresponding to the length of 1 ° of the arc of each particular parallel. To determine the distance in kilometers between two points lying on the same meridian, the distance between them in degrees is calculated, and then the number of degrees is multiplied by 111 km. To determine the distance between two points on the equator, you also need to determine the distance between them in degrees, and then multiply by 111 km.

INTRODUCTION

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

6.1. TYPES OF SCALE TOPOGRAPHIC MAPS

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

Map scale (plan) - the ratio of the length of the line on the map (plan) to the length of the horizontal distance of the corresponding terrain line

m = l K: d M

The scale of the image of small areas throughout the topographic map is practically constant. physical surface(on a plain) the length of the horizontal projection of the line differs very little from the length of the oblique line. In these cases, the ratio of the length of the line on the map to the length of the corresponding line on the ground can be considered the scale of length.

The scale is indicated on maps in different options

6.1.1. Numerical scale

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

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

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

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

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

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

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

Scale series of topographic maps

Table 6.1.

Numerical scale	Card name	1cm card matches on terrain distance	1cm2 card matches in the area of the square
	Five thousandth		0.25 hectare
	Ten thousandth
	Twenty-five thousandth		6.25 hectares
	Fifty thousandth
	One hundred thousandth
	Two hundred thousandth
	Five hundred thousandth
	Millionth

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

6.1.2. Named scale

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

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

Example... The named scale of the map is "1 centimeter 2 kilometers". The length of the segment on the map lK= 6.3 cm. Determine the horizontal distance of the line on the ground.
Solution... Multiplying the size of the segment measured on the map in centimeters by the value of the named scale, we get the horizontal distance in kilometers on the ground.
d= 6.3 cm × 2 = 12.6 km.

6.1.3. Graphic scales

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

Linear scale

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

How to use a linear scale:

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

Transverse scale

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

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

To build it, on a straight line segment, several scale bases are laid ( a). Usually the length of the base is 2 cm or 1 cm.In the obtained points, set perpendiculars to the line AB and draw ten parallel lines through them at regular intervals. The extreme left base above and below is divided into 10 equal segments and connected with oblique lines. The zero point of the lower base is connected to the first point WITH top base and so on. A series of parallel oblique lines is obtained, which are called transversals.
The smallest division of the transverse scale is equal to the line segment C 1 D 1 , (fig. 6.2, but). The adjacent parallel segment differs by this length when moving up the transversal 0C and along the vertical line 0D.
A transverse scale with a base of 2 cm is called normal ... If the base of the transverse scale is divided into ten parts, then it is called centesimal . On a hundredth scale, the smallest division is equal to one hundredth of the base.
The transverse scale is engraved on metal rulers, which are called scale rulers.

How to use the cross scale:

using a caliper to record the length of the line on the map;
put the right leg of the compass on a whole division of the base, and the left leg - on any transversal, while both legs of the compass should be located on a line parallel to the line AB;
the length of the line consists of three counts: counting whole bases, plus counting divisions of the left base, plus counting divisions up the transversal.

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

6.2. VARIETIES OF GRAPHIC ZOOMS

6.2.1. Transitional scale

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

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

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

Now if you build a transverse scale with the length of the base but= 2.29 cm, then one division of the left base will correspond to 40 m (Fig. 6.3).

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

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

6.2.2. Step scale

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

Example... The known distance of 100 m is measured in steps. We walked 137 steps forward and 139 steps backward. Calculate the average length of one stride.
Solution... Total covered: Σ m = 100 m + 100 m = 200 m.The sum of steps is: Σ w = 137 w + 139 w = 276 w. The average length of one step is:

Shsr= 200/276 = 0.72 m.

It is convenient to work with a linear scale when the scale line is marked every 1 - 3 cm, and the divisions are signed with a round number (10, 20, 50, 100). Obviously, the size of one step 0.72 m at any scale will have extremely small values. For a scale of 1: 2,000, the segment on the plan will be 0.72 / 2,000 = 0.00036 m or 0.036 cm. Ten steps, in the corresponding scale, will be expressed as a segment of 0.36 cm. The most convenient basis for these conditions, in the opinion author, there will be a value of 50 steps: 0.036 × 50 = 1.8 cm.
For those who count steps in pairs, a convenient base would be 20 pairs of steps (40 steps) .036 x 40 = 1.44 cm.
The base length of the step scale can also be calculated from proportions or by the formula
but = (Shsr × KSh) / M
where: Shsr - average value of one step in centimeters,
KSh - number of steps at the base of the scale ,
M - denominator of scale.

The base length for 50 steps on a scale of 1: 2,000 with one step length equal to 72 cm will be:
but= 72 × 50/2000 = 1.8 cm.
To build the scale of steps for the above example, you need to divide the horizontal line into segments equal to 1.8 cm, and divide the left base into 5 or 10 equal parts.

Rice. 6.4. Scale of steps.
Measured distance AC = BC + AB = 100 + 20 = 120 sh.

6.3. SCALE ACCURACY

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

Scale accuracy allows two important tasks to be accomplished:

determination of the minimum sizes of objects and terrain items that are depicted at a given scale, and the sizes of objects that cannot be depicted on a given scale;
setting the scale in which to create a map so that it depicts objects and objects of the terrain with a predetermined minimum size.

In practice, it is assumed that the length of a segment on a plan or map can be estimated with an accuracy of 0.2 mm. The horizontal distance on the ground, corresponding on a given scale of 0.2 mm (0.02 cm) on the plan, is called graphic scale accuracy . 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 graphic accuracy, but by the accuracy of the map itself, where errors can be on average 0.5 mm due to the influence of errors other than graphic ones.
If we take into account the error of the map itself and the error of measurements on the map, then we can conclude that the graphic accuracy of determining the distances on the map is 5 - 7 worse than the limiting accuracy of the scale, that is, it is 0.5 - 0.7 mm at the map scale.

6.4. DETERMINING AN UNKNOWN MAP SCALE

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

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

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

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

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

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

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

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

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

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

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

6.5. TECHNIQUE FOR MEASURING AND STAYING DISTANCES ON THE MAP

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

6.5.1. Measuring distances with a millimeter ruler

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

6.5.2. Distance measurement with a caliper

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

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

To get the length broken line the length of each of its links is measured sequentially, and then their values are summed up. Such lines are also measured by extending the compass solution.
Example... To measure the length of a polyline ABCD(fig. 6.6, but), the legs of the compass are first set at the points BUT and IN... Then, rotating the compass around the point IN... move the hind leg out of the point BUT exactly IN"lying on the continuation of the straight line Sun.
Front leg from point IN transfer to point WITH... The result is a compass solution B "C=AB+Sun... Moving the back leg of the compass in the same way from the point IN" exactly WITH", and the front from WITH in D... get a compass solution
C "D = B" C + CD, the length of which is determined using a transverse or linear scale.

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

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

6.5.3. Measuring distances with a curvimeter

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

Rice. 6.7. Mechanical curvimeter

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

Rice. 6.8. Digital curvimeter (electronic)

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

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

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

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

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

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

Table 6.3

Tilt angle

Rules for using the table

1. The first row of the table (0 tens) shows the relative values of the corrections at tilt angles from 0 ° to 9 °, in the second - from 10 ° to 19 °, in the third - from 20 ° to 29 °, in the fourth - from 30 ° up to 39 °.
2. To determine the absolute value of the correction, 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 an integer number of degrees, then it is necessary to find the relative value of the correction by interpolating between the tabular values);
b) calculate the absolute value of the correction to the length of the horizontal distance (i.e., multiply this length by the relative value of the correction and divide the resulting product by 100).
3. To determine the length of the line on the topographic surface, it is necessary to add the calculated absolute value of the correction to the length of the horizontal distance.

Example. On the topographic map, the length of the horizontal distance 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 higher and the nearest lower value multiples of one degree - 8º and 7º:
for 8 ° the relative value of the correction is 0.98%;
for 7 ° 0.75%;
the difference in tabular values is 1º (60 ′) 0.23%;
the difference between the given angle of inclination of the earth's surface 7 ° 15 "and the nearest lower tabular value of 7 ° is 15".
We make up the proportions and find the relative value of the correction for 15 ":

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

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

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

6.6. AREA MEASUREMENT BY MAPS

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

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

6.6.1. Measuring the area of a parcel with straight boundaries

When measuring the area of a site with rectilinear boundaries, the site is divided into simple geometric figures, measure the area of each of them in a geometric way and, summing up the areas of individual sections, calculated taking into account the scale of the map, the total area of the object is obtained.

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

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

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

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

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

Rice. 6.11. Square Grid Palette

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

P = a 2 n,

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

To improve accuracy, the area is determined several times with an arbitrary permutation of the used pallet to any position, including with a rotation relative to its original position. The arithmetic mean of the measurement results is taken as the final area value.

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

Rice. 6.12. Spot palette

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

P = h∑l

Fig 6.13. Palette consisting of a system
parallel lines

Measurement areas of significant plots produced by cards using planimeter.

Rice. 6.14. Polar planimeter

The planimeter is used to determine areas mechanically. The polar planimeter is widespread (Fig. 6.14). It consists of two levers - pole and bypass. Determination of the contour area with a planimeter is reduced to the following steps. After fixing the pole and setting the needle of the bypass lever at the starting point of the contour, take a reading. Then the bypass spire is carefully guided along the contour to the starting point and a second reading is taken. The difference in readings will give the area of the contour in divisions of the planimeter. Knowing the absolute division price of the planimeter, the area of the contour is determined.
The development of technology contributes to the creation of new devices that increase labor productivity when calculating areas, in particular, the use of modern devices, among which are electronic planimeters.

Rice. 6.15. Electronic planimeter

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

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

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

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

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

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

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

We represent expressions (6.1) and (6.2) in general form, denoting by i the ordinal number (i = 1, 2, ..., n) of the vertices of the polygon:
(6.3)
(6.4)
Therefore, the doubled area of the polygon is either the sum of the products of each abscissa and the difference between the ordinates of the next and previous vertices of the polygon, or the sum of the products of each ordinate and the difference between the abscissas of the previous and subsequent vertices of the polygon.
An intermediate control of the calculations is the satisfaction of the conditions:

0 or = 0
Coordinate values and their differences are usually rounded to tenths of a meter, and products - to whole square meters.
Complex formulas for calculating the area of \ u200b \ u200bplot can be easily solved using spreadsheets MicrosoftXL. An example for a polygon (polygon) of 5 points is shown in Tables 6.4, 6.5.
In table 6.4 we enter the initial data and formulas.

Table 6.4.

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

Double area in m 2

SUM (D2: D6)

Area in hectares

In table 6.5 we see the results of the calculations.

Table 6.5.

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






	Double area in m 2
	Area in hectares

6.7. EYE MEASUREMENTS ON THE MAP

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

Video

Scope Tasks

Assignments and questions for self-control

What elements does it include mathematical basis kart?
Expand the concepts: "scale", "horizontal distance", "numerical scale", "linear scale", "scale accuracy", "scale bases".
What is a named map scale and how do I use it?
What is the transverse scale of the map, for what purpose is it intended?
What is the normal transverse scale of the map?
What are the scales of topographic maps and forest management plans used in Ukraine?
What is the Transitional Map Scale?
How is the base of the transition scale calculated?

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

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

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

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

Linear scale

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

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

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

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

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

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

Transverse scale

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

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

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

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

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

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

Fig. 2

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

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

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

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





	Double area in m 2			SUM (D2: D6)
	Area in hectares

Scale. Determination of distances. Solving tasks according to topographic plans Measuring the area of ​​a plot with a curved contour

Measuring distances on the map and plan.

6.1. TYPES OF SCALE TOPOGRAPHIC MAPS

Transverse scale

6.2. VARIETIES OF GRAPHIC ZOOMS

6.3. SCALE ACCURACY

6.4. DETERMINING AN UNKNOWN MAP SCALE

6.5. TECHNIQUE FOR MEASURING AND STAYING DISTANCES ON THE MAP

6.5.1. Measuring distances with a millimeter ruler

6.5.2. Distance measurement with a caliper

6.5.4. Conversion of horizontal distance to slant range

6.6. AREA MEASUREMENT BY MAPS

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

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

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

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

6.7. EYE MEASUREMENTS ON THE MAP

Linear scale

Transverse scale

Scale. Determination of distances. Solving tasks according to topographic plans Measuring the area of a plot with a curved contour

6.6.1. Measuring the area of a parcel with straight boundaries

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

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

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