What types of scale records do you know? Scale, types of scale recording. sides of the horizon. Methodological techniques for constructing a profile on a map - Document

INTRODUCTION

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

6.1. TYPES OF SCALE OF TOPOGRAPHIC MAP

When compiling maps and plans, horizontal projections of segments are depicted on paper in a reduced form. The degree of such a decrease is characterized by scale.

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

m = l K : d M

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

The scale is indicated on the maps in different options

6.1.1. Numerical scale

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

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

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

Solution.
Multiplying the value of the segment on the map in centimeters by the denominator of the numerical scale, we get the horizontal distance in centimeters.
d\u003d 4.0 cm × 50,000 \u003d 200,000 cm, or 2,000 m, or 2 km.

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

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

To meet the needs of the economy, science and defense of the country, maps of various scales are needed. For government topographic maps, forest management plans, plans of forest areas and forest plantations are defined standard scales - scale range(Tables 6.1, 6.2).

Scale series of topographic maps

Table 6.1.

Numerical scale	Map name	1 cm card corresponds on the ground distance	1 cm2 card corresponds on the territory of the square
	five thousandth		0.25 hectare
	ten thousandth
	twenty-five thousandth		6.25 hectares
	fifty thousandth
	hundred thousandth
	two hundred thousandth
	five hundred thousandth
	millionth

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

6.1.2. Named Scale

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

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

Example. The named scale of the map is "2 kilometers in 1 centimeter". The length of the segment on the map lk\u003d 6.3 cm. Determine the horizontal location of the line on the ground.
Solution. Multiplying the value of the segment measured on the map in centimeters by the value of the named scale, we obtain the horizontal distance in kilometers on the ground.
d= 6.3 cm × 2 = 12.6 km.

6.1.3. Graphic scales

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

Linear scale

For building linear scale choose the initial segment, convenient for the given scale. This original segment ( a) are called scale base (Fig. 6.1).

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

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

How to use a linear scale:

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

Cross scale

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

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

To build it on a straight line segment, several scale bases are laid ( a). Usually the length of the base is 2 cm or 1 cm. Perpendiculars to the line are set at the points obtained. AB and draw through them ten parallel lines at regular intervals. The leftmost base from above and below is divided into 10 equal segments and connected by oblique lines. The zero point of the lower base is connected to the first point FROM upper base and so on. Get a series of parallel inclined lines, which are called transversals.
The smallest division of the transverse scale is equal to the segment C 1 D 1 , (fig. 6. 2, a). The adjacent parallel segment differs by this length when moving up the transversal 0C and vertical line 0D.
A transverse scale with a base of 2 cm is called normal . If the base of the transverse scale is divided into ten parts, then it is called hundreds . On a hundredth scale, the price of the smallest division is equal to one hundredth of the base.
The transverse scale is engraved on metal rulers, which are called scale.

How to use the transverse scale:

fix the length of the line on the map with a measuring compass;
put the right leg of the compass on an integer division of the base, and the left leg on any transversal, while both legs of the compass should be located on a line parallel to the line AB;
the length of the line consists of three counts: a count of integer bases, plus a count of divisions of the left base, plus a count of divisions up the transversal.

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

6.2. VARIETY OF GRAPHIC SCALE

6.2.1. transitional scale

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

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

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

If we now construct a transverse scale with a base length a\u003d 2.29 cm, then one division of the left base will correspond to 40 m (Fig. 6.3).

Rice. 6.3. Transitional linear scale.
Measured distance AC \u003d BC + AB \u003d 800 +160 \u003d 960 m.

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

6.2.2. Step scale

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

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

Shsr= 200 / 276 = 0.72 m.

It is convenient to work with a linear scale when the scale line is marked every 1 - 3 cm, and the divisions are signed with a round number (10, 20, 50, 100). Obviously, the value of one step of 0.72 m on any scale will have extremely small values. For a scale of 1: 2,000, the segment on the plan will be 0.72 / 2,000 \u003d 0.00036 m or 0.036 cm. Ten steps, on the appropriate scale, will be expressed as a segment of 0.36 cm. The most convenient basis for these conditions, according to the author, there will be a value of 50 steps: 0.036 × 50 = 1.8 cm.
For those who count steps in pairs, a convenient base would be 20 pairs of steps (40 steps) 0.036 × 40 = 1.44 cm.
The length of the base of the steps scale can also be calculated from proportions or by the formula
a = (Shsr × KSh) / M
where: Shsr - average value of one step in centimeters,
KSh - number of steps at the base of the scale ,
M - scale denominator.

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

Rice. 6.4. Step scale.
Measured distance AC \u003d BC + AB \u003d 100 + 20 \u003d 120 sh.

6.3. SCALE ACCURACY

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

Scale accuracy allows you to solve two important problems:

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

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

6.4. DETERMINATION OF UNKNOWN MAP SCALE

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

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

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

According to the nomenclature of the map sheet. The notation system (nomenclature) of map sheets for each scale is quite definite, therefore, knowing the notation system, it is easy to find out the scale of the map.

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

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

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

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

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

According to the length of the arc of one minute of the meridian . Frames of topographic maps along the meridians and parallels have divisions in minutes of the meridian and parallel arcs.

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

6.5. TECHNIQUE FOR MEASURING AND PUTTING DISTANCES ON A MAP

To measure distances on a map, millimeter or scale bar, a compass-meter, and for measuring curved lines - a curvimeter.

6.5.1. Measuring distances with a millimeter ruler

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

6.5.2. Measuring distances with a compass

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

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

To get the length broken line sequentially measure the length of each of its links, and then summarize their values. Such lines are also measured by increasing the compass solution.
Example. To measure the length of a polyline ABCD(Fig. 6.6, a), the legs of the compass are first placed at points BUT and AT. Then, rotating the compass around the point AT. move the back leg from the point BUT exactly AT" lying on the continuation of the line sun.
Front leg from point AT transferred to a point FROM. The result is a solution of the compass B "C"=AB+sun. Moving the back leg of the compass in the same way from the point AT" exactly FROM", and the front of FROM in D. get a solution of the compass
C "D \u003d B" C + CD, the length of which is determined using a transverse or linear scale.

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

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

6.5.3. Measuring distances with a curvimeter

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

Rice. 6.7. Curvimeter mechanical

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

Rice. 6.8. Curvimeter digital (electronic)

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

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

Rice. 6.9. Slant Range ( S) and horizontal spacing ( d)

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

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

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

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

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

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

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

6.6. MEASUREMENT OF AREA BY MAP

The determination of the areas of plots from topographic maps is based on the geometric relationship between the area of the figure and its linear elements. The area scale 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 with a scale of 1:10,000 (in 1 cm 100 m), the area scale will be (1: 10,000) 2, or in 1 cm 2 there will be 100 m × 100 m = 10,000 m 2 or 1 ha, and on a map of scale 1 : 1,000,000 in 1 cm 2 - 100 km 2.

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

6.6.1. Measuring the area of a parcel with straight boundaries

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

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

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

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

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

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

Rice. 6.11. Square Mesh Palette

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

P \u003d a 2 n,

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

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

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

Rice. 6.12. dot palette

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

P = h∑l

Figure 6.13. Palette consisting of a system
parallel lines

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

Rice. 6.14. polar planimeter

The planimeter is used to determine areas mechanically. The polar planimeter is widely used (Fig. 6.14). It consists of two levers - pole and bypass. Determining the contour area with a planimeter comes down to the following steps. After fixing the pole and setting the needle of the bypass lever at the starting point of the circuit, a reading is taken. Then the bypass spire is carefully guided along the contour to the starting point and a second reading is taken. The difference in readings will give the area of the contour in divisions of the planimeter. Knowing the absolute value of the division of the planimeter, determine the area of the contour.
The development of technology contributes to the creation of new devices that increase labor productivity in calculating areas, in particular, the use of modern 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 a site of any configuration, i.e. with any number of vertices whose coordinates (x, y) are known. In this case, the numbering of the vertices should be done in a clockwise direction.
As can be seen from fig. 6.16, the area S of the polygon 1-2-3-4 can be considered as the difference between the areas S "of the figure 1y-1-2-3-3y and S" of the figure 1y-1-4-3-3y
S = S" - S".

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

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

In this way,
2S= (x 1 + x 2) (y 2 - y 1) + (x 2 + x 3) (y 3 - y 2) - (x 1 + x 4) (y 4 - y 1) - (x 4 + x 3) (y 3 - y 4). Expanding the brackets, we get
2S \u003d x 1 y 2 - x 1 y 4 + x 2 y 3 - x 2 y 1 + x 3 y 4 - x 3 y 2 + x 4 y 1 - x 4 y 3

From here
2S = x 1 (y 2 - y 4) + x 2 (y 3 - y 1) + x 3 (y 4 - y 2) + x 4 (y 1 - y 3) (6.1)
2S \u003d y 1 (x 4 - x 2) + y 2 (x 1 - x 3) + y 3 (x 2 - x 4) + y 4 (x 3 - x 1) (6.2)

Let us represent expressions (6.1) and (6.2) in general view, denoting by i the ordinal number (i = 1, 2, ..., n) of the vertices of the polygon:
(6.3)
(6.4)
Therefore, twice the area of the polygon is equal to either the sum of the products of each abscissa and the difference between the ordinates of the next and previous vertices of the polygon, or the sum of the products of each ordinate and the difference of the abscissas of the previous and subsequent vertices of the polygon.
An intermediate control of calculations is the satisfaction of the following 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 The plot area calculation can be easily solved using MicrosoftXL spreadsheets. An example for a polygon (polygon) of 5 points is given in tables 6.4, 6.5.
In table 6.4 we enter the initial data and formulas.

Table 6.4.

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

Double area in m2

SUM(D2:D6)

Area in hectares

In table 6.5 we see the results of the calculations.

Table 6.5.

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






	Double area in m2
	Area in hectares

6.7. EYE MEASUREMENTS ON THE MAP

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

Video

Scaling tasks

Tasks and questions for self-control

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

Going on an interesting journey or simply looking at maps on the Internet, each person is faced with such a concept as scale. However, not everyone knows what it is, what types of scales are and how to calculate it correctly.

What is scale

The word "scale" came into Russian from the language of accuracy - German - and literally translates as a stick for measuring. However, in cartography this term denotes the number, how many times this map or other image is reduced in comparison with the original. The scale is present on every map, and is also an integral part of any drawing.

What is the scale for?

So why do people need scale in practice? What does scale show? In fact, this concept is connected practically and theoretically with many branches: mathematics, architecture, modeling and, of course, cartography. After all, on no map, even an ultra-modern digital one, it is impossible to display geographical feature in its actual size. Therefore, when drawing the image of certain cities, rivers, mountains, or even entire continents on a map, all these objects are proportionally reduced. And how many times this is done, and is the scale, which is indicated on the margins of the map.

In the old days, when cartography did not yet use a scale, but reduced the depicted objects at its discretion, the resulting maps were very inaccurate and were rather approximate. So travelers using them often got into a mess. Who knows, perhaps the map used by Christopher Columbus also had the wrong scale, and therefore, instead of India, he sailed to America?

Another industry that simply cannot exist without the use of scale is modeling. Indeed, when creating a drawing of a future building or aircraft, an engineer does it on a certain scale, reducing or enlarging the image, depending on the need. So not a single, even the smallest detail, can be made without the use of a drawing, and no drawing can do without a scale.

Main types of scales

Despite the simplicity of the concept of "scale", there are several types of it. On maps, it is usually indicated either with numbers (numerical) or graphically. Graphic scales are divided into two subspecies: linear view scale and cross.

There are also scale subspecies, which are more related to map types. Depending on the size of the scales, maps are distinguished:

Large-scale - from one to two hundred thousand or less.
Medium-scale - from one to a million to one to two hundred thousand.
Small scale - up to one in a million.

Naturally, on small-scale maps, some details are not applied, while at the same time, large-scale maps may contain the names of streets and even small lanes. In modern electronic maps, the user can adjust the scale himself, turning the map from small-scale to large-scale in an instant, and vice versa.

Numerical and named scale

Scale data can be specified different ways. If on a map or drawing the scale is indicated using a fraction (1:200, 1:20,000, etc.), then this type of scale is called numerical. When calculating this size, it is worth taking into account the fact that the scale with the smaller number in the denominator will be larger. In other words, objects on a map with a scale of 1:200 will be larger than on a map with a scale of 1:20,000.

The named scale specifies not just the size of the image reduction, but also names the units of measurement with which this is done. For example, on the map of the area it is indicated that 1 centimeter on it is equal to 1 meter. The named scale is rarely used for small-scale maps, and indeed for maps in general. It is more practical for various drawings. Especially if it is a tiny detail or, on the contrary, a huge residential complex.

Graphic Scale

Graphic views of scales, as already mentioned above, come in two variants.

Linear is a scale depicted as a uniformly divided two-color ruler. As a rule, it is used on large-scale terrain plans and makes it possible to measure the distance on it using a paper strip or compass. This graphical scale option can help you determine the length of rivers, roads, and other curved lines.

Transverse is an improved version of the linear scale. Its purpose is to determine the distance indicated on the plan as accurately as possible. A similar graphical option is usually used on specialized maps.

Drawing scales

Having considered the most common types of scales in cartography, it is worth mentioning that this concept is also inextricably linked with drafting and architectural graphics. Whether they are engineering drawings of tiny mechanical parts or, on the contrary, drawings of huge architectural ensembles, in any case, specialized drawing scales are applied to them. Each drawing form has a column in which, in without fail the scale of the designed product is indicated.

Remarkable is the fact that even if the engineer creates a drawing of the part in life size, all the same, the scale of 1:1 is indicated in the information about it. Unlike maps, in the drawings the scale can be not only reduced (1:5), but also enlarged (5:1) if the depicted product is tiny in size.

To date, only narrow specialists need the ability to correctly calculate the scale without the help of machines. Thanks to modern programs and devices, other people no longer need to be well versed in the scale of a particular map - the computer will do everything for them. But still, everyone should have at least an approximate idea of \u200b\u200bwhat the scale shows, how to calculate it correctly and what types of it exist - after all, this is a component of elementary literacy and human culture.

Scales are numerical, named and graphic.

Numerical scale expressed as an aliquot fraction, the numerator of which equal to one, and the denominator is a round number showing how many times the terrain lines are reduced. The numerical scale is written as a ratio. For example, a numerical scale of 1:1000 shows that one centimeter of the plan contains 1000 centimeters, or 10 meters on the ground.

Named Scale on the maps it is written below the numerical one and indicates how many meters on the ground are contained in one centimeter of the plan. For example, "There are 100 meters in 1 centimeter", which corresponds to a numerical scale of 1: 10,000.

Scales are small and large. The larger the scale denominator, the smaller the scale and vice versa. But the concept of large and small scale is relative. So for the purposes of inventory in urban areas, the scale is 1: 500 large, and 1: 2,000 small. For the inventory of rural settlements, the scale is 1: 2,000 large, and 1: 5,000 small. To display the land of the economy, the scale is 1: 25,000 large, and 1: 50,000 small. The main scales for topographic plans are 1:5000, 1:2000, 1:1000.

The scale for this plan is a constant value. Knowing the numerical scale, one can easily convert the lengths of lines on the ground to the lengths of lines on the plan and vice versa.

Example 1

The length of the line on the ground is 247.56 m. Determine the size of the segment on the plan on a scale of 1: 5,000. For a given scale, one is 5,000 centimeters or 50 meters on the ground. The length of the line on the plan will be equal to 247.56: 50 = 4.95 cm.

Example 2

On a plan drawn up on a scale of 1: 10,000, a segment of 3.15 cm is measured. Determine the length of the line on the ground. For a given scale, the horizontal distance is

3.15 100 = 315 m

To avoid such calculations, use nomograms (graphic scales), which are linear and transverse.

Linear the scale is a horizontal line on which equal segments (usually 1 cm) are plotted, called the base of the scale. The length of the base of the scale is chosen so that it contains a round number of meters (Figure 1).

Example 3

Let it be required to draw a line 285.3 m long on a scale of 1: 10,000. To do this, take two bases from zero division to the right with a compass (Figure 2), (which corresponds to 200 m), and then the left needle of the meter is retracted 8.5 small divisions to the left ( which corresponds to 85 m).

Figure 1 - Linear scale

Figure 2 - Compasses

Using a linear scale, we are forced to evaluate its smallest division by eye, which reduces the accuracy of measurements and is unacceptable when solving engineering geodetic problems on a map. Therefore, to draw up plans, maps and solve problems on the map, they use a transverse scale.

Cross scale is a nomogram printed on a metal plate 10 to 22 cm long (Figure 3).

Figure 3 - Transverse scale with a base of 2 cm

The leftmost base at the top and bottom is divided into 5 or 10 equal parts (α) and the points of this division are connected by inclined lines called transversals. The beginning of the first segment at the bottom is connected to the beginning of the second segment at the top, and so on. The vertical segments are divided into 10 parts (β) and horizontal lines are drawn. The price of the smallest division "a - b" of the ruler depends on the length of the base "A - B" and the number of horizontal and vertical segments.

av = AB / α ∙ β

The measurement of the lengths of lines on the plan according to the transverse scale is carried out using a measuring compass.

To the left of the LPM-1 ruler (the ruler of the transverse scale - 1) it is engraved for which scale it is preferable to use this nomogram, for example: - 1: 5 ... (Figure 3). This means that it is more convenient to determine the length of lines on a scale of 1: 500, 1: 5,000 and other multiples of "5".

Example 4

On the plan, on a scale of 1: 500, a segment taken into the solution of a measuring compass is measured (Figure 3). It is required to determine its length (horizontal distance) on the ground. At this scale 1 corresponds to the centimeter on the plan 500 5 meters. At the base of the line 2 5 10 10 times less, that is 1 10 times less, that is 0.1 meters. There are two full bases (20 m), seven horizontal segments (7 m) and eight vertical segments (0.8 m) in the measuring solution. The straight line segment, in the measuring solution is 27.8 m.

Example 5

On the plan, on a scale of 1: 5,000, a segment taken into the solution of a measuring compass is measured (Figure 3). It is required to determine its length (horizontal distance) on the ground. At this scale 1 corresponds to the centimeter on the plan 5 000 centimeters on the ground or 50 meters. At the base of the line 2 centimeters and corresponds to each 50 meters on the ground. So one base of the ruler is 100 meters on the ground. Then one horizontal segment (horizontal step) on the left base will be in 10 times less, that is 10 10 times less, that is 1 meter. There are two full bases (200 m), seven horizontal segments (70 m) and eight vertical segments (8 m) in the measuring solution. The segment is straight, in the solution of the meter is 278 m.

Example 6

On the plan, on a scale of 1: 1,000, a segment taken into the solution of a measuring compass is measured (Figure 4). It is required to determine its length (horizontal distance) on the ground. At this scale 1 corresponds to the centimeter on the plan 1 000 centimeters on the ground or 10 meters. At the base of the line 1 10 meters on the ground. So one base of the ruler is 10 meters on the ground. Then one horizontal segment (horizontal step) on the left base will be in 10 times less, that is 1 meter. The price of the smallest division (step along the transversal in the vertical direction), respectively, is still in 10 times less, that is 0.1 meters. There are three full bases (30 m), five horizontal segments (5 m) and seven vertical segments (0.7 m) in the measuring solution. The straight line segment, in the measuring solution is 35.7 m.

Figure 4 - Transverse scale with a base of 1 cm

Example 7

On a map on a scale of 1: 10,000, a segment taken into the solution of a measuring compass is measured (Figure 5). It is required to determine its length (horizontal distance) on the ground. At this scale 1 corresponds to the centimeter on the plan 10 000 centimeters on the ground or 100 meters. At the base of the line 1 centimeter and corresponds to it 100 meters on the ground. So one base of the ruler is 100 meters on the ground. Then one horizontal segment (horizontal step) on the left base will be in 10 times less, that is 10 meters. The price of the smallest division (step along the transversal in the vertical direction), respectively, is still in 10 times less, that is 1 meter. There are three full bases (300 m), five horizontal segments (50 m) and seven vertical segments (7 m) in the meter solution. The segment is straight, in the solution of the meter is 357 m.

Figure 5 - Transverse scale with a base of 4 cm

Example 8

On a map on a scale of 1: 25,000, a segment taken into the solution of a measuring compass is measured (Figure 4). It is required to determine its length (horizontal distance) on the ground. At this scale 1 corresponds to the centimeter on the plan 25 000 centimeters on the ground or 250 meters. At the base of the line 4 centimeters and corresponds to each 250 meters on the ground. So one base of the ruler is 1000 meters on the ground. Then one horizontal segment (horizontal step) on the left base will be in 10 times less, that is 100 meters. The price of the smallest division (step along the transversal in the vertical direction), respectively, is still in 10 times less, that is 10 meter. The meter has one full base (1,000 m), four horizontal segments (400 m) and five vertical segments (50 m). The segment is straight, in the solution of the meter is 1450 m.

On the LPM-1 ruler, a nomogram is also engraved for scales that are multiples of two - 1: 2 ... (for example: 1: 2,000) with a base 5 centimeters. Any of the nomograms given here can be used to determine distances according to plans and maps made on any other scale.

Example 9

On the plan, on a scale of 1: 1,000, a segment taken into the solution of a measuring compass is measured (Figure 3). It is required to determine its length (horizontal distance) on the ground according to the nomogram with a multiplicity of 1: 5 ... On a given scale 1 corresponds to the centimeter on the plan 1 000 centimeters on the ground or 10 meters. At the base of the line 2 centimeters and corresponds to each 10 meters on the ground. So one base of the ruler is 20 meters on the ground. Then one horizontal segment (horizontal step) on the left base will be in 10 times less, that is 2 meters. The price of the smallest division (step along the transversal in the vertical direction), respectively, is still in 10 times less, that is 0.2 meters. There are two full bases (40 m), seven horizontal segments (14 m) and eight vertical segments (1.6 m) in the measuring solution. The straight line segment, in the measuring solution is 55.6 m.

Example 10

On a map on a scale of 1: 25,000, a segment taken into the solution of a measuring compass is measured (Figure 4). It is required to determine its length (horizontal distance) on the ground according to the nomogram with a multiplicity of 1: 1 ... On a given scale 1 corresponds to the centimeter on the plan 25 000 centimeters on the ground or 250 meters. At the base of the line 1 centimeter and corresponds to it 250 meters on the ground. So one base of the ruler is 250 meters on the ground. Then one horizontal segment (horizontal step) on the left base will be in 10 times less, that is 25 meters. The price of the smallest division (step along the transversal in the vertical direction), respectively, is still in 10 times less, that is 2.5 meters. There are three full bases (750 m), five horizontal segments (125 m) and seven vertical segments (17.5 m) in the measuring solution. The segment is straight, in the solution of the meter is 892.5 m.

Scale- this is the ratio of the length of the segment on the map, plan or drawing to the corresponding real length on the ground.
The scale shows how many times each line. plotted on the map is reduced in relation to its actual size on the ground.
Reducing the image is a necessity, we rarely think about it, however, we also rarely depict life-size objects. As a rule, in order for them to fit on a sheet of paper, they have to be reduced, less often they have to be increased. This is especially true for the image of the earth's surface, because it is absolutely impossible to depict it one-on-one.
Does any reduced image have a scale? Of course not. The scale is not applicable to the drawing, even if the drawing is of very high quality. In any case, the artist will introduce distortions into the depicted object, and from the definition of the scale, we see that each (!) Line of our image is reduced in relation to the real object equally. Therefore, drawing to scale can be done at least if there is measuring instruments(at least the rulers). As a maximum - with the use of computer technology.

How is scale recorded?

Scale is a ratio. The ratio involves the process of division, which means that the scale is a mathematical fraction in which there is a numerator and a denominator. In the numerator of the fraction, the length of the segment in the image is recorded, and in the denominator, the length of the actual segment being displayed.

Suppose the image is made (although this is impossible for a map) on a one-to-one scale - the length of the displayed segment coincides with the length of the depicted one.
Scale is written as 1:1
If the image is reduced by 3 times, then the scale will be written as 1:3
A reduction of 100,000 times is written as 1:100,000

What does it mean?

If the scale is 1 to 1, then 1 centimeter of our image corresponds to 1 real centimeter of the depicted surface, and if 1:100,000, then 1 centimeter of the image corresponds to 100,000 centimeters. And one meter of the image? 1 meter would then correspond to 100,000 meters. Note that whatever the selected length on the map, the actual length will be larger - in our case, 100,000 times. If the scale is 1:1000 - then a thousand; 1:30,000,000 - thirty million.

Translation

When we say that one centimeter of the map corresponds to thirty million centimeters, no one will understand anything. So, you need to translate this astronomical number into something understandable. We know that there are 100 centimeters in 1 meter. So you can convert centimeters to meters. Divide 30,000,000 centimeters by 100 to get 300,000 meters. Also not very convenient, so you need to translate further. Remember that there are 1000 meters in 1 kilometer. We divide 300,000 meters by 1000. It turned out 300 kilometers. This means that one centimeter of a map at a scale of 1:30,000,000 contains 300 kilometers, and this can already be imagined.
There are simple and reliable way Converting centimeters to kilometers - in the end we divided the number by 100,000 (first by 100 and then by 1000), so you can just mentally close the 5 zeros and translate much faster, but you need to remember that this is only suitable for converting centimeters to kilometers and only if there are enough zeros. For a scale of 1:50,000, it will be enough for us to stop at meters.

Scale types

The scale that is written as a fraction through the sign ":" is called numerical. Numerical Scale Examples: 1:1000 1:1000,000 1:250,000
Regularly, in order not to have to constantly translate the numerical scale on maps (especially school ones), indicate named scale. It shows how much distance is contained in 1 centimeter of the map and is recorded: in 1 cm 1 m; in 1 cm 10 km; in 1 cm 2.5 km, respectively.
Sometimes a linear scale in the form of a measuring ruler is also added under the map. This is convenient, because if it is available, you can use a compass or ruler to measure the distance on the map, apply it to a linear scale and get a result corresponding to the real distance.

Types of maps by scale

Key distinctive feature maps from the figure is the presence of scale. A map without scale is not a map. All cartographic works are usually classified according to the scale in which they are made.
- Small-scale (maps of the world or continents - their scale is smaller than 1: 1000 000)
– Medium-scale (maps of countries, large islands – from 1:100,000 to 1:1,000,000)
- Large-scale (maps of small states, regions, cities - less than 1: 100,000)
Remember: the larger the scale, the less will fit on the map. The fact is that the scale is a fraction, and the smaller the denominator of the fraction, the larger it is.

Not a single geographical object, such as a river, a bridge, a village, can be depicted on a topographic plan in full size. In ancient times, people drew miniature images of the terrain, on which different areas were reduced arbitrarily, in varying degrees. Therefore, ancient drawings of the area do not make it possible to understand, for example, what is the distance between the banks of the river, what is the length of the river, etc. To be more accurate, it is necessary to reduce all distances by the same number of times while maintaining all proportions, to make an image on a scale.

Shows how many times the distances on the plan are reduced in relation to the real distances.

The length of the school on the plan of the school yard is 1000 times less than in reality. This means that on this plan all distances are reduced by 1000 times.

Numerical and named scales

Scale is written differently. In the form of a number, the scale is displayed as follows: 1:100 (this means that 1 cm of the plan replaces 100 cm on the ground). This is a numerical scale. 100 cm is 1 m, so you can simply write: in 1 cm - 1 m. A scale written in this form is a named scale.

Linear scale

Usually on the plans, in addition to the numerical and named scales, a linear scale is placed. It is a line divided into equal segments. The segments to the right of 0 show what distance on the ground correspond to distances on the plan of 1 cm, 2 cm, etc. The segment to the left of 0 is divided into equal small parts. Knowing the distance on the ground, which corresponds to a large segment, and the number of small segments, it is possible to calculate what distance on the ground corresponds to each small segment. For example, the length of the large segment to the left of 0 in the figure is 10m. This segment is divided into 5 small parts, which means that the length of one such part is 10 m: 5 = 2 m.

The linear scale allows you to measure distances on a plan using a measuring compass or a strip of paper.

Using a linear scale, you can determine the length of curved lines, such as rivers, roads. To do this, mark a small distance on a strip of paper or set a small solution between the needles of the measuring compass and rearrange the marked paper or compass along the measured line, counting the number of permutations. Using a linear scale to determine the length of one "step" in meters and multiplying it by the number of permutations, we get the length of the curved line.

Scale selection

The scale is chosen depending on the magnitude of the distances. For example, you need to depict a distance of 6 km. Then the scale of 1 cm - 10 m is not suitable, because this distance is represented by a line of 600 cm, that is, 6 m; but a line of 6 m cannot be placed on regular sheet paper. It is more convenient to take a scale: 1 cm - 1 km. At this scale, a distance of 6 km would correspond to a line of 6 cm.