INTRODUCTION
The topographic map is reduced a generalized image of the area, showing the elements using a system of conventional signs.
In accordance with the requirements, topographic maps are highly geometric accuracy and geographic fit. This is provided by their scale, geodetic base, cartographic projections and a system of symbols.
The geometric properties of a 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 almost constant. At small angles of inclination physical surface(on the plain) the length of the horizontal projection of the line differs very little from the length of the oblique line. In these cases, the length scale can be considered as the ratio of the length of the line on the map to the length of the corresponding line on the ground.
The scale is indicated on the maps in different options
6.1.1. Numerical scale
Numerical scale expressed as a fraction with a numerator equal to 1(aliquot fraction).
Or
Denominator M the numerical scale shows the degree of reduction in the lengths of the lines on the map (plan) in relation to the lengths of the corresponding lines on the ground. Comparing numerical scales, the largest is the one whose denominator is smaller.
Using the numerical scale of the map (plan), you can determine the horizontal distance dm lines on the ground
Example.
Map scale 1:50 000. The length of the segment on the map lk\u003d 4.0 cm. Determine the horizontal location of the line on the ground.
Solution.
Multiplying the value of the segment on the map in centimeters by the denominator of the numerical scale, we get the horizontal distance in centimeters.
d\u003d 4.0 cm × 50,000 \u003d 200,000 cm, or 2,000 m, or 2 km.
note to the fact that the numerical scale is an abstract quantity that does not have specific units of measurement. If the numerator of a fraction is expressed in centimeters, then the denominator will have the same units of measurement, i.e. centimeters.
For example, a scale of 1:25,000 means that 1 centimeter of the map corresponds to 25,000 centimeters of terrain, or 1 inch of the map corresponds to 25,000 inches of terrain.
To meet the needs of the economy, science and defense of the country, maps of various scales are needed. For government topographic maps, forest management plans, forestry plans and forest plantations, standard scales are defined - scale range(Tables 6.1, 6.2).
Scale series of topographic maps
Table 6.1.
| Numerical scale |
Map name |
1 cm card corresponds |
1 cm2 card corresponds |
|---|---|---|---|
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 Scalenamed 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.
To avoid mathematical calculations and speed up work on the map, use graphic scales . There are two such scales: linear and transverse .
Linear scaleTo build a linear scale, choose an initial segment that is convenient for a given scale. This original segment ( a) are called scale base (Fig. 6.1).
Rice. 6.1. Linear scale. Measured segment on the ground
will be CD = ED + CE = 1000 m + 200 m = 1200 m.
The base is laid on a straight line the required number of times, the leftmost base is divided into parts (segment b), to be the smallest divisions of the linear scale
. The distance on the ground that corresponds to the smallest division of the linear scale is called linear scale accuracy
.
How to use a linear scale:
- put the right leg of the compass on one of the divisions to the right of zero, and the left leg on the left base;
- the length of the line consists of two counts: a count of whole bases and a count of divisions of the left base (Fig. 6.1).
- If the segment on the map is longer than the constructed linear scale, then it is measured in parts.
Cross scale
For more accurate measurements, use transverse scale (Fig. 6.2, b).
Fig 6.2. Cross scale. Measured distance
PK =
TK +
PS +
ST =
1
00 +10
+ 7
=
117
m.
To build it on a straight line segment, several scale bases are laid ( a). Usually the length of the base is 2 cm or 1 cm. Perpendiculars to the line are set at the points obtained. AB and 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 top base and so on. Get a series of parallel inclined lines, which are called transversals.
The smallest division of the transverse scale is equal to the segment C 1
D 1
,
(fig. 6. 2, a). The adjacent parallel segment differs by this length when moving up the transversal 0C and 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 scaleSometimes 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 scaleUse 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 the step scale for the example above, you need horizontal line divide into segments equal to 1.8 cm, and divide the left base into 5 or 10 equal parts.
Rice. 6.4. Step scale.
Measured distance AC \u003d BC + AB \u003d 100 + 20 \u003d 120 sh.
6.3. SCALE ACCURACY
Scale Accuracy
(maximum scale accuracy) is a segment of the horizontal line, corresponding to 0.1 mm on the plan. The value of 0.1 mm for determining the accuracy of the scale is adopted due to the fact that this is the minimum segment that a person can distinguish with the naked eye.
For example, for a scale of 1:10,000, the scale accuracy will be 1 m. In this scale, 1 cm on the plan corresponds to 10,000 cm (100 m) on the ground, 1 mm - 1,000 cm (10 m), 0.1 mm - 100 cm (1m). From the above example, it follows that if the denominator of the numerical scale is divided by 10,000, then we get the maximum scale accuracy in meters.
For example, for a numerical scale of 1:5,000, the maximum scale accuracy will be 5,000 / 10,000 =
0.5 m
Scale accuracy allows you to solve two important tasks:
- definition minimum dimensions objects and objects of the area that are depicted on a given scale, and the sizes of objects that cannot be depicted on 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, a millimeter or scale ruler, a compass-meter is used, and a curvimeter is used to measure curved lines.
6.5.1. Measuring distances with a millimeter ruler
Use a millimeter ruler to measure the distance between given points on the map with an accuracy of 0.1 cm. Multiply the resulting number of centimeters by the value of the named scale. For flat terrain, the result will correspond to the distance on the ground in meters or kilometers.
Example. On a map of scale 1: 50,000 (in 1 cm - 500 m) the distance between two points is 3.4 cm.
Determine the distance between these points.
Solution. Named scale: in 1 cm 500 m. The distance on the ground between the points will be 3.4 × 500 = 1700 m.
At angles of inclination of the earth's surface more than 10º, it is necessary to introduce an appropriate correction (see below).
6.5.2. Measuring distances with a compass
When measuring distance in a straight line, the needles of the compass are set at the end points, then, without changing the solution of the compass, the distance is read off on a linear or transverse scale. In the case when the opening of the compass exceeds the length of the linear or transverse scale, the integer number of kilometers is determined by the squares of the coordinate grid, and the remainder - by the usual scale order.
Rice. 6.5. Measuring distances with a compass-meter on a linear scale.
To get the length broken line
sequentially measure the length of each of its links, and then summarize their values. Such lines are also measured by increasing the compass solution.
Example. To measure the length of a polyline ABCD(Fig. 6.6, a), the legs of the compass are first placed at points BUT and AT. Then, rotating the compass around the point AT. move the back leg from the point BUT exactly AT" lying on the continuation of the line sun.
Front leg from point AT transferred to a point FROM. The result is a solution of the compass B "C"=AB+sun. Moving the back leg of the compass in the same way from the point AT" exactly FROM", and the front of FROM in D. get a solution of the compass
C "D \u003d B" C + CD, the length of which is determined using a transverse or linear scale.
Rice. 6.6. Line length measurement: a - broken line ABCD; b - curve A 1 B 1 C 1;
B"C" - auxiliary points
Long curves measured along the chords with compass steps (see Fig. 6.6, b). The step of the compass, equal to an integer number of hundreds or tens of meters, is set using a transverse or linear scale. When rearranging the legs of the compass along the measured line in the directions shown in fig. 6.6, b arrows, count the steps. The total length of the line A 1 C 1 is 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)
Actual distance on 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 absolute value amendments, it is necessary:
a) in the table, by the angle of inclination, find the relative value of the correction (if the angle of inclination of the topographic surface is not given by an integer number of degrees, then the relative value of the correction must be found by interpolation between the tabular values);
b) calculate the absolute value of the correction to the length of the horizontal span (i.e., multiply this length by the relative value of the correction and divide the resulting product by 100).
3. To determine the length of a line on a topographic surface, the calculated absolute value of the correction must be added to the length of the horizontal distance.
Example. On the topographic map, the length of the horizontal laying is 1735 m, the angle of inclination of the topographic surface is 7°15′. In the table, the relative values of the corrections are given for whole degrees. Therefore, for 7°15" it is necessary to determine the nearest larger and nearest smaller multiples of one degree - 8º and 7º:
for 8° relative correction value 0.98%;
for 7° 0.75%;
difference in tabular values in 1º (60') 0.23%;
difference between given angle the slope of the earth's surface 7 ° 15 "and the nearest smaller tabular value of 7º is 15".
We make proportions and find the relative amount of the correction for 15 ":
For 60' the correction is 0.23%;
For 15′ the correction is x%
x% = = 0.0575 ≈ 0.06%
Relative value corrections for tilt angle 7°15"
0,75%+0,06% = 0,81%
Then you need to determine the absolute value of the correction:
= 14.05 m approximately 14 m.
The length of the inclined line on the topographic surface will be:
1735 m + 14 m = 1749 m.
At small angles of inclination (less than 4° - 5°), the difference in the length of the inclined line and its horizontal projection is very small and may not be taken into account.
6.6. MEASUREMENT OF AREA BY MAP
The determination of the areas of plots from topographic maps is based on the geometric relationship between the area of the figure and its linear elements. 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 total area object.
6.6.2. Measuring the area of a plot with a curved contour
An object with a curvilinear contour is divided into geometric shapes, having previously straightened the boundaries in such a way that the sum of the cut-off sections and the sum of the excesses mutually compensate each other (Fig. 6.10). The measurement results will be approximate to some extent.
Rice. 6.10. Straightening curvilinear site boundaries and
breakdown of its area into simple geometric shapes
6.6.3. Measurement of the area of a plot with a complex configuration
Measurement of plot areas, having a complex irregular configuration, more often produced using 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:
Where: a - the side of the square, expressed on the scale of the map;
n- the number of squares that fall within the contour of the measured area
To improve accuracy, the area is determined several times with an arbitrary permutation of the palette used in any position, including rotation relative to its original position. The arithmetic mean of the measurement results is taken as the final value of the area.
In addition to grid palettes, dot and parallel palettes are used, which are transparent plates with engraved dots or lines. Points are placed in one of the corners of the cells of the grid palette with a known division value, then the grid lines are removed (Fig. 6.12).
Rice. 6.12. dot palette
Weight of each point equal to the price dividing the palette. The area of the measured area is determined by counting the number of points inside the contour, and multiplying this number by the weight of the point.
Equidistant parallel lines are engraved on the parallel palette (Fig. 6.13). The measured area, when applied to it with a palette, will be divided into a series of trapezoids with the same height h. Segments of parallel lines inside the contour (in the middle between the lines) are the middle lines of the trapezoid. To determine the area of a plot using this palette, it is necessary to multiply the sum of all measured middle lines by the distance between the parallel lines of the palette h(taking into account the scale).
P = h∑l
Figure 6.13. Palette consisting of a system
parallel lines
Measurement areas of significant plots made on cards with the help of planimeter.
Rice. 6.14. polar planimeter
The planimeter is used to determine areas mechanically. The polar planimeter is widely used (Fig. 6.14). It consists of two levers - pole and bypass. Determining the contour area with a planimeter comes down to the following steps. After fixing the pole and setting the needle of the bypass lever at the starting point of the circuit, a reading is taken. Then the bypass spire is carefully guided along the contour to the starting point and a second reading is taken. The difference in readings will give the area of the contour in divisions of the planimeter. Knowing the absolute value of the division of the planimeter, determine the area of the contour.
The development of technology contributes to the creation of new devices that increase labor productivity in calculating areas, in particular, the use of modern 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 serial number(i = 1, 2, ..., n) polygon vertices: 
(6.3)
(6.4)
Therefore, twice the area of the polygon is equal to either the sum of the products of each abscissa and the difference between the ordinates of the next and previous vertices of the polygon, or the sum of the products of each ordinate and the difference of the abscissas of the previous and subsequent vertices of the polygon.
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 lot area formulas can be easily solved using Microsoft XL spreadsheets. An example for a polygon (polygon) of 5 points is given in tables 6.4, 6.5.
In table 6.4 we enter the initial data and formulas.
Table 6.4.
y i (x i-1 - x i+1) |
||||
|---|---|---|---|---|
Double area in m2 |
SUM(D2:D6) |
|||
Area in hectares |
||||
In table 6.5 we see the results of the calculations.
Table 6.5.
y i (x i-1 -x i+1) |
||||
|---|---|---|---|---|
Double area in m2 |
||||
Area in hectares |
||||
6.7. EYE MEASUREMENTS ON THE MAP
In the practice of cartometric work, eye measurements are widely used, which give approximate results. However, the ability to visually determine distances, directions, areas, slope steepness and other characteristics of objects on a map contributes to mastering the skills correct understanding cartographic image. The accuracy of eye measurements increases with experience. Eye skills prevent gross miscalculations in instrument measurements.To determine the length of linear objects on the map, one should visually compare the size of these objects with segments of a kilometer grid or divisions of a linear scale.
To determine the areas of objects, squares of a kilometer grid are used as a kind of palette. Each square of the grid of maps of scales 1:10,000 - 1:50,000 on the ground corresponds to 1 km 2 (100 ha), scale 1:100,000 - 4 km 2, 1:200,000 - 16 km 2.
The accuracy of quantitative determinations on the map, with the development of the eye, is 10-15% of the measured value.
Video
Scaling tasksTasks 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? Previous
Measure the corresponding segment with a ruler. Preferably, it is made from the thinnest possible sheet material. In case the surface on which it is spread is not flat, a tailor's meter will help. And in the absence of a thin ruler, and if the card is not a pity to pierce, it is convenient to use a compass for measuring, preferably with two needles. Then it can be transferred to graph paper and measure the length of the segment on it.
Roads between two points are rarely straight. A convenient device - a curvimeter - will help you measure the length of the line. To use it, first rotate the roller to align the arrow with zero. If the curvimeter is electronic, it is not necessary to set it to zero manually - just press the reset button. While holding the roller, press it against the starting point of the line so that the notch on the body (it is located above the roller) points directly to this point. Then drive the roller along the line until the line is aligned with the end point. Read the statements. Please note that some curvimeters have two scales, one of which is graduated in centimeters and the other in inches.
Find the scale indicator on the map - it is usually located in the lower right corner. Sometimes this pointer is a segment of a calibrated length, next to which it is indicated what distance it corresponds to. Measure the length of this segment with a ruler. If it turns out, for example, that it has a length of 4 centimeters, and next to it it is indicated that it corresponds to 200 meters, divide the second number by the first, and you will find out that each on the map corresponds to 50 meters on the ground. On some, instead of a segment, there is a ready-made phrase, which may look, for example, as follows: "There are 150 meters in one centimeter." Also, the scale can be specified as a ratio the following kind: 1:100000. In this case, you can calculate that a centimeter on the map corresponds to 1000 meters on the ground, since 100,000/100 (centimeters in a meter) = 1000 m.
Multiply the distance measured with a ruler or curvimeter, expressed in centimeters, by the number indicated on the map or the calculated number of meters or in one centimeter. The result is a real distance, expressed, respectively, or kilometers.
Any map is a miniature image of some territory. The coefficient showing how much the image is reduced in relation to the real object is called the scale. Knowing it, one can determine distance on . For real existing maps on a paper basis, the scale is a fixed value. For virtual electronic cards this value changes along with the change in the magnification of the map image on the monitor screen.
Instruction
If yours is based, then find it, which is called a legend. Most often, it is in the marginal design. The legend must necessarily indicate the scale of the map, which will tell you, measured in distance according to this will be in reality, on . So, if the scale is 1:15000, then this means that 1 cm on map is equal to 150 meters on the ground. If the scale of the map is 1:200000, then 1 cm plotted on it is equal to 2 km in reality
That distance that interests you. Please note that if you want to determine how fast you will walk or drive from one house to another in or from one locality to another, then your route will consist of straight segments. You will not move in a straight line, but along a route that runs along streets and roads.
1.1 Map scales
map scale shows how many times the length of the line on the map is less than the corresponding length on the ground. It is expressed as a ratio of two numbers. For example, a scale of 1:50,000 means that all terrain lines are shown on the map with a reduction of 50,000 times, i.e. 1 cm on the map corresponds to 50,000 cm (or 500 m) on the ground.
Rice. 1. Registration of numerical and linear scales on topographic maps and city plans
The scale is indicated under the lower side of the map frame in numerical terms (numerical scale) and in the form of a straight line (linear scale), on the segments of which the corresponding distances on the ground are signed (Fig. 1). The scale value is also indicated here - the distance in meters (or kilometers) on the ground, corresponding to one centimeter on the map.
It is useful to remember the rule: if you cross out the last two zeros on the right side of the ratio, then the remaining number will show how many meters on the ground correspond to 1 cm on the map, that is, the scale value.
When comparing several scales, the larger one will be the one with the smaller number on the right side of the ratio. Let's assume that there are maps of 1:25000, 1:50000 and 1:100000 scales for the same area. Of these, the 1:25000 scale will be the largest, and the 1:100,000 scale will be the smallest.
The larger the scale of the map, the more detailed the terrain is shown on it. With a decrease in the scale of the map, the number of terrain details applied to it also decreases.
The detail of the image of the area on topographic maps depends on its nature: the less details the area contains, the more fully they are displayed on maps of smaller scales.
In our country and many other countries, the main scales of topographic maps are: 1:10000, 1:25000, 1:50000, 1:100000, 1:200000, 1:500000 and 1:1000000.
The cards used in the troops are divided into large scale, medium scale and small scale.
| map scale | Card name | Map classification | |
| scale | by main purpose | ||
| 1:10 000 (in 1 cm 100 m) | ten thousandth | large scale | tactical |
| 1:25 000 (in 1 cm 250 m) | twenty-five thousandth | ||
| 1:50 000 (in 1 cm 500 m) | five thousandth | ||
| 1:100,000 (in 1 cm 1 km) | hundred thousandth | medium scale | |
| 1:200,000 (in 1 cm 2 km) | two hundred thousandth | operational | |
| 1:500,000 (in 1 cm 5 km) | five hundred thousandth | small scale | |
| 1:1 000 000 (in 1 cm 10 km) | millionth | ||
1.2. Measurement on a map of straight and winding lines
To determine the distance between points of the terrain (objects, objects) on the map, using a numerical scale, it is necessary to measure the distance between these points in centimeters on the map and multiply the resulting number by the scale value.
For example, on a map with a scale of 1:25000, we measure the distance between the bridge and the windmill with a ruler (Fig. 2); it is equal to 7.3 cm, multiply 250 m by 7.3 and get the desired distance; it is equal to 1825 meters (250x7.3=1825).
Rice. 2. Determine the distance between points on the map using a ruler.
A small distance between two points in a straight line is easier to determine using a linear scale (Fig. 3). To do this, it is enough to apply a compass-meter, the solution of which is equal to the distance between given points on the map, to a linear scale and take a reading in meters or kilometers. On fig. 3 the measured distance is 1070 m.
Rice. 3. Measurement on a map of distances with a compass-meter on a linear scale
Rice. 4. Measurement on the map of distances with a compass-meter along winding lines
Large distances between points along straight lines are usually measured using a long ruler or measuring compass.
In the first case, a numerical scale is used to determine the distance on the map using a ruler (see Fig. 2).
In the second case, the “step” solution of the measuring compass is set so that it corresponds to an integer number of kilometers, and an integer number of “steps” is set aside on the segment measured on the map. The distance that does not fit into an integer number of “steps” of the measuring compass is determined using a linear scale and added to the resulting number of kilometers.
In the same way, distances are measured along winding lines (Fig. 4). In this case, the "step" of the measuring compass should be taken as 0.5 or 1 cm, depending on the length and degree of sinuosity of the measured line.
Rice. 5. Distance measurements with a curvimeter
To determine the length of the route on the map, a special device is used, called a curvimeter (Fig. 5), which is especially convenient for measuring winding and long lines.
The device has a wheel, which is connected by a gear system with an arrow.
When measuring distance with an odometer, you need to set its arrow to division 99. Holding the odometer in vertical position guide it along the measured line, without taking it off the map along the route so that the scale readings increase. Bringing to the end point, count the measured distance and multiply it by the denominator of the numerical scale. (In this example 34x25000=850000, or 8500 m)
1.3. The accuracy of measuring distances on the map. Distance corrections for slope and tortuosity of lines
Map Distance Accuracy depends on the scale of the map, the nature of the measured lines (straight, winding), the chosen measurement method, the terrain and other factors.
The most accurate way to determine the distance on the map is in a straight line.
When measuring distances using a measuring compass or a ruler with millimeter divisions, the average measurement error on flat terrain usually does not exceed 0.7-1 mm on the map scale, which is 17.5-25 m for a 1:25000 scale map, scale 1:50000 - 35-50 m, scale 1:100000 - 70-100 m.
In mountainous areas, with a large steepness of the slopes, errors will be greater. This is explained by the fact that when surveying the terrain, it is not the length of the lines on the surface of the Earth that is plotted on the map, but the length of the projections of these lines on the plane.
For example, With a slope slope of 20 ° (Fig. 6) and a distance on the ground of 2120 m, its projection on the plane (distance on the map) is 2000 m, i.e. 120 m less.
It has been calculated that at an inclination angle (slope slope) of 20°, the obtained result of measuring the distance on the map should be increased by 6% (add 6 m per 100 m), by 15% at an inclination angle of 30°, and by 23 at an angle of 40°. %.
Rice. 6. Projection of the slope length on a plane (map)
When determining the length of the route on the map, it should be borne in mind that the distances along the roads, measured on the map using a compass or curvimeter, in most cases are shorter than the actual distances.
This is explained not only by the presence of descents and ascents on the roads, but also by some generalization of the meanders of the roads on the maps.
Therefore, the result of measuring the length of the route obtained from the map should be multiplied by the coefficient indicated in the table, taking into account the nature of the terrain and the scale of the map.
1.4. The simplest ways to measure areas on a map
An approximate estimate of the size of the areas is made by eye on the squares of the kilometer grid available on the map. Each square of the grid of maps at scales 1:10000 - 1:50000 on the ground corresponds to 1 km2, a square of the grid of maps at a scale of 1 : 100000 - 4 km2, to the square of the grid of maps at a scale of 1:200000 - 16 km2.
Areas are measured more accurately palette, which is a sheet of transparent plastic with a grid of squares with a side of 10 mm applied to it (depending on the scale of the map and the required measurement accuracy).
Having superimposed such a palette on the measured object on the map, it first calculates the number of squares that completely fit inside the contour of the object, and then the number of squares intersected by the contour of the object. Each of the incomplete squares is taken as half a square. As a result of multiplying the area of one square by the sum of the squares, the area of \u200b\u200bthe object is obtained.
Using squares of scales 1:25000 and 1:50000, it is convenient to measure the areas of small areas with an officer's ruler, which has special cutouts. rectangular shape. The areas of these rectangles (in hectares) are indicated on the ruler for each hart scale.
2. Azimuths and directional angle. Magnetic declination, meridian convergence and direction correction
true azimuth(Ai) - horizontal angle measured clockwise from 0° to 360° between the north direction of the true meridian of a given point and the direction to the object (see Fig. 7).
Magnetic azimuth(Am) - horizontal angle measured clockwise from 0e to 360° between the north direction of the magnetic meridian of the given point and the direction to the object.
Directional angle(α; DN) - horizontal angle measured clockwise from 0° to 360° between the north direction of the vertical grid line of the given point and the direction to the object.
Magnetic declination(δ; Sk) - the angle between the northern direction of the true and magnetic meridians at a given point.
If the magnetic needle deviates from the true meridian to the east, then the declination is east (taken into account with the + sign), if the magnetic needle deviates to the west, it is western (taken into account with the - sign).
Rice. 7. Angles, directions and their relationship on the map
convergence of meridians(γ; Sat) - the angle between the northern direction of the true meridian and the vertical line of the coordinate grid at a given point. When the grid line deviates to the east, the approach of the meridian is east (taken into account with the + sign), when the grid line deviates to the west, it is western (taken into account with the - sign).
Direction correction(PN) - the angle between the northern direction of the vertical grid line and the direction of the magnetic meridian. It is equal to the algebraic difference between the magnetic declination and the approach of the meridians:
3. Measurement and construction of directional angles on the map. Transition from directional angle to magnetic azimuth and vice versa
On the ground using a compass (compass) measure magnetic azimuths directions, from which they then move to directional angles.
On the map on the contrary, they measure directional angles and from them they pass to the magnetic azimuths of directions on the ground.
Rice. 8. Changing the directional angles on the map with a protractor
Directional angles on the map are measured with a protractor or a chordogonometer.
Measurement of directional angles with a protractor is carried out in the following sequence:
- the landmark on which the directional angle is measured is connected by a straight line to the standing point so that this straight line is greater than the radius of the protractor and intersects at least one vertical line of the coordinate grid;
- combine the center of the protractor with the intersection point, as shown in Fig. 8 and count the value of the directional angle along the protractor. In our example, the directional angle from point A to point B is 274° (Fig. 8, a), and from point A to point C - 65° (Fig. 8, b).
In practice, it often becomes necessary to determine the magnetic AM from a known directional angle ά, or, conversely, the angle ά to a known magnetic azimuth.
Transition from directional angle to magnetic azimuth and vice versa
The transition from the directional angle to the magnetic azimuth and back is performed when it is necessary to find the direction on the ground using a compass (compass), the directional angle of which is measured on the map, or vice versa, when it is necessary to plot the direction on the map, the magnetic azimuth of which is measured, on the terrain with compass.
To solve this problem, it is necessary to know the magnitude of the deviation of the magnetic meridian of a given point from the vertical kilometer line. This value is called the directional correction (PN).
Rice. 10. Determination of the correction for the transition from the directional angle to the magnetic azimuth and vice versa
The direction correction and its constituent angles - the convergence of the meridians and the magnetic declination - are indicated on the map under the south side of the frame in the form of a diagram that looks like the one shown in fig. 9.
convergence of meridians(g) - the angle between the true meridian of the point and the vertical kilometer line depends on the distance of this point from the axial meridian of the zone and can have a value from 0 to ±3°. The diagram shows the average convergence of meridians for a given sheet of the map.
Magnetic declination(d) - the angle between the true and magnetic meridians is indicated on the diagram for the year of surveying (updating) the map. The text placed next to the diagram provides information about the direction and magnitude of the annual change in magnetic declination.
To avoid errors in determining the magnitude and sign of the direction correction, the following method is recommended.
Draw an arbitrary direction OM from the top of the corners in the diagram (Fig. 10) and designate the directional angle ά and the magnetic azimuth Am of this direction with arcs. Then it will immediately be seen what the magnitude and sign of the direction correction are.
If, for example, ά = 97°12", then Am = 97°12" - (2°10"+10°15") = 84°47 " .
4. Preparation on the data map for movement in azimuths
Movement in azimuths- this is the main way of orienting in terrain poor in landmarks, especially at night and with limited visibility.
Its essence lies in maintaining on the ground the directions given by magnetic azimuths, and the distances determined on the map between the turning points of the intended route. The directions of movement are maintained with the help of a compass, distances are measured in steps or on a speedometer.
The initial data for movement in azimuths (magnetic azimuths and distances) are determined on the map, and the time of movement is determined according to the standard and drawn up in the form of a diagram (Fig. 11) or entered in a table (Table 1). Data in this form is issued to commanders who do not have topographic maps. If the commander has his own work map, then he draws up the initial data for movement in azimuths directly on the work map.
Rice. 11. Scheme for movement in azimuth
The route of movement in azimuths is chosen taking into account the terrain, its protective and camouflage properties, so that it provides a quick and covert exit to the specified point in a combat situation.
The route usually includes roads, clearings and other linear landmarks that make it easier to maintain the direction of movement. Turning points are chosen from landmarks that are easily identifiable on the ground (for example, tower-type buildings, road intersections, bridges, overpasses, geodetic points, etc.).
It has been experimentally established that the distances between landmarks at the turning points of the route should not exceed 1 km when driving during the day on foot, and when driving by car - 6–10 km.
For movement at night, landmarks are marked along the route more often.
In order to provide a secret exit to the specified point, the route is planned along hollows, vegetation massifs and other objects that provide movement masking. It is necessary to avoid movement on the ridges of hills and open areas.
The distances between the landmarks chosen on the route at the turning points are measured along straight lines using a measuring compass and a linear scale, or perhaps more precisely, with a ruler with millimeter divisions. If the route is planned along a hilly (mountainous) area, then a relief correction is introduced into the distances measured on the map.
Table 1
5. Compliance with regulations
| no. | Name of the standard | Conditions (order) for fulfilling the standard | Category of trainees | Time estimate | ||
| "excellent" | "hor." | "ud." | ||||
| 1 | Determining the direction (azimuth) on the ground | A direction azimuth (landmark) is given. Indicate the direction corresponding to the given azimuth on the ground, or determine the azimuth to the specified landmark. The time to fulfill the standard is counted from the setting of the task to the report on the direction (azimuth value). Compliance with the standard is assessed |
Serviceman | 40 s | 45 s | 55 s |
| 5 | Preparing data for moving along azimuths | On the M 1:50000 map, two points are indicated at a distance of at least 4 km. Study the terrain on the map, outline the route of movement, select at least three intermediate landmarks, determine the directional angles and the distances between them. Draw up a scheme (table) of data for movement along azimuths (translate directional angles into magnetic azimuths, and distances into pairs of steps). Errors that reduce the rating to "unsatisfactory":
The time to fulfill the standard is counted from the moment the card is issued to the presentation of the scheme (table). |
officers | 8 min | 9 min | 11 min |
When creating topographic maps, the linear dimensions of all terrain objects projected onto a level surface are reduced by a certain number of times. The degree of such reduction is called the scale of the map. The scale of the map can be expressed in numerical form (numerical scale) or in graphical form (linear, transverse scales), in the form of a graph.
Distances on a map are usually measured using a numerical or linear scale. More accurate measurements are made using a transverse scale.
On the scale of the linear scale, the segments corresponding to the distances on the ground in meters or kilometers are digitized. This makes it easier to measure distances as no calculations are required.
Determination of distances and areas on the map. Measurement of distances.
When using a numerical scale, the distance measured on the map in centimeters is multiplied by the denominator of the numerical scale in meters.
For example, the distance from the GGS point elev. 174.3 (square 3909) to the fork in the road (square 4314) on the map is 13.96 cm, on the ground it will be: 13.96 x 500 = 6980 m. (map scale 1: 50,000 U-34-85 -BUT).
If the distance measured on the ground must be plotted on the map, then it must be divided by the denominator of the numerical scale. For example, the distance measured on the ground is 1550 m, on a map at a scale of 1: 50,000 it will be 3.1 cm.
Measurements on a linear scale are performed using a measuring compass. With a compass solution, two contour points on the map are connected, between which it is necessary to determine the distance, then applied to a linear scale and the distance on the ground is obtained. Curvilinear sections are determined in parts or using a curvimeter.
Determination of areas.
The area of a piece of terrain is determined from the map most often by counting the squares of the coordinate grid covering this area. The size of the shares of squares is determined by eye or using a special palette. Each square formed by the lines of the coordinate grid corresponds to: 1: 25,000 and 1: 50,000 - 1 km.sq., 1: 100,000 - 4 km.sq., 1: 200,000 - 16 km.sq.It is useful to remember that the following 2 x 2 mm ratios are appropriate for scales:
1: 25,000 - 0.25 ha = 0.0025 km2
1: 50,000 - 1 ha = 0.01 km2
1: 100,000 - 4 ha = 0.04 km2
1: 200,000 - 16 ha = 0.16 km2
The determination of the areas of individual plots is carried out during the alienation land plots for the Department of Defense.
The accuracy of determining distances on the map. Correction for route length.
The accuracy of measuring lines, areas on a topographic map. Buy truck tractors and trucks at the most the best prices, you can visit auto-holland.ru. All trucks have passed pre-sale preparation and inspection control (instrumental, computer and visual).
The accuracy of measuring lines and areas primarily depends on the scale of the map. The larger the scale of the map, the more accurately the lengths of lines and areas are determined from it. At the same time, the accuracy depends not only on the accuracy of measurements, but also on the error of the map itself, which is inevitable when it is compiled and printed. Errors can reach 0.5 mm for flat areas, and up to 0.7 mm in mountains. The source of measurement errors is also the deformation of the map and the measurements themselves.
Absolutely with the same error, flat rectangular coordinates are determined from topographic maps of the above scales.
Distance correction for line slope.
For example, the distance between two points, measured on the map, on a terrain with an inclination angle of 12 degrees is 9270 m. The actual distance between these points will be 9270 x 1.02 = 9455 m. Thus, when measuring distances on the map, it is necessary to introduce corrections for the slope lines (relief).
Long-range straight-line distances in one six-degree zone can be calculated using the formula:
This method of determining the distance is used mainly in the preparation of artillery firing and when launching missiles at ground targets.
A topographic map is a two-dimensional map that depicts a three-dimensional area, while the height of the earth's surface is indicated using contour lines. As in the case of any other map, the distance between two points on a topographic map is measured along a straight line connecting them, as if a bird flies between these points. This is done first, and only then the surface topography and other terrain features that may affect the overall length of the route are taken into account. Learn how to measure distance along a straight line.
Steps
Measuring distance on a linear scale
- Take a strip of paper long enough to cover the distance between the points you are interested in. Please note that this method better suited for measuring relatively short linear distances.
- Press a strip of paper against the map and try to mark the location of two points on it as accurately as possible.
-
Attach a strip of paper to a linear scale. Find a linear scale on a topographic map - as a rule, it is located in the lower left corner of the map. Attach a strip of paper with two marks to it to determine the distance between them. Use this method to measure small distances that fit on a linear scale.
Determine b about most of the distance on the main scale. Attach a strip of paper to the scale so that the right mark coincides with the whole number on the scale. In this case, the left label should be on an additional scale.
- The point of the main scale, in which the right mark will appear, is determined by the condition that the left mark must fall on the additional scale. In this case, it is necessary to combine the right label with an integer on the main scale.
- The integer corresponding to the right label on the main scale indicates that the measured distance is at least so many meters or kilometers. The rest of the distance can be more accurately determined by the additional scale.
-
Go to the additional scale, on which the base of the scale is divided into parts. Determine the length of the smaller part of the distance on the additional scale. The left mark will match the integer on the secondary scale - this number should be divided by ten and added to the distance determined on the main scale.
Measuring distance on a numerical scale
-
Mark the distance on a strip of paper. Place a strip of paper with a straight edge on the map and align that edge with the points you want to measure. Mark "Point A" and "Point B" on paper.
- Press the strip of paper against the card and do not bend it to get the most accurate results possible.
- If desired, you can use a ruler or measuring tape instead of paper. In this case, write down the measured distance between the dots in millimeters.
-
Measure the distance with a ruler. Attach a ruler or measuring tape to the paper and determine the distance between the two marks. Use this method to measure large distances that are outside the linear scale, or if you want to calculate the distance as accurately as possible.
- Try to determine the distance to the nearest millimeter.
- Find the scale at the bottom of the map. Here the ratio of lengths should be given, as well as a segment (linear scale) with centimeters plotted on it. As a rule, for convenience, the scale is chosen in whole numbers, for example, 1 centimeter = 1 kilometer.
-
Calculate the distance along a straight line. To do this, use the distance measured on the map in millimeters and the numerical scale, which is the ratio of lengths. Multiply the measured distance by the scale denominator.
-
Attach a strip of paper to the map and mark the points on it. Lay a strip of paper with a straight edge over the card. Align this edge simultaneously with the first (“Point A”) and the second (“Point B”) points that you want to measure the distance between, and mark on paper the location of these points.
