Numerical scale

scale is a scale, which is expressed by a fraction. Its numerator equals 1 (=1) and its denominator shows how many times the horizontal distance line in the image area is reduced while representing a horizontal line on a plan or map.

Numerical scale – is an unnamed value. . It is written in the form: 1:1000, 1:2000, 1: 5000, etc. Numbers 1000, 2000 and 5000 are called the denominator of scale M there.

Numerical scale shows, that one unit of length of the line on the plan (map) contains the same number of units of length of line, located on the ground. For example, one unit of the length of the line on the plan of scale 1:5000 contains exactly 5000 units of the same length on the ground. It means, that one centimeter on the plan 1:5000 corresponds to 5000 centimeters or 50 meters on the ground.

In some cases, the linear scale can be used for solving certain tasks. A linear scale is a special graphical representation of numerical scale (see Fig. 1). The segment AB is a basis of the linear scale. Usually it equals 2 cm. It is transferred into the appropriate length on the ground and signed. The leftmost base scale is divided into 10 equal parts.


Example (Fig. 1): This linear scale is used for work on topographic plan of scale 1:2000. Its basis equals 2 cm which is equivalent to 40 meters on the ground for this scale. The smallest part of the basis equals 2 millimeters or 4 meters on the ground.

The segment CD is taken from the topographic plan of scale 1:2000. It consists of two bases and two smallest parts which, as a result, correspond to 2*40m+2*2m = 84 m. (Fig. 1)

More accurate graphical definition and construction of line lengths can be done using another graphics - on the transverse scale (see fig. 2).

The transverse scale

– scale is a graph for the most accurate measurement of distances on the topographic plan (map).

The base of scale AB equals 2 cm. The smallest segment of base is CD=0.1*AB=2mm. The smallest part of transverse scale’s base is cd=0.1*CD=0.1*AB=0.2 mm. This relation follows from the similarity of triangles BCD and Bcd (see fig. 2).

Thus, base of transverse scale will correspond to 40 m for the numerical scale 1:2000, the one-tenth of base will equal 4 m, and the hundredth of AB will equal 0.4 m.

Example: the segment AB (Fig. 2), taken from the plan of 1:2000 scale, corresponds to the length 137.6 m. It equals 3 bases of transverse scale (3x40 = 120), four bases and the small segment of base (4x4 = 16 m ) and one smallest segment of the base of scale equal 0.4*4 or 1.6 m.

Let’s consider the most important characteristic of the concept of "scale".

The precision of scale is the horizontal segment on the ground, which corresponds to segment 0.01 cm on the plan of this scale. This characteristic depends on resolving power of naked human eye. Resolving power allows to consider minimal distance, equal to 0.1 mm, on the topographic plan. This value will be equal to 0.1 mm*M on the terrain, where M is a denominator of scale.

Fig. 2

Transverse scale allows to measure the length of the line on the plan (map) scale of 1:2000 accurate to precision of this scale.

Example: 1 mm of 1:2000 plan contains 2000 mm of the terrain and 0.1 mm on the terrain contains, accordingly, 0,1xM (mm) = 0.1 x 2000 mm = 200 mm = 20 cm or 0.2 m.

The value of measurement of the line length on the plan should be rounded to the precision of scale. Example: The length of the line equals 58,37 meters. It is built in scale 1:2000. The length of the line is rounded up to 58,4 m, because precision of scale equals 0,2 meters. If line is built in scale 1:500, its length is rounded up to 58.35 (precision of scale 1:500 equals 0,05).


of topographical plans

To use topographical plans the conventional symbols adopted for scale should be studied. Conventional symbols are graphic symbols, showing the location of objects and phenomena, and their quantitative and qualitative characteristics. They are issued in the form of individual tables or tables on instructional plans. Symbols are divided into large-scale (outline), and out of scale.

Large-scale symbols are symbols, which illustrate items in the scale of plan. These items show large objects, such as arable lands, meadows, forests, seas, lakes, etc.

Out of scale symbols – are symbols, which illustrate items out the scale of plan. These symbols show small objects, which can’t be shown in the scale of plan due to their small size - width of roads, wells, springs, bridges, electric poles, power poles, etc. The magnitude of these symbols does not match to the true size of the illustrated objects.

Linear signs - cartographic symbols, which are used for illustrating linear objects. The length of these objects is expressed in the scale of plan, but their width exceeds their actual width, considerably.

Areal conventional symbols are cartographic conventional symbols, which are used for filling the area of objects, which are expressed in the scale of plan.

Out of scale linear symbols are cartographic symbols, which are used for illustrating linear objects, which area can’t be expressed in the scale of plan.

Out of scale areal symbols are cartographic symbols which are used for illustrating objects, which area can’t be expressed in the scale of plan.

Explanatory inscriptions are inscriptions, which explain species or genus of objects illustrated on a map, and their quantitative and qualitative characteristics.

Stipples of the map (plan) are elements of map (plan), which are created of lines, dots or dashes.

Background elements of the map (plan) are elements of the map (plan), which are filled with any background colour.

Tasks solved by
the topographical plan

The topographical plan can solve a number of tasks, including the determination of: rectangular coordinates of the point, the length of the line; the directional corner and the rhumb of the line, the mark of point, slope, slope gradient, etc. The procedure for solving these tasks is exemplified by the test plan of 1:2000 scale.

Determination of rectangular
coordinates of the points

The coordinate grid, which is forming squares with sides of 10 cm, is plotted on the topographical plan. The vertical lines of the grid are parallel to the x-axis and the horizontal lines are parallel to the y-axis. The cordinates of the vertices of the angles of the grid squares are inscribed.

Example: Record 79, 2 means that the abscissa of the grid lines X = 79, 2 km, and it is located 79200 m from the origin of the coordinate system. Record 66, 2 means that the ordinate of the grid lines Y = 66, 2 km, and it is located 66200 m from the origin of the coordinate system.

For quick searching of any point on the topographical plan the lower left angle of the appropriate corner of the grid is inscribed.

Example: By using the coordinate grid, a measuring instrument and a transverse scale, on the topographical plan one can define the rectangular coordinates of the point A (Fig. 4), which is placed in square 79.2 - 66.2. It must be remembered that the abscissa increase to the north, and ordinates - to the east.

First, abscissa X southern line of grid is recorded in meters, X (southern line of grid) = 79200.0 m. This is the abscissa of the lower (southern) line of the square, which contains point A. The distance Δh = a-A is determined in meters, rounded up to the precision of the scale. This distance is determined by using a measuring instrument and transverse scale. The resulting value Δh = 64.8 m is summed up with the abscissa of the lower (southern) line of the square (X (southern line of grid) = 79,200.0 m) and we get the abscissa of point A: X (A) = 79200.0 + 64.8 = 79264.8 m.


The ordinate of point A is defined similarly: the values of the ordinate Y of the western line of the grid = 66200.0 m is summed up with the length of segment Δ = B-A equal to 141.6 m, and we get Y(A)= 66200.0 + 141.6 = 66341.6 m.

Measuring the length of the lines

The distance between points A and B (Fig. 4) is measured by a measuring instrument. The length of the line AB is calculated in transverse scale and recorded with precision and scale.

Determination of
the directional angle and the rhumb of the line

The directional angle α is a horizontal angle, which is measured from the northern direction OX of the coordinate grid clockwise to the direction of the line.

Directional angle α of line AB (Fig. 4) can be measured with a protractor. Fig. 5 shows directional angles α1, α2, α3, and α4 of four lines М-1, М-2, М-3, М-4.

Fig. 5

The direction angle of a given direction α dir is called direct. The directional angle of opposite the direction αrev is called reverse (Fig. 6).


Communication between direct and reverse directional angles looks like


Rhumb R is an acute horizontal angle between the nearest direction of OX axis of the coordinate of the coordinate grid and the direction of the line. Rhumbs can have values from 0 to 90 and are accompanied by the name of the quarter, in which the line is placed. Rhumbs of four lines M-1, M-2, M-3, M-4 are shown in Fig. 7: NE:r1, SE:r2, SW:r3, and NW:r4. NE is the name of the rhumb, and r1 is the value of the rhumb.


The rhumb of a given direction r dir is called direct.The rhumb of the opposite direction r rev is called reverse. Direct and reverse rhumbs are equal and differ only in name (Fig. 8).

For example, if the direct rhumb is r dir= NE:35, then the reverse rhumb is r rev=SW:35.

Fig 8

The table of the transition from directional angles α to rhumbs r is given below.

Formulae for the transition from directional angles to rhumbs

Fig 8

Determination of the marks of points
and the slope gradient of the line on terrain

Height H of point on terrain is a distance in the direction of the vertical line to zero surface.

For example, N (A) equals line Aa is the height of the point A over zero surface PQ, H(b) equal line Bb is the height of point B over zero surface PQ (Fig. 9).

The mark of the point is called a numerical value of the height of the point. For example, H(A)=150 m, H(B)=149 m

On the topographical plan the relief is illustrated by inscriptions of the marks of individual characteristic points, conventional symbols (scour, break, etc.) and horizontal lines.

Contours are closed curves , connecting terrain points with the same marks. Contours are formed by the intersection of the surface area by horizontal planes, drawn through a given distance. This distance (h) is called the vertical interval.

The contour interval (d) is a distance between two adjacent contours (Fig. 9 - 11).

Fig 9

The mark of the point, lying between two adjacent contours, can be determined by their marks. For example: the mark of the first point B of the lower horizontal line H 1 equals 161 m (Fig. 10), the mark of the second point A on the top of the horizontal line H 2 equals 162 m (vertical interval h = 1 m), contour interval d = 16.8 m. The horizontal distance from the first point C is equal to c = 7.6 m (Fig. 10). Then (with precision to 0.1 m) mark H(c) of point C is calculated by the following formula


Fig 10


The slope gradient is an angle between the direction of the slope and a horizontal plane at a given point A.The slope of the line of terrain (u) is a tangent angle (ν) of the line of terrain (tangent of slope gradient) to horizontal line (Fig. 11).

Fig 11


The longer the angle of the bank, the steeper is the slope.

For our example, the slope line of terrain is equal to:



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