RU2511205C2 - Method of determining area - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: invention can be used to determine the area of flat figures, e.g. in physics, thermodynamics, pulse engineering, mapping, determining scaled areas of land surfaces etc. Said technical result is achieved due to that, in the method of calculating areas of compound contours by determining the volume of liquid displaced by a body, the body is made in form of a plate of constant thickness, the carrier of the area to be determined is mounted thereon, which is cut on outer and optional inner contours, followed by dividing the displaced liquid volume by the thickness of the plate. The material of the plate used has plasticity and liquid unwettability properties.

EFFECT: easy determination of area.

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Description

The invention relates to methods for determining the areas of complex contours and can be used for these purposes in physics, thermodynamics, pulsed technology, cartography, determining scaled sections of land surfaces, etc.

There are various methods of measuring the area of flat figures based on mathematical methods, for example, numerical and graphic integration, dividing the figures into parts (squares, rectangles, circles, etc.), the areas of which are determined by simple formulas and arithmetic operations [1].

For example, with the graphical method, the measured area is divided into thin lines into geometric shapes, usually triangles (less often into other shapes), in which their elements — the base and height — are determined graphically from the plan taking into account the scale (it is desirable that the resulting triangles are close to equilateral) [1, p. 73]. Then, using the geometry formulas, the area of a flat figure is calculated. To increase accuracy, as well as to control, the area of each figure is determined a second time, but by other measured elements. The total area of the figure (section) is obtained by summing all the geometric figures [1, p. 73-74].

The disadvantages of these methods are the cumbersomeness and low accuracy of calculating areas, especially with strongly rugged contours and internal residue of surfaces, and in some cases the impossibility of using these traditional methods, for example, for curved surfaces.

There are also mechanical methods for calculating the area of plane figures. For example, using polar and roller planimeters [1, p. 74-88; 2] used to determine the areas of flat figures of irregular shape and to find the numerical values of integrals of a certain kind. The determination of the area is carried out manually by tracing the outline of the figure with a pin connected with a calculating-decisive mechanism.

At present, electronic-mechanical methods for calculating the areas of planar figures are used, implemented using two types of planimeters: polar compensation and linear (roller) and their modifications [3].

The disadvantages of these methods is that they are implemented using complex and expensive electronic devices and mechanical devices and can only be used to calculate the area of plane figures. In addition, these methods for determining the area are quite inaccurate.

The closest technical solution to the method selected as a prototype is the Archimedes hydrostatic method based on measuring the volume of bodies with the volume of liquid displaced by immersed bodies of arbitrary shape. This method Archimedes (according to legend) applied in solving the problem of the king of Syracuse Hieron [4, 5].

The disadvantage of this method is its functional limitation, since it is used only to find volumes of small bodies.

An object of the invention is to expand the functionality of the method of Archimedes.

The essence of the invention is to determine the volumes of bodies of arbitrary shape in order to use it to calculate surface areas limited by arbitrary (incorrect) external and possible internal closed contours applied on thin media (on paper, films, etc.).

The technical result is achieved by the fact that in the method for calculating the areas of complex contours by determining the volume of fluid displaced by the body, the body is made in the form of a plate of constant thickness, the carrier of the defined surface is fixed on it, cut off along the outer and possible inner contours, and then dividing the displaced volume of liquid into the thickness of the plate, while using a plate material with the properties of plasticity and non-wetting with a liquid.

The method is as follows. The calculation of the surface areas of flat and curved surfaces is presented in the drawing. The method is implemented using a thin and elastic carrier of the measured surface 1, a plastic plate 2, a prism 3, a vessel with a liquid 4 and a scale 5.

An example of calculating surface area.

The determination of the surface area (flat or curved) can be divided into stages (see drawing):

- a transparent flexible material is applied to the surface to be measured (for example, an mylar film), the external and possible internal contours of the surface are applied to it, obtaining a surface carrier 1;

- fix the carrier surface 1 on a plastic plate 2 of constant thickness h;

- cut along the contours of the carrier 1 from the plate 2 a direct prism 3 of height h;

- completely immerse prism 3 in a vessel with liquid 4;

- on the scale of the vessel 5 calculate the volume ΔV displaced by the prism 3;

находят искомую площадь поверхности S.- according to the formula $$\frac{\Delta V}{h}=S$$

find the desired surface area S.

The described hydrostatic method for determining the areas of flat and curved surfaces is simple, inexpensive and fairly accurate. It can be widely used in solving various kinds of technical problems.

Information sources

1. Surveying: Textbook for universities. - M .: Academic Project; Gaudeamus, 2011 .-- 409 p. - (Gaudeamus: library of a surveyor and cartographer). S. 84-87.

2. Polytechnical Dictionary / Editorial Board: A.Yu. Ishlinsky (Ch. Ed.) And others. - M.: Big Russian Encyclopedia, 2000. - 656 p. - S. 383, fig. p. 385.

3. Kleine Enzyklopadie. Matematik. - Leipzig: VEB bibliographisches Institut, 1971.- 950 S. - Bildtafel 43: Mathematische Gerate I.

4. Danko P.E. Higher mathematics in exercises and tasks. At 2 hours 4.1. Textbook manual for universities / P.E. Danko et al. - Moscow: Izdat. Onyx 21 Century House: Peace and Education, 2003. - 304 p. - S.260-261 (Theorems 1 and 2).

5. Kudryavtsev P.S. A course on the history of physics: Textbook. manual for students of ped. in-to physical specialist. - M .: Education, 1982. - 448 p. - S. 32.

Claims (1)

A method for calculating the areas of complex contours by determining the volume of liquid displaced by the body, characterized in that the body is made in the form of a plate of constant thickness, a carrier of a defined surface is fixed on it, cut off along the outer and possible internal contours, and then dividing the displaced volume of liquid into the plate thickness, they use plate material with the properties of plasticity and non-wetting with liquid.

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