Planimeter

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Polar planimeters are the most common

A planimeter is a mathematical instrument and an analog computer , i.e. a mechanical measuring device , for determining any surface area in maps or drawings. You move the edge of the area to be measured with a driving pen or a magnifying glass with crosshairs or similar, with a measuring wheel integrating the area . The edge is a closed curve, the start and end points of the measurement run are the same. The sign of the measurement is determined by whether the edge is traversed clockwise or counterclockwise.

The surfaces to be measured can have almost any shape (irregular polygons , other irregular surfaces, contour lines or plots of land ), but must have continuous and piecewise smooth edges. The entire area must be within reach of the travel arm, i.e. H. have a diameter that is less than twice the length of the travel arm (usually 20–30 cm). The accuracy is typically approx. 1 ‰, with special devices also higher.

Construction principle of the polar planimeter with pole, polar arm, measuring roller on the driving arm and measuring magnifier
Amsler polar planimeter

The polar planimeter

There are several types of planimeters such as polar planimeters, disc and roller planimeters, of which the former is the most common.

The polar planimeter was invented in 1854 by the Schaffhausen engineer and entrepreneur Jakob Amsler-Laffon .

The device has a fixed pole that is placed near the surface to be measured. Then you drive along the edge of the area as precisely as possible, which is possible, for example, on maps or cadastral plans with inaccuracies of about 0.1 to 0.3 mm.

Working principle

The mechanical construction of planimeters is based on a principle developed by Leonhard Euler , which the well-known mathematician developed for calculating infinitesimal areas, and on Green's theorem .

Euler's principle states that an infinitesimal surface can be composed of an infinitesimal parallelogram plus an infinitesimal triangular surface . The triangular area in turn consists of a (finite) radius and an infinitesimal angle .

In the planimeter, the small angles are continuously added up (integrated) while the area is bypassed with the measuring pen or a measuring magnifier with crosshairs. It is crucial in this process that axial displacements of the roller (i.e. displacements in the direction of the axis of rotation) on the plane base ideally have no influence on the rotation of the roller ( perfect slip in the axial direction, so to speak ).

The surface area can be read off directly on a scale that is provided with a vernier or other optical device to increase the reading accuracy .

Amsler roller planimeter
Roller planimeter for measuring elongated surfaces

Adjustments and accuracy

The measure of the area read on the roller is in particular also proportional to the length of the travel arm. By adjusting the travel arm length, a planimeter can be adapted to different units of measurement (square centimeters, square inch, ...) or to different drawing scales. The inaccuracy in this setting is directly reflected in the end result. Some planimeters are provided with an adjustable length arm with a corresponding setting scale, some with a fixed length arm .

The accuracy of the area determination can be increased within certain limits by enlarging the area to be determined and after planimization dividing the result by the square of the linear enlargement factor.

application

The application of the method is only limited by the size of the planimeter. In the case of very large areas, however, the pole can be placed in the middle ("Pole inside" method), which means it can be almost 4 times as large as in the normal "Pole outside" mode .

Furthermore, planimeters were manufactured in significantly larger versions. For example, in leather production, planimeters about 1.5 meters in size were used to determine the area of ​​animal hides.

There are also special planimeters for determining the static moment , the moment of inertia or higher moments of any surface (so-called moment planimeters ). These were used, for example, in shipbuilding when designing steam engines and diesel engines in order to minimize the machine- related rolling movements of the ship.

Moment planimeters were also used in railway and road construction to calculate the additions and removals of soil material.

literature

  • Jakob Amsler: About the mechanical determination of the surface area, the static moments and the moments of inertia of flat figures, especially with a new planimeter . Quarterly journal of the Zurich Natural Research Society, Schaffhausen 1856
  • Alfred Amsler: About the area and volume of curves and surfaces generated by movement and about mechanical integration . Meier, Schaffhausen, 1880.
  • Robert Amsler, Theodor H. Erismann: Jakob Amsler-Laffon 1823–1912, Alfred Amsler 1857–1940. Pioneers of testing and precision . Association for economic history studies, miles 1993, ISBN 3-909059-04-X
  • Andreas Galle: Mathematical Instruments . Teubner, Leipzig 1912
  • Felix Klein : Elementary Mathematics from a Higher Viewpoint, Part II: Geometry. Lecture held in the summer semester of 1908. Elaborated by E. Hellinger . Teubner, Leipzig 1909, pp. 22–31
  • Hermann Pieper: The Oldenburg Planimeter . Herald and electoral staff, Lüneburg 1825 ( digitized version )
  • Henry Selby Hele-Shaw : The Theory of Continuous Calculating Machines . In: Philosophical Transactions of the Royal Society, Part II, Vol. 176 (1885), pp. 367-402
  • Simon Stampfer : About Caspar Wetli's new planimeter . In: Journal of the Austrian Engineering Association, Volume 11, No. 7, Vienna 1850
  • William Thomson (Lord Kelvin): Mechanical Integration of the General Linear Differential Equation of any Order with Variable Coefficients (Paper VI) . In: Proceedings of the Royal Society, 24: 271-275 (1876)
  • Friedrich Adolf Willers: Mathematical machines and instruments . Akademie-Verlag, Berlin 1951

Web links

Commons : Planimeter  - collection of images, videos and audio files
Wiktionary: Planimeter  - explanations of meanings, word origins, synonyms, translations

Individual evidence

  1. ↑ G. Coradi: Instructions for using the leather knife by G. Coradi . Zurich 6, approx. 1913, 2 pages, instruction manual.
  2. ↑ Jacob Amsler: Use of the integrator (moment planimeter) for calculating the application and removal when installing railways, streets and canals . Orell Füssli & Co., Zurich, 1875.