Planimeter

A planimeter ( due to its shape in professional circles " Mogelkutsche " ) is a mathematical instrument and analog computer, so a mechanical gauge to determine any surface content in maps or drawings. You drive the edge of the surface to be measured with a tracer or a magnifying glass with crosshairs, etc. along with a measuring wheel integrates the surface. The edge is a closed curve starting and end point of the measurement run is the same. The sign of the measurement is determined by whether the edge in a clockwise or counterclockwise direction is traversed.

The areas to be measured can be almost any shape ( irregular polygons, other irregular surfaces, contours or land), but must have continuous and piecewise smooth edges. The entire surface should be in the range of the driving arm, i.e., have a diameter which is smaller than twice the Fahrarmlänge (usually 20-30 cm). The accuracy is also higher in typically about 1 ‰, with special equipment.

The polar planimeter

There are various types of such planimeters polar planimeter, disc and Rollenplanimeter, of which the former is the most prevalent.

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

The unit has a fixed terminal which is set in the vicinity of the surface to be measured. Then you drive to the edge of the area as accurately as possible from what is possible for example on maps or cadastral plans with inaccuracies of about 0.1 to 0.3 mm.

Principle of operation

The mechanical design of planimeters is based on a principle of Leonhard Euler, which the famous mathematician has developed for the calculation of infinitesimal surfaces, as well as on the set of Green.

Euler's principle states that an infinitesimal area of an infinitesimal parallelogram plus an infinitesimal triangular face is assembled. The triangular area in turn consists of a (finite) radius and an infinitesimal angle.

In planimeter the area with the measuring stick or a measuring magnifier with reticle are now during bypass the small angles summed consecutively (integrated). Is critical in this process that the axial displacements of the roller ( i.e. displacements in the direction of the rotation axis ) on the plane support, ideally, does not affect the rotation of the roller have (as it were perfect slip in the axial direction).

On a scale that is provided with a vernier or other optical device for increasing the accuracy of reading, the surface area can be read directly.

Adjustments and accuracy

The depth reading on the role of measure of the surface is particularly proportional to the length of the tracer arm. By adjusting the Fahrarmlänge a planimeter can therefore to different units of measurement ( square centimeter square inch, ...) or be adapted to different underwriting standards. The inaccuracy in this setting, but goes directly to the bottom line. Some planimeter are provided with a tracer arm of adjustable length with a corresponding adjustment scale, some with a tracer arm of fixed length.

The accuracy of determination of the area can be increased within certain limits, by increasing the surface to be determined and divided by the Planimetrierung the result by the square of the linear magnification factor.

Application

The application of the method is limited only by the size of the planimeter. For very large areas can, however, the pole in the middle set (procedure " pole inside" ), thus it can be almost four times as large as in the normal mode " pole outside."

There are also Spezialplanimeter for the determination of the static moment, the moment of inertia or higher moments of arbitrary surfaces ( so-called Momentenplanimeter ). These were used in the shipbuilding industry, for example, in the design of steam engines and diesel engines in order to minimize the engine -related rolling movements of the ship.