planimeter

planimeter. A planimeter is a mechanical measuring instrument used to determine the area of an arbitrary two-dimensional shape by tracing its perimeter. It is a classic tool of analog computers, enabling the integration of area under a curve or within an irregular boundary without complex mathematical calculation. Historically vital in fields like cartography, surveying, and engineering, it operates on elegant geometric and integral calculus principles, translating a path into a measurable rotation.

History

The invention of the planimeter is attributed to the Swiss mathematician Jakob Amsler-Laffon in 1854, whose polar design became the most widely used model. Earlier conceptual work, however, was done by Johann Martin Hermann and others exploring the mechanical integration of area. The device saw rapid adoption by the Prussian and other European land registries for property assessment, and later by the United States Coast and Geodetic Survey for calculating drainage basins from topographic maps. Its development paralleled that of other mechanical integrators like the integraph and was a precursor to more complex analog computers used for solving differential equations in fields like naval architecture and ballistics.

Principle of operation

The core principle relies on the geometric theorem known as Green's theorem, which relates a line integral around a closed curve to a double integral over the enclosed area. In mechanical terms, as the tracer point is guided around the shape's boundary, the measuring wheel partly rolls and partly slides on the drawing surface. This combined motion causes the wheel to rotate by an amount proportional to the area swept out by the instrument's linkage. The final rotation, read from a vernier scale or mechanical counter, is directly converted into area units using a known scale factor, effectively performing a mechanical integration.

Types of planimeters

The most common historical type is the **polar planimeter**, where one arm is anchored at a fixed pole on the drawing board. The **linear planimeter** or **roller planimeter** moves within straight guides, suitable for long, narrow areas. For measuring areas from closed curves plotted as graphs, the **hatchet planimeter** or **Amsler planimeter** uses a sharp point instead of a wheel. High-precision versions like the **coordinate planimeter** or those built by firms like Coradi in Zurich were used in scientific laboratories. Specialized types include the **platometer** for map work and the **integraph**, which directly draws the integral of a plotted function.

Mathematical basis

The operation is a direct physical implementation of the planimeter equation, derived from Green's theorem in the plane. The mathematics models the linkage as a system of vectors; the area is computed from the path of the tracer and the constrained motion of the joints. The key result, known as the **Shoelace formula** for polygonal approximation, is related conceptually, as the planimeter effectively performs a continuous version of this calculation. The underlying theory involves properties of the determinant and the concept of the **moment of a line segment**, connecting geometry to integral calculus.

Applications

Planimeters were indispensable in cartography for measuring the area of watersheds, countries, or lakes from topographic maps produced by organizations like the Ordnance Survey. In surveying, they calculated land parcels for the Dominion Land Survey and property taxation. Engineers used them to find cross-sectional areas from blueprints for earthworks or the moment of inertia of ship hulls from lines plan drawings. In science, they analyzed indicator diagrams from steam engine trials to calculate work done, and in meteorology, to estimate the area covered by precipitation on weather charts. Their use declined with the advent of digital image processing and GIS software like ArcGIS.

Category:Measuring instruments Category:Analog computers Category:Cartography