 This Month's Feature Column
The Mathematics of Surveying: Part II. The Planimeter
The second of two columns on the mathematics of surveying (the first is here). . . .
Introduction
A planimeter is a table-top instrument for measuring areas, usually the areas of irregular regions on a map or photograph. They were once common, but have now largely been replaced by digital tools.
The following picture gives some idea of the setup. The pole arm rotates freely around the pole, which is fixed on the table. The tracer arm rotates around the pivot, which is where it joins the polar arm. You trace a curve in the clockwise direction with the tracer, and as you do so the measuring wheel rolls along, and the total distance it rolls is accumulated on the dial. The support wheel keeps the thing from flopping over. At the end, you read off a number from the dial, and after multiplication by a factor depending only on the particular configuration of the planimeter, you get the area inside the curve.
The next figure gives you a better view of the mechanism.
We call the carriage the assembly of wheels, dial, and pivot. In the next picture you get a better look at it, and can see the worm drive that causes the dial to rotate as the measuring wheel moves.
Here is the diagram of the original planimeter, from the article by Jakob Amsler that introduced it:
How can such a simple thing measure areas?
Bill Casselman
University of British Columbia, Vancouver, Canada
cass at math.ubc.ca
John Eggers
University of California, San Diego
jeggers at ucsd.edu
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