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# Planimeter

[pluh-nim-i-ter]

A planimeter is a measuring instrument used to measure the area of an arbitrary two-dimensional shape. The most common use is to measure the area of a plane shape.

There are many different kinds of planimeters but all operate in a similar way. A pointer on the planimeter is used to trace around the boundary of the shape. This induces a movement in another part of the instrument and a reading of this is used to establish the area of the shape. The precise way in which they are constructed varies, the main types of mechanical planimeter being polar; linear; and Prytz or "hatchet" planimeters.

In the linear and polar planimeter, as one point on a linkage is traced along the shape's perimeter, that linkage rolls a wheel along the drawing. The area of the shape is proportional to the number of turns through which the measuring wheel rotates when the planimeter is traced along the complete perimeter of the shape. The concept having been pioneered by Hermann in 1814, Swiss mathematician Jakob Amsler-Laffon built the first modern planimeter in 1854, the operation of which can be justified by appealing to Green's theorem. Applying:

$oint_\left\{C\right\}M,dx + N,dy = int_\left\{S\right\}left\left(frac\left\{partial N\right\}\left\{partial x\right\}-frac\left\{partial M\right\}\left\{partial y\right\}right\right),dx,dy$

to:

$oint_\left\{C\right\}x,dy - y,dx$

(rewritten as:

$oint_\left\{C\right\}- y,dx + x,dy$

for easier identification of corresponding terms) yields:

$int_\left\{S\right\}left\left(frac\left\{partial left\left[xright\right]\right\}\left\{partial x\right\}-frac\left\{partial left\left[-yright\right]\right\}\left\{partial y\right\}right\right),dx,dy = int_\left\{S\right\}2,dA$

The right hand side of this equation is proportional to the area enclosed by the contour. The left hand side is equal to

$oint_\left\{C\right\}- y,dx + x,dy = oint_\left\{C\right\} \left(-y, x\right)cdot\left(dx, dy\right)$

The integrand has the form of a dot product, which means that it is the integral of the projection of (dx, dy) onto (-y, x). The vector (-y, x) is orthogonal to (x, y) because $\left(x, y\right) cdot \left(-y, x\right) = \left(x\right)\left(-y\right) + \left(y\right)\left(x\right) = 0$.

The planimeter contains a measuring wheel that rolls along the drawing as the operator traces the contour. When the planimeter's measuring wheel moves perpendicular to its axis, it rolls, and this movement is recorded. When the measuring wheel moves parallel to its axis, the wheel skids without rolling, so this movement is ignored. That means the planimeter measures the distance that its measuring wheel travels, projected perpendicularly to the measuring wheel's axis of rotation.

In a planimeter, the linkage determines the orientation of the measuring wheel, as a function of its position on the drawing. It is therefore somewhat intuitive that by counting the number of turns through which the measuring wheel rotates, we can evaluate that line integral. If that is true, then the turns count is proportional to the area inside the contour.

The first planimeter was invented by Hermann in 1814 and there were many subsequent developments before Amsler's famous planimeter. Electronic versions also exist.

There was a book written in 1903 on the planimeter, The Polar Planimeter by J. Y. Wheatley. See also Chapter 8 of How Round is your Circle by C. J. Sangwin and J. Bryant, 2007.

Developments of the planimeter can establish the position of the Centre of mass or the second moment.