The Planimeter and the Theorem of Green

Oliver Knill, Harvard University, Department of Mathematics
12/9/2000
The polar planimeter is a mechanical device for measuring areas of regions in the plane which are bounded by smooth boundaries. The measurement is based directly on Green's theorem in multi-variable calculus: the planimeter integrates a line integral of a vector field which has constant curl.

What is a planimeter ?

Planimeters are mechanical instruments which can measure the area of closed regions in the plane. Planimeters are used in medicine for example to measure the size of the cross-sections of tumors or organs, in biology to measure the area of leaves or wing sizes of insects, in agriculture to measure the area of forests, in ingeneering it is used to measure the size of profiles.

How does the planimeter work?

The polar planimeter has the shape of a ruler with two legs. One leg of length $r$ connects the origin $(0,0)$ to $(a,b)$. The second leg of length $r$ connects $(a,b)$ with $(x,y)$. During the measurement process, the planimeter is fixed at $(0,0)$ and the point $(x,y)$ is at any time located at the boundary of the region. If $(x,y)$ is given, then $(a,b)$ is determined as an intersection point of two circles of radius $r$ around $(0,0)$ and $(x,y)$. If we require that the angle between the two legs of the planimeter is smaller than $\pi$, then the intersection point is uniquely determined and we could write $(a,b)=(a(x,y),b(x,y))$ to emphasis that $a,b$ are functions of $x$ and $y$. The measurement consists of dragging the end point $(x,y)$ along the boundary of a region $R$. A wheel attached to the second arm measures the motion of $(x,y)$ in the direction orthogonal to the leg. After completing the path along the boundary $\gamma$ of the region $R$, the total wheel rotation indicates the area of the region $R$.

Why does the planimeter work?

Let $F(x,y)=(P(x,y),Q(x,y))$ be the vector field defined by attaching a unit vector orthogonal to at the arm $(x-a,y-b)$ at $(x,y)$. The weel rotation measures the line integral $\int_{\gamma} F(x,y) \; \cdot \; ds$ along the piecewise smooth path $\gamma$. We call the vector field $F$ the "planimeter Vector field". Let $R$ by the region enclosed by $\gamma$. By Green's theorem, this is equal to $\int \int_R {\rm curl}(F) \; dx dy$.

If $r$ is the length of the legs of the planimeter, the planimeter vector field is explicitely given by

\begin{displaymath}F(x,y)=(P(x,y),Q(x,y))=(-(y-b(x,y))/r,(x-a(x,y))/r) \; . \end{displaymath}

The curl ${\rm curl}(F)=Q_x-P_y$ of $F$ is equal to $2/r+(-a_x-b_y)/r$ which is $2/r$ plus the curl of the vector field $(b(x,y)/r,-a(x,y)/r)$. The later has the curl $-1/r$:
Proof. This is a calculation. The equations $(x-a)^2+(y-b)^2=r^2,a^2+b^2=r^2$ give $x^2-2ax+y^2-2by=0,a^2+b^2=r^2$ and from $x^2-2ax+y^2+2 \sqrt{r^2-a^2} y = 0$, $x^2-2by+y^2+2 \sqrt{r^2-b^2} x = 0$, we get after solving for $a,b$ $a_x+b_y=1/r+(2r^2 xy/R^2) (1/\sqrt{R^2(R^2-4r^2)}-1/\sqrt{R^2(R^2-4r^2)})=1/r$ where $R^2=x^2+y^2$.
Because ${\rm curl}(v)=1/r$, the planimeter line integral $\int_{\gamma} F(x,y) \; \cdot \; ds$ is $1/r$ times the area of the enclosed region. In units where $r=1$, the curl is $1$ and consequently, we measured the area of the region.

Trying it out in Class

A planimeter can be assembled for classroom demonstration with a few pieces. We took the bars from a desk folder hanger and joined them with a screw. At both ends we added a wheel, one which turns at the origin, the other, orthogonal to the bar at the other end. The device could be used at the blackboard. We gauged the planimeter using a square of $10x10 {\rm in}^2$ on the blackboard. A good demonstration is to measure then the area of a disc of radius 10 inches.

Literature

Green's theorem is the classic way to explain the planimeter. The explanation of the planimeter through Green's theorem seems have been given first by G. Ascoli in 1947 [1]. It is further discussed in classroom notes [4,2]. A web source is the page of Paul Kunkel [3], which contains an other explanation of the planimeter. The formulation given in the present document can be presented in class without consuming too much time because the calculation of the curl can be left to a computer algebra package: In Mathematica for example, we get the intersection points $(a,b)$ with "Solve" and the curl by differentiation.

Bibliography/Links

Planimeter exposition

1
Guido Ascoli.
Vedute sintetiche sugli strumenti integratori. (italian).
Rend. Sem. Mat. Fis. Milano, 18:36, 1947.

2
R.W. Gatterdam.
The planimeter as an example of green's theorem.
American Mathematical Monthly, 88:701-704, 1981.

3
P. Kunkel.
The planimeter. Web document (no more available) http://www.nas.com/ kunkel/planimeter/planimeter.htm, 1999.

4
L.I. Lowell.
Comments on the polar planimeter.
American Mathematical Monthly, 61:467-469, 1954.