How does the planimeter work?

Oliver Knill, 12/9/2000, Back to the applet
[Geometric Setup] The planimeter is a mechanical device for measuring areas in the plane. It has the shape of a ruler with two legs. One leg of length 1 connects the fixed origin (0,0) to (a,b). A second leg of length 1 connects (a,b) with the end point (x,y). The point (x,y) determines (a,b) as the intersection of two unit circles centered at (0,0) and (x,y). The intersection is unique if the angle between the two planimeter legs is smaller than 180 degrees.
The measurement consists of dragging (x,y) along the boundary of the region R. The wheel at (x,y) measures the motion in the direction orthogonal to the leg. After completing the path along the boundary of the region R. the total wheel rotation indicates the area of the region. Why?
[The planimeter vector field Let F(x,y)=(P(x,y),Q(x,y)) be the Planimeter vector field. It is defined by attaching a unit vector orthogonal to the vector (x-a,y-b) at (x,y). The weel rotation is the line integral of F along the boundary of R. Green's theorem tells that this integral is the double integral of curl(F) over the region R. The planimeter vector field is explicitely given by F(x,y)=(P(x,y),Q(x,y))=(-(y-b(x,y)),(x-a(x,y))). Furthermore, curl(F)=Qx-Py is equal to 2+(-ax-by) which is 2 plus the curl of the vector field G(x,y)=(b(x,y),-a(x,y)). A direct verification shows that curl(G)=-1. The Planimeter lineintegral is therefore the area of the enclosed region.
Bibliography. 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:
  • Guido Ascoli. Vedute sintetiche sugli strumenti integratori (Italian).
    Rend. Sem. Mat. Fis. Milano, 18:36, 1947.
  • R.W. Gatterdam. The planimeter as an example of Green's theorem.
    American Mathematical Monthly, 88:701-704, 1981.
  • L.I. Lowell. Comments on the polar planimeter. American Mathematical Monthly, 61:467-469, 1954.
  • CPP Module at Duke by Oliver Knill and Dale Winter

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