The planimeter


Exhibit: table of content

Mathematics Maths21a, Summer 2005
Multivariable Calculus
Oliver Knill, SciCtr 434, knill.harvard.edu
The planimter connects is fixed at (0,0) has an "elbow" at (a,b) and points to (x,y). Both segments of length 1. Given (x,y), we can find (a,b) as a function of (x,y). The planimeter vector field F(x,y) = (-(y-b(x,y),x-a(x,y)) has curl 1. The weel rotation of the planimeter when (x(t),y(t)) traces a closed curve is the line integral of the vector field F along the curve. The key observation is that the field F has curl(F)=1 as can be verified with a simple computation (see the four lines of Mathematica below). By Greens theorem, the total wheel rotation is the area of the enclosed region R.
Links: Mathematica Code:
 (* Mathematica verfies that the planimeter vector field has curl(F) = 1   *)
 (* Oliver Knill, Harvard University, 9/2000, final form 8/2005            *)
 
 s=Simplify[Solve[{ (x-a)^2+(y-b)^2==1, a^2+b^2==1 }, {a,b}]];
 aa[x_,y_]:=a /. s[[1,1]]; bb[x_,y_]:=b /. s[[1,2]];
 F[x_,y_]:={-(y-bb[x,y]),x-aa[x,y]};
 curlF=D[F[x,y][[2]],x] - D[F[x,y][[1]],y]; 
 Simplify[curlF==1]
 
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Oliver Knill, Maths21a, Multivariable Calculus, Summer 2005, Department of Mathematics, Faculty of Art and Sciences, Harvard University