The planimeter


Exhibit: table of content

Mathematics Math21a, Fall 2005
Multivariable Calculus
Oliver Knill, SciCtr 434, knill@math.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, Math21a, Multivariable Calculus, Fall 2005, Department of Mathematics, Faculty of Art and Sciences, Harvard University