As a straight line rolls on a circle without slipping, a point on the line moves along a curve called the involute of the circle. The properties of this curve are often used in mechanical engineering to produce gear profiles. The other end of the straight line also traces an involute (the other curve), which is produced also when a string is being wound on a circle.
In[]:=
Dynamic[Manipulate[r=1;circ=ParametricPlot[r{Sin[x],Cos[x]},{x,0,2Pi},PlotStyle->Thick][[1]];i=rα;j=Pir;Px=rSin[α];Py=rCos[α];vP={Px,Py};Ex=Px-iCos[α];Ey=Py+iSin[α];vE={Ex,Ey};Fx=Px+(j-i)Cos[α];Fy=Py-(j-i)Sin[α];vF={Fx,Fy};EF=Line[{vE,vF}];iv={ParametricPlot[r{Sin[v],Cos[v]},{v,0,α},PlotStyle{Thick,Red}][[1]]};pP=Point[vP];pE=Point[vE];radius={Dashed,Line[{{0,0},vP}]};evE={ParametricPlot[r{(Sin[w]-wCos[w]),(Cos[w]+wSin[w])},{w,α,0},PlotStyle{Thick,RGBColor[0.2,0.5,0.2]}][[1]]};evF={ParametricPlot[r{rSin[w]+(j-w)Cos[w],rCos[w]-(j-w)Sin[w]},{w,α,0},PlotStyle{Thick,RGBColor[0.2,0.5,1]}][[1]]};vT={Ex-extSin[α],Ey-extCos[α]};vU={Ex+extSin[α],Ey+extCos[α]};TU={Dashed,Line[{vT,vU}]};Graphics[{TU,evE,If[evFs,evF,{}],EF,circ,iv,pP,pE,radius},ImageSize{450,400},AxesTrue,PlotRange{{-r,r+2.4},{-r-1,r+1}}],{{α,2,"roll line on circle"},0.01,Pi},{{ext,1,"length of tangent line"},0.1,2},{{evFs,False,"show other curve"},{True,False}},TrackedSymbolsManipulate]]
Out[]=
