Parametrization by Arclength

Math 3335

Helmut  Knaust - Department of Mathematical Sciences, University of Texas at El Paso, hknaust@utep.edu
10/18/16; edits: 1/24/20

The plot below shows the curve f(t) = (sin t, cos 3t), t∈ [0,2π]. The dots are spaced at constant increments of time.

f[t_]={Sin[t],Cos[3t]};n=50;
p1=ParametricPlot[f[t],{t,0,2π},PlotStyle->Blue,Epilog->Table[{Blue,AbsolutePointSize[8],Point[f[t]]},{t,0,2π,2π/n}],ImageSize->600]

We are now going to parametrize by arclength. The plot on the right shows the dots at constant increments of time and length.

L=NIntegrate[Norm[f'[x]],{x,0,2π}];
Clear[a];
a[0]=0;
a[k_]:=a[k]=FixedPoint[Function[t,t+(L/n-NIntegrate[Norm[f'[x]],{x,a[k-1],t}])2π/L],a[k-1]+2π/n,20]
list=Table[a[k],{k,0,n}];
p2=ParametricPlot[f[t],{t,0,2π},PlotStyle->Blue,Epilog->Table[{Red,AbsolutePointSize[8],Point[f[t]]},{t,list}]];
Show[p2,ImageSize->600]