Arc Length Reparameterization

Define Curve r(t) = (e^t)sin(t)(OverHat[i])+(e^t)cos(t)(OverHat[j])

r[t_]:={Exp[t]*Sin[t],Exp[t]*Cos[t]};

Plot Curve r(t) from t = 0 to t = 5:

ParaPlot = ParametricPlot[r[t],{t,0,5}, AxesLabel->{"x","y"},PlotLabel->"r(t), 0⩽t⩽5", PlotStyle->{Thick,Red}]

Re-parameterize r(t) in terms of arc length using t = t(s):

t[s_]:=Log[1+Sqrt[2]*s/2];

arcr[s_]:=r[t[s]];

Plot curve r(s) from s = 0 to s = 5.  Superimpose on original graph:

ParaArcPlot = ParametricPlot[arcr[s],{s,0,5}, PlotStyle->{Thick,Blue}];

Show[ParaPlot,ParaArcPlot, PlotLabel->"r(t), 0⩽t⩽5 and r(s), 0⩽s⩽5"]

Plot curve r(s) from s = 0 to s = 50.  Superimpose on original graph:

ParaArcPlot = ParametricPlot[arcr[s],{s,0,50}, PlotStyle->{Thick,Blue}];

Show[ParaPlot,ParaArcPlot, PlotLabel->"r(t), 0⩽t⩽5 and r(s), 0⩽s⩽50"]

Plot curve r(s) from s = 0 to s = 100.  Superimpose on original graph:

ParaArcPlot = ParametricPlot[arcr[s],{s,0,100}, PlotStyle->{Thick,Blue},PlotRange->All];

Show[ParaPlot,ParaArcPlot, PlotLabel->"r(t), 0⩽t⩽5 and r(s), 0⩽s⩽100"]

Plot curve r(s) from s = 0 to s = 150.  Superimpose on original graph:

ParaArcPlot = ParametricPlot[arcr[s],{s,0,150}, PlotStyle->{Thick,Blue}];

Show[ParaPlot,ParaArcPlot, PlotLabel->"r(t), 0⩽t⩽5 and r(s), 0⩽s⩽150"]

Plot curve r(s) from s = 0 to s = 200.  Superimpose on original graph:

ParaArcPlot = ParametricPlot[arcr[s],{s,0,200}, PlotStyle->{Thick,Blue}];

Show[ParaPlot,ParaArcPlot, PlotLabel->"r(t), 0⩽t⩽5 and r(s), 0⩽s⩽200"]