Length Function, Parametrizing by Arc Length

We can define an arclength function s(t) using
s(t)=Integral from a to t of ||r'(w)|| dw. Then s(t) is the length along the curve from r(a) to an arbitrary point r(t).

ASIDE: Note then, that ds/dt =||r'(t)|| so ds = ||r'(t)|| dt as in the definition of arc length!


  f[t_]:={Cos[t],Sin[t],t/3}


  fplot=ParametricPlot3D[f[t],{t,-4 Pi,4 Pi}];


  s[t_]:=arclength[f,-4 Pi,t]

Under suitable conditions, this length function has an inverse function, t(s).


  Solve[s[t]==s,t]
  t[s_]=t /.%[[1]]

Look at g(s)=f(t(s)), which traces out the same curve over suitable values of s. Since t ranges from -4Pi to 4Pi, s ranges from s(-4Pi) to s(4Pi). When a curve is parametrized in this manner, we say the curve is parametrized by arc length.


  g[s_]:=f[t[s]]


  g[s]


  Simplify[g[s]]


  s[-4 Pi]

Obvious, I hope!


  s[4 Pi]


  gplot=ParametricPlot3D[g[s],{s,s[-4 Pi],s[4 Pi]}];

Is it the same plot?


  Show[GraphicsArray[{fplot,gplot}]];

What's so great about parametrizing a curve by arc length?


  g'[s]


  Simplify[g'[s]]


  Simplify[norm[g'[s]]]

When a curve is parametrized by arc length, g'(s) always has unit length!

Up to Arc Length