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!

In[41]:=

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

In[42]:=

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

In[43]:=

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

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

In[44]:=

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

Out[44]=

           4 Sqrt[10] Pi - 3 s
  {{t -> -(-------------------)}}
                Sqrt[10]

Out[45]=

    4 Sqrt[10] Pi - 3 s
  -(-------------------)
         Sqrt[10]

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.

In[46]:=

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

In[47]:=

  g[s]

Out[47]=

       4 Sqrt[10] Pi - 3 s        4 Sqrt[10] Pi - 3 s
  {Cos[-------------------], -Sin[-------------------], 
            Sqrt[10]                   Sqrt[10]
   
    -(4 Sqrt[10] Pi - 3 s)
    ----------------------}
          3 Sqrt[10]

In[48]:=

  Simplify[g[s]]

Out[48]=

         3 s            3 s      -4 Pi      s
  {Cos[--------], Sin[--------], ----- + --------}
       Sqrt[10]       Sqrt[10]     3     Sqrt[10]

In[49]:=

  s[-4 Pi]

Out[49]=

  0

Obvious, I hope!

In[50]:=

  s[4 Pi]

Out[50]=

  8 Sqrt[10] Pi
  -------------
        3

In[51]:=

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

Is it the same plot?

In[52]:=

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

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

In[53]:=

  g'[s]

Out[53]=

         4 Sqrt[10] Pi - 3 s         4 Sqrt[10] Pi - 3 s
   3 Sin[-------------------]  3 Cos[-------------------]
              Sqrt[10]                    Sqrt[10]
  {--------------------------, --------------------------, 
            Sqrt[10]                    Sqrt[10]
   
       1
    --------}
    Sqrt[10]

In[54]:=

  Simplify[g'[s]]

Out[54]=

            3 s              3 s
   -3 Sin[--------]  3 Cos[--------]
          Sqrt[10]         Sqrt[10]      1
  {----------------, ---------------, --------}
       Sqrt[10]         Sqrt[10]      Sqrt[10]

In[55]:=

  Simplify[norm[g'[s]]]

Out[55]=

  1

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

Up to Arc Length