Curvature

The curvature of a curve is the magnitude of the rate of change of the tangent vector with respect to arc length: ||dT/ds||. As a function of t, the chain rule yields that the curvature is
||T'(t)||/||F'(t)||, where F(t) is the position vector.
The idea is that if the rate of change of the tangent vector is large, the curve is bending sharply. Remember that a vector has magnitude AND direction, so the length may not change, but if the direction does, the vector is changing. Curvature is often denoted by the Greek letter kappa.

I have defined a curvature function in the notebook "initialization.ma". The syntax is
curvature[F_,t_]:=Module[{T,c},
T[t]=D[F[t],t]/Sqrt[D[F[t],t].D[F[t],t]];
c=Sqrt[D[T[t],t].D[T[t],t]]/Sqrt[D[F[t],t].D[F[t],t]];
Return[c]]

Let's try some. Note that if you want to use this in another notebook you create, you must copy the function definition to the new notebook or run the notebook "initialization.ma".

In[76]:=

  r1[t_]:={Cos[t]+t Sin[t],Sin[t]-t Cos[t], t^2}

In[77]:=

  ParametricPlot3D[r1[t],{t,-6,16}];

Does the curve seem to bend less as t gets larger? The curvature should agree with this.

In[78]:=

  Simplify[curvature[r1,t]]

Out[78]=

      1
  ----------
          2
  5 Sqrt[t ]

What about a circle? How does it bend?

In[79]:=

  r2[t_]:={Cos[t],Sin[t]}

In[80]:=

  ParametricPlot[r2[t],{t,0,2 Pi}, AspectRatio->Automatic];

In[81]:=

  Simplify[curvature[r2,t]]

Out[81]=

  1

Another Circle?

In[82]:=

  r3[t_]:={5 Cos[t],5 Sin[t]}

In[83]:=

  ParametricPlot[r3[t],{t,0,2 Pi}, AspectRatio->Automatic];

In[84]:=

  Simplify[curvature[r3,t]]

Out[84]=

  1
  -
  5

Do you see a pattern? Try some more for yourself.

A helix is another curve of constant curvature.

In[85]:=

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

In[86]:=

  ParametricPlot3D[helix[t],{t,0,6 Pi}];

In[87]:=

  Simplify[curvature[helix,t]]

Out[87]=

  9
  --
  10

Up to Vector Differential Calculus