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MA 227 Standard C04


Arclength Parameter and the TNB Frame

At the end of the course, each student should be able to…

  • C04: VectFuncSTNB. Compute and apply the arclength parameter and TNB frame for a vector function.

C04. Arclength Parameter and the TNB Frame

  • The arclength parameter s(t) is defined to be the directed arclength from \vect r(0) to \vect r(t).
    • The arclength of a curve from t=a to t=b is given by L=\int_a^b\|\vect{r}’(t)\|\,dt.
    • Thus the arclength parameter is given by s(t)=\int_0^t\|\vect{r}’(\tau)\|\,d\tau. Note that \frac{ds}{dt}=\|\vect{r}’(t)\|.
    • Usually s is left in terms of t and used indirectly, but in special cases a curve may be parameterized directly in terms of s.
  • The arclength parameter is used to define the TNB frame for a curve: three mutually orthogonal unit vectors defined at each point of a curve independent from the curve’s parameterization.
    • The unit tangent vector is given by \vect T=\frac{d\vect r}{ds}=\frac{d\vect r/dt}{\|d\vect r/dt\|}. It gives the direction of motion.
    • The unit normal vector is given by \vect N=\frac{d\vect T/ds}{\|d\vect T/ds\|}= \frac{d\vect T/dt}{\|d\vect T/dt\|}. It gives the direction of curvature.
      • The orthogonality of \vect T,\vect N follows from the fact that \vect T\cdot\vect T=1, and therefore (by the product rule) \vect T\cdot\frac{d\vect T}{dt}=0.
    • The unit binormal vector is given by \vect B=\vect T\times\vect N. If \vect T points forward, and \vect N points leftward, then \vect B points upward.
  • The rate measuring how a curve turns at a point with respect to its arclength is known as its curvature \kappa.
    • It is computed as \kappa=\|\frac{d\vect T}{ds}\|= \frac{\|d\vect T/dt\|}{\|d\vect r/dt\|}.

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 12.3 (exercises 1-8, 11-14)
    • 12.4 (exercises 1-4, 9-16)
    • 12.5 (exercises 7-8)