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)