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)