# 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)