8. Properties of Curves

h. Summary & Examples of Curve Computations

2. Examples of Curve Computations

Here are summaries of the complete computations for the 3 curves we have looked at throughout this chapter: the circle, the helix and the twisted cubic, plus the computations for a new curve, the cubic spiral helix. Then in the exercises, you will be asked to compute everything for several other curves.

Curve Examples

1. Circle
2. Helix
3. Twisted Cubic
4. Cubic Spiral Helix

Curve Exercises

Pick one of the following curves:

1. \(\vec{r}(t)=(e^{t}\cos t,e^{t}\sin t,e^{t})\)
2. \(\vec{r}(t)=(3t^2,4t^3,3t^4)\)
3. \(\vec{r}(t)=(e^{t},\sqrt{2}t,e^{-t})\)
4. \(\vec{r}(t)=\langle t^2,2t,\ln(t)\rangle\)
5. \(\vec{r}(t)=(\sinh(t),\cosh(t),t)\)
   where \(\sinh(t)=\dfrac{e^t-e^{-t}}{2}\) and \(\cosh(t)=\dfrac{e^t+e^{-t}}{2}\).

Compute all of the following:

1. Velocity Vector: \(\vec{v}(t)=\dfrac{d\vec{r}}{dt}\)
2. Acceleration Vector: \(\vec{a}(t)=\dfrac{d\vec{v}}{dt}\)
3. Jerk Vector: \(\vec{j}(t)=\dfrac{d\vec{a}}{dt}\)
4. Speed: \(\dfrac{ds}{dt}=|\vec{v}|\)
5. Arclength from \(\vec{r}(0)\) to \(\vec{r}(1)\): \(\displaystyle L=\int_{\vec{r}(0)}^{\vec{r}(1)} ds=\int_0^1 |\vec{v}|\,dt\)
6. Unit Tangent Vector: \(\hat{T}=\dfrac{\vec{v}}{|\vec{v}|}\)
7. Velocity \(\times\) Acceleration: \(\vec{v}\times\vec{a}\dfrac{}{}\)
8. its Length: \(|\vec{v}\times\vec{a}|\dfrac{}{}\)
9. Unit Binormal Vector: \(\hat{B}=\dfrac{\vec{v}\times\vec{a}}{|\vec{v}\times\vec{a}|}\)
10. Unit Normal Vector: \(\hat{N}=\hat{B}\times\hat{T}=\dfrac{\hat{T}'(t)}{\;|\hat{T}'(t)|\;}\)
11. Curvature: \(\kappa=\dfrac{|\vec{v}\times\vec{a}|}{|\vec{v}|^{3}} =\dfrac{\;|\hat{T}'(t)|\;}{|\vec{v}|}\)
12. Torsion: \(\tau=\dfrac{\vec{v}\times\vec{a}\cdot\vec{j}}{|\vec{v}\times\vec{a}|^2}\)
13. Tangential Acceleration: \(a_{T}=\vec{a}\cdot\hat{T}=\dfrac{d}{dt}|\vec{v}|\)
     (Compute \(2\) ways.)
14. Normal Acceleration: \(a_{N}=\vec{a}\cdot\hat{N}=\kappa|\vec{v}|^2=\dfrac{|\vec{v}|^2}{R}\)
     (Compute \(2\) ways.)

We have intentionally not given the solutions to these exercises so that they can be assigned as a take-home quiz.