8. Properties of Curves

Alternate Homework

In these alternate homework problems, you will be asked to pick one of \(5\) parametric curves and compute all \(14\) curve properties for that curve. There are some computational hints at the bottom of this page.

At the instructor's discretion, you may be assigned one of the curves as follows:
Curve (A):   Students whose last names begin with A-E.
Curve (B):   Students whose last names begin with G-J.
Curve (C):   Students whose last names begin with K-0.
Curve (D):   Students whose last names begin with P-T.
Curve (E):   Students whose last names begin with U-Z.
Curve (F):   Any student who has learned about hyperbolic functions.
As a sample, all of these curves have the same level of difficulty as the Twised Spiral Cubic solved here, except that curve (f) is easier than the others if you understand hyperbolic functions.

Alternate Problems:

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. \(\vec{r}(t)=(t^2,\dfrac{2}{3}t^3,\dfrac{1}{4}t^4)\)
6. \(\vec{r}(t)=(\sinh(t),\cosh(t),t)\)
   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.)

• For every one of these curves, when computing the speed, the quantity inside the square root is a perfect square.
• The cross product satisfies \[ (a\vec u)\times(b\,\vec v)=(ab)\,\vec u\times\vec v \] So when computing \(\hat{B}\times\hat{T}\), first factor out an overall coefficient from \(\hat B\) and from \(\hat T\).
• For the curves involving \(e^t\), you may need \(e^t e^{-t}=1\).
• To simplify formulas, you may need to factor quantities such as: \[\begin{aligned} 4t^4-1 \quad&\qquad t^4+4t^2+4 \\ 1+8t^2+12t^4 &\qquad 4t^2+4+\dfrac{1}{t^2} \end{aligned}\]
• Check your work by computing: \[\begin{aligned} |\hat T| \quad &|\hat N| \quad |\hat B| \\ \hat T\cdot\hat N \quad \hat T&\cdot\hat B \quad \hat N\cdot\hat B \end{aligned}\]