8. Properties of Curves

Alternate Homework

In these alternate homework problems, you will be asked to pick one of \(6\) parametric curves and compute all \(14\) curve properties for that curve. There are some computational hints at the bottom of this page.

As a reference, here is a sample curve fully worked out, so you know what is expected. It is at the same level of difficulty as all the exercises, except for Curve F which is easier if you know hyperbolic functions.

\(\rule{0pt}{15pt}\vec{r}(t) =\left\langle 2t\cos t,2t\sin t,\dfrac{1}{3}t^3\right\rangle\)

At the instructor's discretion, you may be assigned a curve based on the first letter of your last name:

Students whose last names begin with A-E.
\(\rule{0pt}{15pt}\vec{r}(t) =\left\langle e^{t}\cos t,e^{t}\sin t,e^{t}\right\rangle\)

Students whose last names begin with G-J.
\(\rule{0pt}{15pt}\vec{r}(t) =\left\langle 3t^2,4t^3,3t^4\right\rangle\)

Students whose last names begin with K-0.
\(\rule{0pt}{15pt}\vec{r}(t) =\left\langle e^{t},\sqrt{2}t,e^{-t}\right\rangle\)

Students whose last names begin with P-T.
\(\rule{0pt}{15pt}\vec{r}(t) =\left\langle t^2,2t,\ln(t)\right\rangle\)

Students whose last names begin with U-Z.
\(\rule{0pt}{18pt}\vec{r}(t) =\left\langle t^2,\dfrac{2}{3}t^3,\dfrac{1}{4}t^4\right\rangle\)

Students who have learned about hyperbolic functions.
i.e. \(\rule{0pt}{18pt}\sinh(t)=\dfrac{e^t-e^{-t} }{2}\) and \(\cosh(t)=\dfrac{e^t+e^{-t} }{2}\)
\(\rule{0pt}{15pt}\vec{r}(t) =\left\langle \sinh(t),\cosh(t),t\right\rangle\)

Alternate Problems:

Compute all of the following:

Velocity Vector: \(\vec{v}(t)=\dfrac{d\vec{r}}{dt}\)
Acceleration Vector: \(\vec{a}(t)=\dfrac{d\vec{v}}{dt}\)
Jerk Vector: \(\vec{j}(t)=\dfrac{d\vec{a}}{dt}\)
Speed: \(\dfrac{ds}{dt}=|\vec{v}|\)
Arclength from \(\vec{r}(0)\) to \(\vec{r}(1)\): \(\displaystyle L=\int_{\vec{r}(0)}^{\vec{r}(1)} ds=\int_0^1 |\vec{v}|\,dt\)
Unit Tangent Vector: \(\hat{T}=\dfrac{\vec{v}}{|\vec{v}|}\)
Velocity \(\times\) Acceleration: \(\vec{v}\times\vec{a}\dfrac{}{}\)
Its Length: \(|\vec{v}\times\vec{a}|\dfrac{}{}\)
Unit Binormal Vector: \(\hat{B}=\dfrac{\vec{v}\times\vec{a}}{|\vec{v}\times\vec{a}|}\)
Unit Normal Vector: \(\hat{N}=\hat{B}\times\hat{T}=\dfrac{\hat{T}'(t)}{\;|\hat{T}'(t)|\;}\)
Curvature: \(\kappa=\dfrac{|\vec{v}\times\vec{a}|}{|\vec{v}|^{3}} =\dfrac{\;|\hat{T}'(t)|\;}{|\vec{v}|}\)
Torsion: \(\tau=\dfrac{\vec{v}\times\vec{a}\cdot\vec{j}}{|\vec{v}\times\vec{a}|^2}\)
Tangential Acceleration: \(a_{T}=\vec{a}\cdot\hat{T}=\dfrac{d}{dt}|\vec{v}|\)
(Compute \(2\) ways.)
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: \[\begin{aligned} |\hat T| \quad &|\hat N| \quad |\hat B| \\ \hat T\cdot\hat N \quad \hat T&\cdot\hat B \quad \hat N\cdot\hat B \end{aligned}\]

8. Properties of Curves

Alternate Homework

Alternate Problems:

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