8. Properties of Curves

Homework

This page has a standard set of problems in which you will be asked to compute various curve properties for various curves.

There is also an alternate homework, in which you will be asked to compute all the curve properties for one of a few specific curves. The choice is up to your instructor.

1. Consider the 2D parametric curve \(\vec r(t)=\langle t^2-1,2t^2+3\rangle\) for \(t \ge 0\).

   1. Compute the velocity, acceleration and jerk for a general \(t\) and for \(t=2\).

   2. Find the arc length between \(t=0\) and \(t=2\).

   3. Find the curvature for a general \(t\) and for \(t=2\).
      Note: To compute a cross product, set the third component to \(0\).

   4. Describe the curve and plot it. What does this say about the curvature?

2. Consider the position vector \(\vec{r}(t)=\langle 3t, 2\cos t,2\sin t\rangle\).

   1. Is this a helix or circle? How can you tell?

   2. Compute the velocity, acceleration and jerk for a general \(t\) and for \(t=\dfrac{\pi}{3}\).

   3. Find the arc length between \(t=0\) and \(t=\dfrac{\pi}{3}\).

   4. Find the curvature and torsion for a general \(t\) and for \(t=\dfrac{\pi}{3}\). Is this curve right or left handed?

3. Reparametrize the curve \(\vec r(t)=\langle 3t^2+4,4t^2-3\rangle\) for \(0 \le t \le 1\) with respect to arc length, \(s\), starting from \(t=0\). Plot the curve. Be sure to give the parameter range.

4. Consider the parametric curve \(\vec{r}(t)=\langle e^t, \sqrt{2}\,t,e^{-t}\rangle\).

   1. Compute the velocity, speed, acceleration and jerk.

   2. Find the arc length between \(t=0\) and \(t=1\).

   3. Find \(\hat T\), \(\hat N\) and \(\hat B\).

   4. Find the curvature and torsion. Is this curve right or left handed?

   5. Find the tangential and normal accelerations using \(2\) methods for each.

5. (Optional) Again consider the parametric curve \(\vec{r}(t)=\langle e^t, \sqrt{2}\,t,e^{-t}\rangle\).
   You can use the results from the previous problem.

   1. Verify the Frenet equations along the curve for the \(t\) derivatives of \(\hat{T}\), \(\hat{N}\) and \(\hat{B}\).

   2. Given \(\vec F=y\hat{T}+xz\hat{N}+y\hat{B}\), find \(\dfrac{d\vec F}{dt}\) along the curve using the Frenet formulas.