Task 11.1

We'll use Mathematica to create a program to compute all the quantities related to the TNB frame and curvature.

  • Download the Mathematica file TNB-intro.nb.
  • Read through the introductory examples, evaluating each block of code.
  • Adapt the code to compute the TNB frame and curvature for a given curve.

Task 11.2

Recall that the curvature vector is $\kappa = \dfrac{d\vec T}{ds}$, with curvature being a the length of this vector. This vector tells us how much the direction of motion ($\vec T$) changes, as we increase the distance moved along the curve. As such, a tight corner will result in a large change of direction and hence a large curvature. Large corners will result in a small curvature.

The radius of curvature at a point, namely $1/\kappa$, provides the radius of a circle that approximates the shape of curve at that point. Large turns results in a large radius of curvature, while tight turns results in a small radius of curvature. This circle lies in the plane formed by $\vec T$ and $\vec N$ (so a normal vector to this plane is $\vec B$). We call this plane the osculating plane. The center of the circle can be found by following $\vec N$ from the point on the curve.

  1. For the curve $\vec r(t) = (3\cos t, 3\sin t, 4t)$, we have already computed $\vec T$, $\vec N$, $\vec B$, and $\kappa$. At $t=\pi/2$, evaluate these quantities.
  2. Give an equation of the the osculating plane at $t=\pi/2$. You'll need to identify a normal vector to the plane, and a point on the plane.
  3. Explain why the center of curvature is given by $\vec r + \frac{1}{\kappa}\vec N$.
  4. Give the location of the center of curvature for $\vec r(t) = (3\cos t, 3\sin t, 4t)$ at $t=\pi/2$.

Task 11.3

There are many ways to compute the TNB frame and curvature. In this problem, we'll develop a few others.

  1. Explain why $ \vec N = \dfrac{\vec r^{\prime\prime}_{\perp \vec r^{\prime}}}{ |\vec r^{\prime\prime}_{\perp \vec r^{\prime} }| } $.
  2. Explain why $ \vec B = \dfrac{\vec r^{\prime}\times \vec r^{\prime\prime}}{|\vec r^{\prime}\times \vec r^{\prime\prime}|}$.
  3. For a function of the form $\vec r(x) = (x, f(x))$, show that $\kappa = \dfrac{|f''(x)|}{(1+(f'(x))^2)^{3/2}}$.

Task 11.4

The last problem for prep each day will point to relevant problems from OpenStax. Spend 30 minutes working on problems from the sections below.