Activity 10.5.2.
In this activity we will use our previous work in finding unit speed parameterizations of lines and circles to make sense of the definition of curvature.
(a)
Recall that in Example 10.3.7 we found that the unit speed parameterization of a line through the points \((1,2,3)\) and \((2,0,5)\) to be \(\vr_1(t)= \langle 1,2,3\rangle + \frac{t}{3} \langle 1,-2,2 \rangle\text{.}\) Calculate \(\vT(s)\text{,}\) \(\frac{d\vT}{ds}\text{,}\) and \(\vecmag{ \frac{d\vT}{ds} }\) for the unit speed parameterization of this line.
(b)
Write a few sentences to explain why your result for the calculation of the curvature \(\kappa=\vecmag{ \frac{d\vT}{ds} }\) for this line makes sense. Exercise 8 asks you to do the calculations to confirm this property of an arbitrary line.
(c)
Recall that we concluded Activity 10.3.2 by noting giving a parameterization of circle in 2-space of radius \(R\) centered at the origin as \(\langle R\cos(t),R\sin(t)\text{.}\) A quick calculation can verify that this parameterization has constant speed and that speed is \(R\text{.}\) Modifying this parameterization by multiplying the parameter by \(\frac{1}{\text{speed}}=\frac{1}{R}\) gives the following parameterization:
\begin{equation*}
\vr(s) = \left\langle R \cos\left(\frac{s}{R}\right), R \sin\left(\frac{s}{R}\right)\right\rangle
\end{equation*}
Compute the speed of this parameterization to verify that this has unit speed. This will also show that \(\vr\, '(s)=\vv(s)=\vT(s)\text{.}\) (Recall that this fact is not true in general!)
(d)
Calculate \(\vT(s)\text{,}\) \(\frac{d\vT}{ds}\text{,}\) and \(\vecmag{ \frac{d\vT}{ds} }\) for the unit speed parameterization of a circle of radius \(R\) centered at the origin.
(e)
Write a few sentences to explain why your result for the calculation of \(\kappa=\vecmag{ \frac{d\vT}{ds} }\) of circle will be constant. You should also address why circles with larger radii will have have smaller curvature.
