Activity 10.5.3.
In this activity, we calculate the curvature of an ellipse and of a helix and use graphs to help with our understanding
(a)
Consider the ellipse shown below. We have omitted the scale on the axes because we want to think of this as the general ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\text{,}\) which has parameterization
\begin{equation*}
\vr_1(t) = \langle a\cos(t), b\sin(t) \rangle\text{.}
\end{equation*}
(The ellipse pictured below has \(a\gt b\text{.}\))
An ellipse centered at the origin. The ellipse is wider than it is tall.
On the graph, identify the points at which you believe the curvature to be the largest and the points at which you believe the curvature is smallest.
(b)
For the given parameterization of the ellipse, find \(\vv(t)\) and \(\va(t)\text{.}\) Also identify the parameter values that correspond to the points of the ellipse with the largest and smallest \(x\) or \(y\) coordinates.
(c)
Using your calculations for \(\vv(t)\) and \(\va(t)\) from part b in Equation (10.5.1) to find a formula for the curvature of the ellipse with the given parameterization. Use your formula to find the curvature of the ellipse at the parameter values you identified in part b as corresponding to the extreme points of the ellipse.
(d)
While it is possible to use single-variable calculus techniques to find the parameter values that maximize and minimize \(\kappa(t)\) on the interval \([0,2\pi]\text{,}\) the algebra involved masks much of the insight. Use graphing technology such as Desmos to plot the curvature function you found above, preferably with sliders that allow you to adjust the values of \(a\) and \(b\text{.}\) Write a couple of sentences about what you observe about the parameter values that maximize and minimize curvature and how they depend on \(a\) and \(b\text{.}\) Then use the general form of the curvature function from above to find the curvature at these parameter values, simplifying as much as possible.
(e)
The standard helix has parameterization \(\vr_2(t) = \cos(t) \vi + \sin(t) \vj + t \vk\text{.}\) Find the curvature of the helix using this parameterization.
(f)
Write a few sentences to describe why the curvature of the helix given above is constant. You may want to include a plot of the curve.
