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\lhead{Math 4441 Activity 3}
\rhead{<your name(s)>}


\begin{document}
\title{Math 4441 Activity 3}
\author{<your name(s)>}
\date{September 14, 2022}
\maketitle
\vspace{5em}

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\begin{exercise}\label{evolute:curvature-formula}
If $\vec{\alpha}(t)=(x(t),y(t))$ is a regular curve in $\mathbb{R}^2$, show that its planar curvature is given by
\[
k(t) = \dfrac{\langle\vec{\alpha}''(t),J(\vec{\alpha}'(t))\rangle}{\|\vec{\alpha}'(t)\|^3} = \dfrac{x'(t)y''(t)-x''(t)y'(t)}{((x'(t))^2+(y'(t))^2)^{3/2}}.
\]
\emph{Hint:} Be careful!  In the definition of planar curvature, we need the derivative of $\vec{t}$ \emph{with respect to arclength}, not with respect to the parameter $t$.  So we'll need to use the chain rule, which will give us something like:
\[
\frac{d}{ds}(\vec{t}(t)) = \bigstar\,\vec{t}'(t),
\]
where the prime indicates a derivative with respect to $t$.  What is $\bigstar$?  Can we write it in terms of $\vec{\alpha}$?
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Sketch the curve $\vec{\alpha}(t)=(t+\sin t,1+\cos t)$, $-\pi<t<\pi$, which is called a \textbf{cycloid}.  Try to do this by hand, to help build intuition about the curve.
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}\label{evolute:curvature-of-cycloid}
Using the formula you proved in Exercise~\ref{evolute:curvature-formula}, compute the curvature of the cycloid as a function of $t$.  Use this computation to sketch a few osculating circles (at, say, $t=0,\pm\pi/4$).
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Find an expression for the unit normal vector to the cycloid as a function of $t$.  Use this, as well as your work in Exercise~\ref{evolute:curvature-of-cycloid}, to produce an expression for the center of curvature of $\vec{\alpha}(t)$ in terms of $t$.  Plot the curve parametrized by this expression.
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Prove that
\[
\vec{\varepsilon}(t) = \vec{\alpha}(t) + \dfrac{\langle\vec{\alpha}'(t),\vec{\alpha}'(t)\rangle}{\langle\vec{\alpha}''(t),J(\vec{\alpha}'(t))\rangle}J(\vec{\alpha}'(t)).
\]
Use this to give expressions for $X(t)$ and $Y(t)$, the $x-$ and $y-$components of $\vec{\varepsilon}(t)$.
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Compute the evolute of the parabola $\vec{\alpha}(t)=(t,t^2)$, $-\infty<t<\infty$.  Sketch both the parabola and its evolute.
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Find the evolute of the ellipse $\vec{\alpha}(t)=(a\cos t,b\sin t)$, $0\leq t\leq 2\pi$.  Sketch both the ellipse and its evolute, for some choice of values $a\neq b$.
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Use \verb|evolute.nb| to generate plots for five of the curves below, along with their evolutes.  That is, plot the curve, its evolute, and several normal lines connecting the original curve to its evolute.  You can choose which curves you want to consider.
\begin{enumerate}[label=(\alph*)]
	\item $(6\cos t-4\cos^3 t,4\sin^3 t)$, $0\leq t\leq 2\pi$ (a nephroid);
	\item $(3\cos t + \cos 3t, 3\sin t - \sin 3t)$, $0\leq t\leq 2\pi$ (an astroid);
	\item $(2(1-\cos t)\cos t,2(1-\cos t)\sin t)$, $0\leq t\leq 2\pi$ (a cardioid);
	\item $(2\cos t+\cos 2t,2\sin t-\sin 2t)$, $0\leq t\leq 2\pi$ (a deltoid);
	\item $(7\cos t - \cos 7t,7\sin t-\sin 7t)$, $0\leq t\leq 2\pi$ (an epicycloid);
	\item $(6\cos t - 2\cos 6t,6\sin t-2\sin 6t)$, $0\leq t\leq 2\pi$ (an epitrochoid);
	\item $(2\cos t + 3\cos(2t/3),2\sin t-3\sin(2t/3))$, $0\leq t\leq 6\pi$ (a hypocycloid);
	\item $((1+3\cos t)\cos t,(1+3\cos t)\sin t)$, $0\leq t\leq 2\pi$ (a lima\c{c}on);
	\item $(e^{t/4}\cos t,e^{t/4}\sin t)$, $-\infty<t<\infty$ (a logarithmic spiral);
	\item $(t-\tanh t,1/\cosh t)$, $-\infty<t<\infty$ (a tractrix).
\end{enumerate}
Don't worry about the curvature occasionally vanishing.  The plots work out.
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\end{document}