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\large{\textbf{Math 4250/6250 Homework \#3}}
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\bigskip
This homework assignment covers our notes on integral geometry (5) and on rotation index (6). Please pick 3 of the following problems. Remember that undergraduate students should average \textbf{one} challenge problem per assignment, while graduate students should average \textbf{two} challenge problems per assignment.
\section{Regular Problems}
\noindent Do Carmo, p 47. \#8, \#12 (\textbf{There is a misprint in this problem! The ratio $M_1/M_2 = 1/2$, not $1/3$ as stated in the text}), \#13.
In \#12, DoCarmo asks for the ``measure'' of a set of lines. What is measure? We established a system of coordinates for the set of lines when we proved the Cauchy-Crofton formula: $\ell(p,\theta)$ is the line at distance $p$ from the origin which is normal to angle $\theta$. In these coordinates, we can compute the ``area'' of a set of lines by integrating the function $1$ over the set with respect to $p$ and $\theta$. So the ``measure'' or area of a set $S$ of lines is just
\begin{equation*}
\operatorname{Measure}(S) = \int_S 1 \, \mathrm{d}p \, \mathrm{d}\theta.
\end{equation*}
\begin{problems}
\item The curve $\alpha(t) = \left( (2a \cos t + b) \cos t, (2a \cos t + b) \sin t \right)$ with $t \in [0,2\pi)$ is called a limacon. Compute the rotation index of this curve.
\end{problems}
\section{Challenge Problems}
\noindent
\begin{problems}
\newcommand{\R}{\mathbf{R}}
\item (Curves of Finite Total Curvature). Suppose $a(s) : S^1 \rightarrow \R^2$ is a smooth, regular closed curve of length $\ell$ parametrized by arclength. Let a subdivision $\mathcal{S}_n$ of $a$ be a collection of parameter values $x_0 = 0 < x_1 < \dots < x_n < \ell$. Let the mesh size $\operatorname{Mesh}(\mathcal{S}_n)$ of the subdivision $\mathcal{S}_n$ be the maximum of $x_i - x_{i-1}$. The exterior angle or turning angle $\theta_i$ of the subdivision at $i$ is the angle formed by $a(x_{i-1})a(x_i)$ and $a(x_i)a(x_{i+1})$.
If $\kappa(s)$ is the curvature of $a(s)$, then the total curvature of $a$ is given by
\begin{equation*}
K = \int \kappa(s) \, \mathrm{d}s.
\end{equation*}
Prove that
\begin{equation*}
K = \lim_{\operatorname{Mesh}(\mathcal{S}_n) \rightarrow 0} \sum_{i=0}^{n} \theta_i.
\end{equation*}
\item Prove Istvan Fary's integralgeometric formula
for curvature. If $a(s)$ is a space curve and $a_v(s)$ is the
projection of $a(s)$ to the plane through the origin normal to
$v$, let $\kappa(s)$ denote the curvature of $a(s)$ and $\kappa_v(s)$ denote
the curvature of $a_v(s)$. And let $K_v$ be the total curvature of $a_v(s)$ and $K$ be the total curvature of so that
\begin{equation*}
K_v = \int \kappa_v(s) \, \mathrm{d}s \quad \text{ and } \quad K = \int \kappa(s) \, \mathrm{d}s.
\end{equation*}
Now show that
\begin{equation*}
K = C \int_{S^2} K_v \, \mathrm{dArea} \end{equation*}
where $C$ is a constant, and $v$ is integrated over $S^2$.
\textbf{Hint}: Use problem 1 to reduce the problem to the case where $a$ is a polygon. Show first that the total curvature of such a curve formed by two line segments $w_1$ and $w_2$ is the angle between the tangents to $w_1$ and $w_2$.
\textbf{Hint 2}: Suppose that $\theta = \angle x_1 x_2 x_3$ is the angle between $x_2 x_1$ and $x_2 x_3$, and that $\theta_v$ is the angle between the projection of $x_1$, $x_2$, and $x_3$ into the plane normal to $v$. To complete hint 1, you must show that
\begin{equation*}
\theta = C \operatorname{Avg(\theta)} = C \int_{v \in S^2} \theta_v \,\mathrm{dArea}.
\end{equation*}
Instead of doing the integral on the right directly, try to prove that the function $\operatorname{Avg}(\theta)$ is a linear function of $\theta$. Can you compute $\operatorname{Avg}(0)$ and $\operatorname{Avg}(\pi)$?
\item Prove Milnor's integralgeometric formula for curvature. If
$a(s)$ is a space curve with curvature $\kappa(s)$, let $a_v(s)$ be
the projection of $a(s)$ to a straight line. This is a nonregular
curve with total curvature $K_v = \pi \cdot \text{(the \# of times
the curve changes direction)}$. Prove that
\begin{equation*}
\int \kappa(s) \, \mathrm{ds} = K \int_{v \in S^2} k_v \, \mathrm{dArea}.
\end{equation*}
\newcommand{\Len}{\operatorname{Length}}
\item In exercise \#6 on page \#47 of DoCarmo we showed that one could define the \textbf{parallel curve} to a \textbf{smooth} convex curve $\alpha(t)$ by constructing the curve
\begin{equation*}
\beta(t) = \alpha(t) - r N(t)
\end{equation*}
and that we could prove $\Len(\beta) = \Len(\alpha) + 2\pi r$ using differential geometry.
Suppose now that the curve $\alpha(t)$ is convex, but not smooth (like a square) and redefine the parallel to $\alpha$ to be the outer boundary curve of the set of points within distance $r$ of the curve $\alpha$. Prove that (as before)
\begin{equation*}
\Len(\beta) = \Len(\alpha) + 2 \pi r,
\end{equation*}
this time using integral geometry.
\end{problems}
\end{document}