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\large{\textbf{Math 4250/6250 Homework \#2}}
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This homework assignment covers our notes on the Bishop frame (3), Link, Twist, and Writhe (3a) and on the four-vertex theorem. Please pick 5 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. \#3, p 23. \#4 (assume the curve is regular), \#5, \#6, \#7
\section{Challenge Problems}
\noindent
Do Carmo, p 25. \#11, p. 26 \#18, p. 49 \#5, \#13.
The Bishop frame defines two ``curvatures'' $\kappa_1$ and $\kappa_2(s)$ of a space curve $\alpha(s)$. The curve $(\kappa_1(s),\kappa_2(s))$ is called the ``normal development'' of the curve $\alpha$.
\begin{problems}
\item Compute the Bishop frame and normal development
of a helix.
\textbf{Hint:}Compute the Frenet frame
for the helix, and write the Bishop frame $V(s)$ in the
form:
\begin{equation*}
V(s) = \cos g(s) N(s) + \sin g(s) B(s),
\end{equation*}
where $g(s)$ is an unknown function of $s$. Then use the
fact that $V'(s)$ is supposed to be a scalar multiple of
the tangent vector $T(s)$ to find a differential
equation involving $g'(s)$. Solve that ODE by
integration to find $g(s)$, and plug it back into the
equation above to find $V(s)$ explicitly.
\item Prove that a curve lies on a sphere if and only
if its normal development (remember the Bishop frame!)
lies on a line not through the origin.
\item Compute the Gauss linking integral explicitly for the unit circle and the $z$-axis. (Notice that this is an improper integral over $\mathbf{R} \times S^1$, not $S^1 \times S^1$. But it should still converge.)
\item Suppose a curve $\alpha(s)$ has the property that it's normals pass through a fixed point, as in problem \#4 on page 23. However, this time do \emph{not} assume that the curve is regular. What can you say about $\alpha(s)$?
\end{problems}
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