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{\bf MATH. 433\\
DIFFERENTIAL GEOMETRY\,\,I.
MAKEUP EXAM}\\
{\bf October 20, 1999 Wednsday 14.40-16.00 , SAZ-21}
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[30]{\bf 1.}\,\, Let $\alpha: (a,b) \rightarrow {\bf R^2}$ be a
regular parametrized plane curve.
Assume that there exists $t_{0}$ , $a<t<b$ , such that the distance
$|\alpha(t)|$ from the origin to the trace of $\alpha$ will be
maximum at $t_{0}$. Prove that the curvature $k$ of $\alpha$ at $t_{0}$
satisfies $|k(t_{0}| \ge {1 \over |\alpha(t_{0})|}$.

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[40]{\bf 2.}\,\, Let $\alpha : I \rightarrow {\bf R^2}$ be a regular
parametrized plane curve. Assume that $k(s) \ne 0, s \in I$. In this situation
, the curves $ \beta(s)=\alpha(s)+{1 \over k(s)}\, {\bf n}(s)$,
 $s \in I$ , is called
the {\it evolute} of $\alpha$ and $\gamma(s)=\alpha(s)-s\, {\bf t}(s)$,
$s \in I$ is called the {\it involute} of $\alpha$. Prove the following
{\it Theorem: (I). The involute $\gamma$ of a $C^2$ curve $\alpha$ is $C^3$
curve as a function of its arclength $s_{I}$. The curvature of the involute
is $k_{I}(s_{I})={1 \over s}$, and $\alpha$ is the evolute of its involute ,
$\alpha=evolute of \gamma$. (II). The evolute $\beta$ of a $C^3$ curve
$\alpha(s)$ with strictly monotone curvature is a $C^2$ curve as a function
of its arclength parameter $s_{E}={1 \over k(s)}+const.$. $\alpha$ is
the involute of it evolute , $\alpha=involute of \beta$.}


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[30]{\bf 3.a}\,\, Let $\alpha : I \rightarrow {\bf R^3}$ be a regular curve
paremterized by its arclength $s\in I$. Assume that the position vector
$\alpha(s)$ is always a linear combination of the binormal and normal vectors
 ${\bf b}(s), {\bf n}(s)$. Show that the curve doesnot pass through the origin.\\
 {\bf 3.b}\,\,Suppose $\alpha(t)$ parametrizes a space curve with curvature
 function $k(t)$. Define a new curve $\gamma$ by stting , for each
 $t \in {\Bbb{R}}$, $\gamma=c\, \alpha({t \ovr c})$, where $c$ is an arbitrary
 fixed constant. Derive the curvature of the this new curve.


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