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{\bf MATH. 433\\
DIFFERENTIAL GEOMETRY\,\,I.
SECOND MIDTERM EXAM}\\
{\bf October 12, 1999 Tuesday 14.40-16.00 , SAZ-21}
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[30]{\bf 1.}\,\, Let $\alpha:I \rightarrow {\Bbb{R^3}}$ be a regular curve
with the principal normal vector ${\bf n(s)=(\cos a s, \sin a s, 0)$.
Here $a$ is a nonzero constant and $s \varepsilon I$ is the arclength
parameter. Determine the curve completely (find $\alpha$, $k(s)$ the
curvature and $\tau(s)$ the torsion).


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[30]{\bf  2.}\,\, If a closed plane curve $C$ is contained inside
a disc of radius $r$, prove that there exists a point $p \varepsilon C$
such that the curvature $k$ of $C$ at $p$ satisfies $|k| > 1/r$.


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[40]{\bf 3.}\,\,Show that the curvature of $k(t) \ne 0$ of a regular
parametrized curve $\alpha : I \rightarrow {\Bbb{R^3}}$ is the curvature
at $t$ of the curve $\pi \circ \alpha$ , where $\pi$ is the normal projection
of $\alpha$ over the osculating plane at $t$.


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