$\textstyle \parbox{0.45\textwidth}{\begin{flushleft}\vspace{-\baselineskip} {\L... ...tet}\\ \textbf{Matematiska Institutionen}\\ Thomas Erlandsson \end{flushleft}}$



DIFFERENTIAL GEOMETRY MN1 FALL 1999


PROBLEM 1-5


1. Show that the curvature of a plane curve is in general given by the formula

\begin{displaymath}\kappa (t) = \frac{\det(\dot c (t),\,\ddot c (t))}{\vert\dot c (t)\vert^3}.\end{displaymath}


2. Show that the curvature and torsion of a space curve are in general given by the formulae

\begin{displaymath}\kappa (t) = \frac{\vert\dot c (t) \times \ddot c(t)\vert}{\vert\dot c (t)\vert^3},\end{displaymath}


\begin{displaymath}\tau (t) = \frac{\det (\dot c (t),\, \ddot c (t),\, c\,(t))}{\vert\dot c (t) \times \ddot c (t)\vert^2},\end{displaymath}

where $ x \times y$ is the vector product in R3.


3. Let $\,c(t)\,$ be a curve in $\, R^n\,$ parametrized by arc length with the property that $\,\vert c(t)\vert^2\,$ has a local maximum at $\,t_0.$ Let $\,p_0=c(t_0)\,$ and $\,\rho^2=\vert p_0\vert^2.$ Show that

\begin{displaymath}\,\kappa(t_0)\ge \frac{1}{\rho}\end{displaymath}

where $\,\kappa(t_0)=\vert\ddot c(t_0\vert\,$ (which is equal to the first curvature of $\,c(t)\,$ at $\,t_0\,$ if it is defined).

4. Let $\,A,\,B,\,C,\,D,\,E,\,G\,$ be constants such that

\begin{displaymath}AD+BE+CG \ne 0.\end{displaymath}

Consider the curves $\,c(t) = (x(t),y(t),z(t))\in R^3\,$ whose tangent vectors at each point $\,P = (x,y,z)\,$ in space are in the plane $\,L_P\,$ through $\,P\,$ whose normal is

(Bz-Cy+D,Cx-Az+E,Ay-Bx+G).

It is clear that such curves satisfy

\begin{displaymath}(Bz(t)-Cy(t)+D)\dot x (t)+(Cx(t)-Az(t)+E)\dot y(t)+(Ay(t)-Bx(t)+G)\dot z (t) = 0. \end{displaymath}

Let $\,c(t)\,$ be a curve which satisfies such a condition and assume that $\,\dot c (t), \, \ddot c (t)\,$ are linearly independent at $\,P.$ Let $\,\tau \,$ be the torsion of the curve at $\,P =(x,y,z).$



\begin{narrower}\par a) Show that $\,L_P\,$\space is the osculating plane of the curve at $\,P.$\end{narrower}


\begin{narrower}\par b) Show that $\,\tau\,$\space at $\,P\,$\space satisfies th... ...(AD+BE+CG)}{(Bz-Cy+D)^2+(Cx-Az+E)^2+(Ay-Bx+G)^2}.\end{displaymath}\end{narrower}

5. Let $\,c(t)\,$ be a curve in $\, R^3\,$ parametrized by arc length and let $\,t_0=0\in I.$ Let the Frenet-frame at $\,c(0)\,$ be $\,e_i(0)=e_i\,$ and let $\,\kappa(0)=\kappa_0,\,\tau(0)=\tau_0.$ We then have the following well known series expansion for $\,t\,$ close to $\,0\,$

\begin{displaymath}c(t)-c(0)=te_1+\frac{1}{2}\kappa_0t^2e_2+\frac{1}{6}\kappa_0\tau_0t^3e_3+o(t^3).\end{displaymath}

The projections of the curve in a small neighbourhood of $\,c(0)\,$ in the planes of the Frenet-frame at that point are therefore approximated by the following curves:



\begin{narrower}\par a) the projection onto the $\,(e_1,e_3)-$ plane, the rectif... ...th}x=t,\quad y=\frac{1}{2}\kappa_0 t^2,\quad z=0.\end{displaymath}\end{narrower}

\begin{narrower}i) Draw these projektions and their orientations in the case $\... ...alpha\,$\space in the cases $\,\tau_0>0\,$\space and $\,\tau_0<0.$\end{narrower}

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