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

DIFFERENTIAL GEOMETRY MN1 FALL 2001

PROBLEMS

1. Consider the circular helix
   
   $$c(t)=(a\cos t,a \sin t, bt), \quad a>0,\, b\ne 0.$$
   
   
   Show that its curvature $\,\kappa\,$ and torsion $\,\tau$ are given by the formulae
   $\,\kappa=\frac{a}{a^2+b^2},$         $\tau=\frac{b}{a^2+b^2}$
   Hint: Use the formulae in Exercise 1.6.3 in our paper on curves.

   Note that if $\,b>0\,$ (so that $\,\tau>0$) the helix is a ``right-handed'' curve and if $\,b<0\,$
   (so that $\,\tau<0$) it is ``left-handed''. Draw a picture to illustrate the meaning of right and left-handed. Be careful with the choice of orientations of the curves. Prove that it is not possible to find an isometry of $\mathbf R^3\,$ which maps a right-handed helix onto a left-handed.

2. Prove that a regular, nice curve $c: I \to \mathbf{R^3}$ is a plane curve if and only if its torsion vanishes identically.

   Hint: It is no restriction to assume the curve is parameterized by arc-length. Let $(e_1(s),e_2(s),e_3(s))$ be the distinguished Frenet frame. Use the Frenet equations to conclude that $e_3(s)\equiv v_0,$ a constant vector, and from the fact that $\, \dot c\,$ and $\,v_0\,$ are orthogonal prove that $c(s)\cdot v_0\equiv constant\,$ by integration.

3. Prove that a regular, nice curve $c: I \to \mathbf{R^3}$ is a straight line if $\,\dot c(t)\,$ and $\,\ddot c(t)$ are linearly dependent for all $\,t.$

   Hint: What is the torsion and the curvature of such a curve? We also need a certain characterization of straight lines to be found in our paper on curves.

4. Find the most general function $\,f(t)\,$ so that the curve $c(t)=(a\cos t,a\sin t, f(t))\,$ will be a plane curve.

   Answer: $f(t)=A\sin t + B\cos t + C,\quad A,B,C= constants.$

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

   
   

   a)

   the projection onto the $\,(e_1,e_3)-$plane, the rectifying plane, is described by the cubical parabola
   

   $$x=t,\quad y=0,\quad z=\frac{1}{6}\kappa_0\tau_0 t^3;$$

   
   

   b)

   the projection onto the $\,(e_2,e_3)-$plane, the normal plane, is described by the semi-cubical parabola with a cusp at origo
   

   $$x=0,\quad y=\frac{1}{2}\kappa_0 t^2,\quad z=\frac{1}{6}\kappa_0\tau_0 t^3;$$

   
   

   c)

   the projection onto the $\,(e_1,e_2)-$plane, the osculating plane, is decribed by the parabola
   

   $$x=t,\quad y=\frac{1}{2}\kappa_0 t^2,\quad z=0.$$

   
   

   i)

   Draw these projektions and their orientations in the case $\,\tau_0>0$ and $\,\tau_0<0.$

   
   

   ii)

   (For VG only) We shall now study what the curve looks like along the negative $\,e_1-$axis if we raise or lower our eyes somewhat above or under the osculating plane. This means that we shall find the projection of the curve in a $\,(f_2,f_3)-$plane where the $\,ON-$system $\,(f_1,f_2,f_3)\,$ is created from the system $\,(e_1,e_2,e_3)\,$ by letting the $\,e_1,e_3-$plane rotate a small angle $\,\alpha\,$ $(- \varepsilon<\alpha<\varepsilon)\,$ with the $\,e_2-$axis as axis of rotation. Derive the analytical expression of the projection and draw pictures of how it looks for different values of $\,\alpha\,$ in the cases $\,\tau_0>0\,$ and $\,\tau_0<0.$

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